Jean-Claude Luneno
email: jean-claude.luneno@ltu.se

Jan-Olov Aidanpää
email: jan-olov.aidanpaa@ltu.se

Use of Nonlinear Journal Bearing Impedance Descriptions to Evaluate Linear Analysis of the Steady-State Unbalance Response for a Rigid Symmetric Rotor Supported by Two Identical Finite-length Hydrodynamic Journal Bearings at High Eccentricities

Keywords: Nonlinear model, linear model, journal bearing, impedance descriptions, dynamics, unbalance response, rotor. 0

Use of nonlinear journal-bearing impedance descriptions to evaluate linear analysis of the steady-state imbalance response for a rigid symmetric rotor supported by two identical finite-length hydrodynamic journal bearings at high eccentricities

Jean-Claude Luneno, Jan-Olov Aidanpää

Nomenclature: Introduction:

C r C = Bearing radial clearance [m] Y X j i C ij , ,
Undercritical industrial heavy rotating machinery sometimes operates at high eccentricities, for example hydropower or steam turbine with horizontal rotors. Therefore high eccentricity dynamic analysis of rotor-bearing systems is of industrial interest. Linear models are widely used in most industrial design processes due to their simplicity and the ease with which it is possible to interpret the results. In the area of rotor-bearing system dynamics, most developed linear bearing models derived from the numerical differentiation approach are proved valid enough at low eccentricities.

In predicting the critical mass for a rigid symmetric rotor, Ram Turaga, A. S. Sekhar and B. C. Majumdara [START_REF] Turaga | Comparison between Linear and Nonlinear Transient analysis Techniques to Find the Stability of a Rigid Rotor[END_REF] used the finite element method to calculate the bearing coefficients and found that linear models were valid for eccentricities less than 0.6. A. K. Tieu and Z. L. Qiu [START_REF] Tieu | Stability of Finite Journal Bearings from Linear and Nonlinear Bearing Forces[END_REF] used Lund's infinitesimal perturbation method [START_REF] Lund | A Calculation Method and Data for the Dynamic Coefficients of Oil-Lubricated Journal Bearings[END_REF] and found that the rotor trajectory by the linear analysis assumes a significant error already at eccentricity 0.6 when the whirl amplitude is greater than 20 percent of the bearing clearance. D. Childs [START_REF] Childs | Turbomachinery Rotordynamics: Phenomena, Modeling, & Analysis[END_REF] and Lund (1966) have demonstrated that the rotor is completely stable for eccentricities grater than 75 percent of the bearing clearance and for all L/D ratios.

Bearings with low L/D ratios in particular are more stable than others [START_REF] Childs | Journal bearing impedance descriptions for rotordynamic applications[END_REF], therefore they will be the object of the present work, using L/D = 0.25. R D Brown, G Drummond and P S Addison [START_REF] Brown | Chaotic response of a short journal bearing[END_REF] showed that a rigid rotor supported on a hydrodynamic bearing film at high eccentricity satisfied the conditions for chaos. Linear models are not valid for motions other than period one, therefore a bifurcation diagram is needed to locate such operating conditions and validate linear models outside these regions. The purpose of this paper is to use the journal bearing impedance descriptions method for both linear and nonlinear models of bearing reaction forces to investigate how well linear models can approximate nonlinear models at high eccentricities.

The high nonlinear characteristics of bearing reaction forces at large eccentricities makes both numerical differentiation and integration approaches less accurate to linearise the bearing reaction forces [START_REF] Childs | Journal bearing impedance descriptions for rotordynamic applications[END_REF].

Method

D. Childs, H. Moes, H. van Leeuwen [START_REF] Childs | Journal bearing impedance descriptions for rotordynamic applications[END_REF], D. Childs [START_REF] Childs | Turbomachinery Rotordynamics: Phenomena, Modeling, & Analysis[END_REF] and H. Moes and R. Bosma [7] derived the journal impedance descriptions for rotordynamic applications, which consist of defining the bearing reaction force components as analytical nonlinear functions of the journal motion (displacement and velocity). They also derived the analytical bearing stiffness and damping coefficients by linearisation of the bearing reaction forces using Taylor series expansion; where second and higher order differential terms have been dropped. These analytical expressions for bearing coefficients are particularly suitable for rotordynamics work because they yield more accurate results and less computational time than the existing numerical differentiation and pressure-integration approaches. They also provide very accurate models for all eccentricities and L/D ratios. The Moes cavitated ( film -π

) finite-length bearing model [START_REF] Moes | Mobility and impedance definitions for plain journal bearings[END_REF] will be used for present studies, with the following impedance (dimensionless bearing-load-force vector due to pure squeezing) components:

( ) ( ) { } ( ) { }( ) ( ) ( ) { } { }( ) ( ) ( ) ( ) ( )( ) 1 2 2 1 2 1 2 2 1 2 1 2 1 2 1 2 2 1 1 1 2 1 2 1 2 2 1 2 2 1 1 2 1 2 1 1 1 1 tan 1 2 1 3 1 2 4 3 1 1 6 1 2 1 3 1 1 6 - - - - - - - - - - - - - - - - - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ - - - - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - - + + - + + - ≈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + - + + + - - ≈ y E x y T D L E xT E xT y W D L x T y x x T y W y x ε ε ε ε π ε ε ε 1 2 2 < + = = y x C e ε
The dimensional bearing reaction forces in x and y directions are:

y s y x s x W C R L V F W C R L V F 3 3 2 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = µ µ
Transformation to the rotor fixed reference coordinates Z Y X , , is carried out as follows:

( ) ( ) { } 2 1 2 2 1 tan sin cos sin cos X Y Y X C k V V V Y X X Y C X Y y C Y X x j S s ω ω ε ω ω ω ς ς ς ς ς - + + = × - = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = - = + = - [ ] T j Y X V 0 ˆ = , [ ] T k ω ω 0 0 ˆ= , [ ] T Y X C 0 ˆ= ε 2 ω ω = Vectors F W C V V S j , , , , ˆε
and angles ς γ , are illustrated in Fig 17 in Appendix A.

The dimensional bearing reaction forces in X and Y directions are:

) cos( ) sin( ) sin( ) cos( ς ς ς ς y x Y y x X F F F F F F + = - = … (1)
The above bearing reaction forces are nonlinear functions of rotor displacement and velocity in the bearing clearance.

Equations of motion

For an unbalanced rigid symmetric rotor supported by two identical journal bearings, the equations of motion of the rotor mass centre are:

( )

( ) g t E F M Y t E F M X Y X - + = + = ω ω ω ω sin 2 cos 2 2 2 ... (2) where g t E F F M Y X , , , , , ,
ω are rotor mass (kg), bearing reaction force (N) in the X direction, bearing reaction force (N) in the Y direction, rotor mass eccentricity (m), rotor rotational speed (rad/s), time (s) and gravity acceleration (

2 / s m
) respectively.

2) Linear bearing model

The bearing reaction forces 2) are linearised about the equilibrium position and are expressed as linear functions of rotor displacement and velocity.

Y X F F , in equations (

Equations of motion about the equilibrium position

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ t t E Y X K K K K Y X C C C C M Y X YY YX XY XX YY YX XY XX ω ω ω sin cos 2 2 … (3)
Introducing the dimensionless variables:

⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ = = ′ ′ = ′ = = ′ ′ = ′ = t C Y Y C Y Y C Y Y C X X C X X C X X ω τ ω ω ω ω 2 2
, , , , Equations ( 3) can be written in non-dimensional form as follows:

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ′ ′ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ′ ′ ′ ′ τ τ sin cos 2 2 C E p Y X k k k k Y X c c c c Y X p YY YX XY XX YY YX XY XX … (4) Where g C p 2 2 ω =
, and the 8 non-dimensional bearing coefficients are obtained according to [START_REF] Childs | Journal bearing impedance descriptions for rotordynamic applications[END_REF] as follows: The partial derivatives required to evaluate the above coefficients (equations ( 5)) are found in Appendix B and the complete expressions for 0 0 ,W γ are found in Appendix A.

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ - = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ - = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ - - = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ - = ε ε ε ε α α α α α γ γ ε α γ γ ε α γ ε γ α γ ε γ ε γ ε γ ε γ ε γ ε γ γ ε γ ε γ γ ε γ W W c W W c c c W W k W W k k k YY YX
At static equilibrium, the Sommerfeld number is:

0 1 2 0 0 0 2 > = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = W F DL C R S πε π ω μ , Or 0 1 2 ) ( 0 0 0 2 0 = - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = W F DL C R f πε π ω μ ε … (6)
Equation ( 6) is a nonlinear function of 0 ε to be solved in the interval ] [ 1 ... 0 .

A case example

Some of the following data used for simulations are taken from [START_REF] Jeng Chen | Introduction to Dynamics of Rotor-Bearing Systems[END_REF] where a short-bearing solution was assumed. Results in [START_REF] Jeng Chen | Introduction to Dynamics of Rotor-Bearing Systems[END_REF] for eccentricities (

7 . 0 < ε
) were compared with results obtained using the nonlinear finite-length bearing impedance descriptions method. For investigating high eccentricities (

7 . 0 > ε
) the bearing radial clearance C was doubled, and the short bearing solution was no longer valid:

22.6796 = M [Kg], 2 10 54 . 2 - × = R [m], R D × = 2 , R L × = 5 . 0 , 25 . 0 = D L , 6 10 6 . 101 - × = C [m], 0.0069 = µ [ s Pa × ], 6 10 128 . 8 - × = E [m]

Results

The results shown in Figures 4 to 10 display the steady-state unbalance response solutions of equations (2) for the system with a nonlinear bearing model (blue), and the steady-state unbalance response solutions of equations (4) for the system with a linear bearing model (red). The solutions are presented as rotor trajectories plots for the selected rotational speeds of interest, 1000, 2000, 3000, 6000, 6300, 7700, 11000 rpm.

Figures 11 and12 show the differences between the maximum absolute values of X and Y displacements of steady-state solutions of equations ( 2) and ( 4) for a range of rotational speeds from 5000 rpm to 10000 rpm.

Figures 13 and 14 are the bifurcation diagrams and steady-state unbalance response respectively. These results were obtained by using data from [START_REF] Jeng Chen | Introduction to Dynamics of Rotor-Bearing Systems[END_REF] where the bearing radial clearance Equations ( 2) and (4) were numerically integrated using the 5th order Runge-Kutta numerical integration method with adaptive integration time steps. Equations (3) were transformed into non-dimensional form (equations (4)) just for numerical convenience using the derived bearing coefficients in their non-dimensional form. The dimensional solutions (X, Y displacements) of equations ( 2) are converted to their nondimensional quantities by dividing by the bearing clearance C.

The initial conditions for X and Y variables were to be chosen in the interval ]0…C[ for the numerical integration of equations (2) and for the numerical integration of equations ( 4) the initial conditions were (

0 0 Y , X
). The remaining initial conditions for velocities 0 Rotor Mass Center Orbits

0 0 0 0 = ′ = ′ = = Y X Y X .
X-displacement [-] Y-displacement [-]
Fig 14 . Rotor steady-state unbalance response (data taken from [START_REF] Jeng Chen | Introduction to Dynamics of Rotor-Bearing Systems[END_REF]); where the bearing radial clearance

6 10 8 . 50 - × = C
[m] is half that used in the case example studied .

Discussions

The results of this investigation show that linear bearing models derived from the nonlinear impedance descriptions of the Moes cavitated ( film -π

) finite-length bearing can predict the steady-state unbalance response of a rigid symmetric rotor supported by two identical journalbearings at high eccentricities. However, this is only the case when operating conditions are below the threshold speed of instability (around 11400 rpm) and when the system has period one solutions. The error will also become larger closer to the resonance speed (3000 rpm), see Fig 6.

When the system has period one solutions, the trajectories predicted by linear models are in acceptable agreement with the nonlinear models. However, deviations are to be expected because the linear models will always predict elliptical trajectories, while the nonlinear models display trajectories of a banana shape and are generally asymmetrical with respect to the static equilibrium position as illustrated in [START_REF] Jeng Chen | Introduction to Dynamics of Rotor-Bearing Systems[END_REF]. As can be seen in Figures 11 and12, the deviations in the maximum magnitudes of the X and Y displacements around the static equilibrium position are less than 30 percent in regions where the system has period one solutions (not at the resonance speed). The greatest difference in the max(abs(X,Y)) is around 0.035 and the corresponding largest max(abs(X,Y)) value is about 0.135. The smallest difference in the max(abs(X,Y)) values is around 0.015 and the corresponding largest max(abs(X,Y)) value is about 0.08. In percent, this makes the deviations into the interval (19%-26%). The deviations are expected, especially for hydrodynamic bearings operating at large eccentricities which exhibit strong nonlinear characteristics. All the other existing numerical methods for linearising the hydrodynamic bearing forces by computing the so-called bearing stiffness and damping coefficients have been proven to be inaccurate for predicting the journal trajectory at large eccentricities (see references [START_REF] Childs | Turbomachinery Rotordynamics: Phenomena, Modeling, & Analysis[END_REF][START_REF] Jeng Chen | Introduction to Dynamics of Rotor-Bearing Systems[END_REF]), and they are therefore limited at (moderate) low eccentricities (less than 0.6 according to several authors). Linear models derived from the impedance descriptions method are valid at large eccentricities. The reason for this is because they do not require any numerical differentiation or integration. They are obtained in a closed analytical form. This paper deals exclusively with journal-bearings at large eccentricity; therefore the other existing linear numerically derived models (methods) are not valid for comparisons. With differences in the range of (19%-26%), the journal trajectories from a linear model could be considered as an acceptable prediction in the present studied case where the original physical model is a strong nonlinear one.

A certain minimum speed is required to generate the hydrodynamic pressure inside the bearing clearance and therefore the rotor speed can not be zero. The case studied in this paper has a resonance at around 3000 rpm. Due to the large error expected, it is not the objective to investigate the linear model in the region around this speed (2200-4000 rpm). At low speeds, less than 2200 rpm, the journal oscillations are small due to the low unbalance force. At 1000 rpm, the journal oscillations are less than 1% of the bearing clearance and can be neglected. Therefore the low speed cases can just be considered as static due to the low unbalance excitation force. The threshold speed of instability of the journal-bearing is around 11400 rpm; therefore the system becomes unstable when running near or above this speed. The hydrodynamic lubrication will fail and metal-to-metal contact will be established if this speed is reached. The highest speed which could be reached, with unbalance force excitation in the system, is 11300 rpm. The simulations were done with rotor speeds up to 11000 rpm.

Between 6300-7500 rpm, the rotor whirls with half of its spinning frequency. Between 9000-10000 rpm, the motions are quasi periodic with regions of phase locking.

Increasing the rotor speed further, close to the threshold speed of instability (11400 rpm), the rotor system reaches instability and the hydrodynamic lubrication fails. Figures 13 and14 show the case with data from [START_REF] Jeng Chen | Introduction to Dynamics of Rotor-Bearing Systems[END_REF] where the system operates at low eccentricities, having only period one solutions at the displayed frequencies under the threshold speed of instability. For this case, a linear model will predict all displayed rotor trajectories (Fig14) obtained from the nonlinear bearing model.

Conclusion

Linear models derived from the nonlinear impedance descriptions of the Moescavitated ( film -π

) finite-length bearing can predict the steady-state unbalance response of a symmetric rigid rotor supported by two identical journal-bearings at high eccentricities. However, this is only the case where operating conditions are below the threshold speed of instability and when the system has period one solutions. The error will also increase closer to the resonance speed. The deviations in the maximum magnitudes of the X and Y displacements about the static equilibrium position are in the interval between approximately 19% and 26% when the system has period one solutions, except at the resonance speed. These deviations depend on the net difference in shapes between the two respective rotor trajectories and also on the fact that the nonlinear models generate trajectories which are generally asymmetrical with respect to the static equilibrium position. This paper concerns the dynamics of a bearing model with L/D = 0.25 at high eccentricities. However, the described procedure is general and may be used in investigations of systems with bearings of other L/D ratios ( )

( ) ( ) ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ = = ′ = ′ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = ′ - + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ′ + ′ ′ - ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ′ + ′ - + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ′
( ) ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ - ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + - + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ - = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ - = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = ⎟ ⎠ ⎞ ⎜

ModellingAFig 1 .

 1 Fig 1. Rigid symmetric rotor supported by two identical plain cylindrical journal bearings

Fig 2 and

 2 Fig 3 are the bifurcation diagrams for the system with a nonlinear bearing model. In these bifurcation diagrams (not maximum or minimum displacements) the rotational speed is increased in steps of 100 rpm and 100 Poincaré sections are plotted after the transients have decayed, 200 periods from start.

  is half of that used in the case example studied. The bifurcation diagrams in Fig 13 differ markedly from those in Fig 2 and Fig 3. The bearing radial clearance is the only parameter changed to obtain the resulting differences.

Fig 2 .Fig 3 .Fig 5 .Fig 6 .Fig 7 .Fig 8 .Fig 11 .Fig 12 .Fig 13 .

 235678111213 Fig 2. X displacements bifurcation diagram for the case example studied

  Fig 15. Kinematic variables for impedances and mobilities (from [6] and [4])

  Bearing reaction force F ˆ[N] components in the x and y directions

	x F F ,	y
	Y X F F ,	Bearing reaction force F ˆ[N] components in the X and Y directions
	g										Gravity acceleration [	m	/ s	2	]
	K ij		,			i	,	j	=	X	,	Y	Dimensional bearing stiffness coefficients [N/m]
	k ij	,	i	,			j	=	X	,	Y	Non-dimensional bearing stiffness coefficients [-]
	k ˆ										Unit vector in z and Z directions. [-]
	L										Bearing length [m]
	M										Rotor mass [kg]
	1 O									Bearing centre
	O	2								Rotor (journal) centre
	R										Bearing inner radius [m]
	rpm						Round per minute
	S										Sommerfeld number [-]
	t										Time [s]
											rotating with ω relative to rotor inertial coordinates	Y X , ,	Z
	X,Y							Rotor displacement components in X and Y directions [m]
	X ,	Y			Rotor velocity components in X and Y directions
	X ,	Y			Rotor acceleration components in X and Y directions
	X ,	Y			Rotor non-dimensional displacement components in X and Y
											directions [-]
	,Y X ′		'	Rotor non-dimensional velocity components in X and Y directions
	X	′ ′,	Y	′ ′	Rotor non-dimensional acceleration components in X and Y
											directions
	γ										Journal (rotor) attitude angle [radians]
				,						=	Dimensional bearing damping coefficients [Ns/m]
	c ij	,	i	,		j	=	X	,	Y	Non-dimensional bearing damping coefficients [-]
	D										Bearing inner diameter [m]
	E										Rotor mass eccentricity [m]
	e										Journal (rotor) dimensional eccentricity [m]

0

F

Half static load (for a symmetrically loaded rotor) [N] j V ˆ Journal (rotor) velocity vector [m/s] S V ˆ Journal's pure-squeeze-velocity vector [m/s] S V Journal's pure-squeeze-velocity magnitude [m/s] 0 W Bearing impedance W ˆmagnitude at the static equilibrium position y x W W ,

Bearing impedance W ˆmagnitude components in x, y directions

x,y,z Coordinate system with abscise (x-axis) fixed to the vector S V ˆ,


Clément Courvoisier

Julien Fatome

Christophe Finot
email: christophe.finot@u-bourgogne.fr

All-optical measurement of residual chromatic dispersion and OSNR using self-phase modulation in optical fiber

Keywords: Optical performance monitoring, Nonlinear optics, Optical fiber, Optical Telecommunication

We describe a new all-fibered device providing high accuracy all-optical measurements of both optical signal to noise ratio and residual chromatic dispersion. Taking advantage of the self-phase modulation driven spectral evolution of pulses propagating in a highly non-linear medium, we have numerically and experimentally validated the benefits of using an anomalous dispersive fiber combined with a central spectral filtering.

INTRODUCTION

Following the development of high-speed optical networks and associated increasing bit-rates and complexity, it has become of a great importance to benefit from efficient devices able to accurately measure the various impairments degrading the data stream during its propagation in optical fiber. Consequently, all-optical solutions enabling a direct performance monitoring are a growing field of investigation to detect and compensate for the performance fluctuations of these systems [START_REF] Pan | Optical performance monitoring for the next generation optical communication networks[END_REF].

Among the large panel of techniques that have already been studied, several methods based on nonlinear devices have so far demonstrated their potential to monitor the level of residual chromatic dispersion (RCD) as well as optical signal to noise ratio (OSNR). We can underline several reported techniques based on semiconductors optical amplifiers [START_REF] Vorreau | Optical noise and dispersion monitoring with SOA-based optical 2R regenerator[END_REF] or nonlinear effects occurring in optical fibers, including stimulated Brillouin scattering [START_REF] Pelusi | Multi-channel in-band OSNR monitoring using stimulated Brillouin scattering[END_REF] and Kerr nonlinearity. In the latter case, several configurations have been experimentally tested such as : Kerr nonlinearity in a nonlinear optical loop mirror [START_REF] Adams | All-optical in-band OSNR monitoring at 40 Gb/s using a nonlinear optical loop mirror[END_REF], parametric processes [START_REF] Ng | In-band OSNR and chromatic dispersion monitoring using a fibre optical parametric amplifier[END_REF], cross-phase modulation [START_REF] Vo | Multi-impairment monitoring at 320 Gb/s based on cross-phase modulation radiofrequency spectrum analyser[END_REF] or self-phase modulation (SPM) taking place within normally dispersive fibers associated with an offset spectral filtering [START_REF] Westbrook | Measurement of residual chromatic dispersion of a 40-Gb/s RZ signal via spectral broadening[END_REF].

In the present contribution, we propose and experimentally demonstrate a new configuration enabling both monitoring of the RCD and OSNR of a return-to-zero telecommunication signal. Compared to ref. [START_REF] Westbrook | Measurement of residual chromatic dispersion of a 40-Gb/s RZ signal via spectral broadening[END_REF], our set-up benefits from the SPM effect occurring in an anomalous dispersion fiber and requires an output centered optical bandpass filtering (OBPF). We first describe the principle of operation as well as the experimental set-up of our device. In the following sections, experimental and numerical results outline a high sensitivity relative to the two impairments.

EXPERIMENTAL SETUP AND PRINCIPLE OF OPERATION

Our experimental setup is illustrated in Fig 1a . A 10-GHz modelocked fiber laser (MLL) delivers ultrashort pulses with a full width at half-maximum (FWHM) of 2.4 ps and with a central wavelength of 1550 nm. A dual intensity/phase programmable optical filter is then inserted in order to emulate the RCD by providing a parabolic phase profile to the incoming pulses. In parallel, the initial signal is also degraded by means of its combination with a spectrally filtered amplified spontaneous emission source (ASE). By tuning the level of spectral attenuation of the dispersion emulator, a direct control of the OSNR is thus achieved. The core of our all-fibered device relies on a 920-m long highly nonlinear fiber (HNLF, fiber from ofs fitel) characterized by a Kerr coefficient γ of 10 W -1 .km -1 and an anomalous dispersion β 2 of 0.7 ps/km/nm. The linear losses (0.5 dB/km) and dispersion slope (0.01 ps/km/nm 2 ) are both reduced so their global impact can be neglected. In order to achieve an adequate working power (WP) and thus to observe the expected nonlinear behavior, an erbium-doped fiber amplifier (EDFA) combined with a variable attenuator (so as to finely tune the WP) is inserted before injection into the HNLF. An additional segment of standard single mode fiber SMF (31 m) is also used and its effect will discussed in § 3.

During their propagation in the HNLF, optical pulses experience a significant spectral expansion provided by the SPM effect, as highlighted in the experimental measurements depicted in Fig. 1b. Note that this spectral broadening is also characterized by an atypical depletion of the central part of the spectrum for input average powers near 13 mW (point B). For higher power levels, solitonic compression leads naturally to general enrichment of the spectrum.

At the fiber output, a 100-GHz Gaussian-shape OBPF centered on the signal wavelength (see dashed white line, Fig. 1b) carves into the resulting expanded spectrum.

The fundamental feature of our system is the shape of its intensity transfer function (TF) that non-linearly maps the output average power obtained after filtering as a function of the input average power. Indeed, according to the choice of fiber parameters, three different behaviors can be obtained : a monotonously increasing TF, a curve exhibiting a saturated plateau area or a raise and fall evolution [START_REF] Fatome | Scaling guidelines of a soliton-based power limiter for 2R-optical regeneration applications[END_REF]. At the opposite of classical applications that usually target an all-optical ultrafast power-limiter and where a plateau zone is suitable [START_REF] Asobe | Noise reduction of 20 Gbit/s pulse train using spectrally filtered optical solitons[END_REF][START_REF] Gay | 170 Gbit/s bit error rate assessment of regeneration using a saturable absorber and a nonlinear fiber based power limiter[END_REF], we voluntarily implement here a strong rise and fall evolution. Indeed, such a TF characteristic enables an enhanced sensitivity towards the initial pulse stream fluctuations and has recently been experimentally exploited in the context of an ultrafast all-optical amplitude jitter magnifier [START_REF] Finot | All-optical fiber-based ultrafast amplitude jitter magnifier[END_REF] and in a bit error monitoring scheme [START_REF] Finot | Experimental demonstration of an ultrafast all-optical bit-error indicating scheme[END_REF].

The experimental data for the TF of our device are plotted in Fig. 1c (open circles) and are in very good agreement with the results obtained from numerical simulations of the standard nonlinear Schrödinger equation (dash grey line), with ψ, z and t being the electrical field, the longitudinal and temporal coordinates respectively :

2 2 2 2 . 2 i z t ψ β ψ γ ψ ψ ∂ ∂ = - ∂ ∂
It is important to notice that the dramatic depletion of the central part in the spectrum observed for input powers close to 13 mW (point B) is responsible for the minimum of output power observed on the TF.

CHROMATIC DISPERSION MONITORING

In this section, we focus on the capability of our device to efficiently highlight the consequences of the presence of RCD. We consider in our discussion an input signal only affected by RCD and not by other sources of impairments such as polarization mode dispersion or cross/self phase modulation. As a result, the parabolic spectral phase imposed by the initial dispersion emulator directly impacts the temporal width of the pulses entering into the HNLF as can be seen in Fig. 2a, where the temporal duration of pulses has been measured by an optical sampling oscilloscope, OSO (see for example insets of Fig. 2a). Note that the fixed 31-m long segment of SMF imposes an additional chirp on the incoming pulses, so that the minimum temporal duration was not obtained for a RCD equals to zero.

For a fixed average power, the amount of RCD directly acts on the input peak power of the pulses so that the resulting spectral broadening would be severely affected. Indeed, the SPM occurring into the HNLF basically depends on the temporal gradient of the intensity profile, in other words, on both peak-power and temporal duration. Fig. 2b illustrates such a dependence for an input average power of 7.25 mW (ie, working point corresponding to the point A on Fig. 1c). This sensitivity is also readily apparent on the TF : Fig. 2c1 summarizes the TF simulated for various amounts of RCD and underlines a continuous expansion of the TF towards higher average input powers. The experimental results, plotted in Fig. 2c2 are once again in good agreement with these numerical predictions and emphasize the high sensitivity of our device towards the RCD. Consequently, if for a practical use of our device we operate at a fixed input average power, we can make out that the output average power will be strongly affected by the level of RCD (Fig. 2d). To illustrate this point, we have considered the evolution of the output average power as a function of RCD for two different initial WP, corresponding to the local extrema of the TF (with no RCD): maximum A (black crosses) and minimum B (grey crosses) as defined in Fig. 1c.

In the case of a WP A (7.25 mW, black crosses), the evolution of the output power as a function of level of RCD is monotonously increasing in the range of -1 to 2 ps/nm. Moreover, for this span, one can not only get access to the absolute value of the RCD, but also to its sign, which was not obvious with previously demonstrated techniques and which could be considered as crucial to ensure a convenient feedback from a tunable dispersion compensator. This extended range of unambiguous measurement is closely linked to the addition of the SMF pre-chirp: without such a fiber, the measurable RCD would have been limited to -0.3 to 2 ps/nm, as shown by numerical results plotted with a dashed black curve in Fig. 2d.

If we now operate at the WP B (12.9 mW, grey crosses), the system exhibits a higher sensitivity to the level of RCD. However, this is at the cost of the information regarding the sign of the RCD. In this context, combining the measurements at working powers A and B could be beneficial to measure accurately low RCD levels: WP A providing the sign of the RCD as well as a rough estimation, while the absolute value can be then more finely determined using WP B.

Fig. 2: (a) FWHM temporal duration of the pulse entering the HNLF according to the RCD induced by the dispersion emulator. Insets : temporal intensity profiles recorded over a time span of 15 ps. (b) Experimental demonstration of the influence of the RCD on the resulting spectrum recorded at the output of the HNLF and for a working average power of 7.25 mW. (c) TF for various amounts of RCD. Numerical simulations (c1) are compared with experimental results (c2). (d)

Influence of the RCD on the output power for two WPs : initial average power of 7.25 mW (black crosses) and 12.9 mW (grey crosses). The results without the SMF (numerical simulations) are plotted using a dashed black line.

IMPACT OF OSNR

In this last section, we investigate the impact of the in-band optical signal to noise ratio (OSNR) on the output average power. In this series of experiments and for a fixed RCD, the OSNR is simply tuned by varying the level of signal mixed with the additional ASE, see Fig. 1a.

Experimental results are summarized in Fig. 3a and show that for a degraded OSNR, a higher input power is required to observe the same TF. This can be physically understood given the fact that, for a fixed temporal pulse profile, the SPM-driven spectral broadening is highly sensitive to the peak-power, thus directly related to the energy contained into the pulse. Consequently, a decreased OSNR will require a higher average power to compensate for the energy contained in the noise and therefore to maintain the same pulse energy. We can make out that by operating at WP B, the output power is directly correlated to the OSNR. As shown by experimental results depicted in Fig. 3b, the output power varies continuously from 0.1 to 0.55 mW for an OSNR ranging from -32 to -15 dB with a remarkably linear part over 10 dB. Hence, the direct measurement of the output power for a fixed WP can enable us to easily monitor the OSNR of the incoming signal. Quite interestingly, if one can record the TF in presence of both RCD and OSNR, one should be able to discriminate the consequences of each effect. More precisely, the easy-to-implement following strategy that focuses on point A (defined as the first local maximum of the TF, see Fig. 1b and1c) could be used. From Fig. 3a, it can be noticed that the output power for this point is nearly not affected by the OSNR. On the contrary, at this point, the RCD has a strong impact on the output power (Fig. 2c). Consequently, the value of the RCD could be extracted from the output power recorded at this maximum of the TF. Once the RCD value is known (and provided calibration of the influence of the OSNR), we can then use the value of the working power required for point A, this WP being highly affected by the OSNR (see Fig. 3a).

CONCLUSIONS

We have proposed and experimentally demonstrated a new practical all-optical architecture that provides direct measurement of both optical signal to noise ratio and level of residual chromatic dispersion. The sensitivity of our device is remarkable and RCD with absolute value below 1 ps/nm can easily be detected. Our proof-ofprinciple experiments were based on a 10-GHz 2.4-ps signal, but additional numerical simulations have revealed the possibility to extend this concept to higher repetition rates and other pulse durations (using scaling laws [START_REF] Fatome | Scaling guidelines of a soliton-based power limiter for 2R-optical regeneration applications[END_REF], longer pulse durations can indeed enable a larger RCD measuring range up to several hundred ps/nm).

Based on a central filtering and anomalous dispersion regime, our experimental set-up has the advantage to require moderate input powers compared to other solutions reported so far. Moreover, the use of an anomalous segment of HNLF limits the detrimental pulse-to-pulse interactions and patterning effects [START_REF] Fatome | Scaling guidelines of a soliton-based power limiter for 2R-optical regeneration applications[END_REF] that may occur in normally dispersive fibers. Finally, from a practical implementation point of view, we believe that the compactness of our device could be greatly improved in a near future by replacing the fiber element by a highly nonlinear waveguides, for which constant progress in dispersion management now allows anomalous dispersion regime of propagation.

Figure 1 .

 1 Figure 1. (a) Experimental setup. PM : powermeter. OVA : optical variable attenuator (b) Evolution of the experimental optical spectrum monitored after the HNLF as a function of input average power. Dashed white line illustrates the FWHM of the OBPF. (c) Transfer function of the device: experimental results (circles) are compared with numerical simulations (dashed grey line).

Fig. 3 :

 3 Fig. 3: (a) Influence of the OSNR on the TF. (b) Output average power as a function of the input signal OSNR for a working power of 12.9 mW.


A Mezaouer
email: mezaouer@univ-blida.dz

J.-F Dupuy

K Boukhetala
email: kboukhetala@usthb.dz

Inverse probability weighted estimation in a reliability model with missing data

Keywords: Inverse probability weighted estimation, Large-sample properties, Linear transformation model, Missing data, Simulation study

The linear transformation model is a useful regression model for the analysis of reliability data. In this paper, we consider the problems of estimation and testing in this model in a context of missing data. Precisely, we consider the situation where explanatory variables are available for every unit in the experiment sample, while the event times and censoring indicators can only be observed on a subset of the sample. We rely on an inverse probability weighted-type estimation approach for approximating the regression parameter of interest. The theoretical and numerical properties of the resulting estimator are investigated. The proposed approach appears to outperform the classical complete-case analysis.

Introduction

Reliability is defined as the "probability that an item can perform its intended function for a specified interval under stated conditions" (see [START_REF] Dovich | Reliability Statistics[END_REF] for example). More generally, the statistical approach to reliability includes the statistical methods and models designed to the analysis of failure time data. Among them, the regression models have received much attention over the past decades. Regression models in reliability relate the time to failure (the response variable) to a set of explanatory variables or covariates (such as the temperature, pressure, voltage, various stresses,. . . ). Usual objectives of a regression analysis of reliability data are to: i) estimate the effects of the explanatory variables on the risk of failure, ii) examine various hypotheses about these effects. A number of regression models have been proposed for that purpose (accelerated failure time models, generalized proportional hazards models, additive models, multiplicative models,. . . ). Numerous examples and applications can be found, for example, in [START_REF] Klein | Survival Analysis: Techniques for Censored and Truncated Data[END_REF], Meeker and Escobar (1998), [START_REF] Bagdonavičius | Accelerated Life Models[END_REF], and Martinussen and Scheike (2006). In this paper, we consider a general class of semi-parametric regression models for failure time data, which is called the class of linear transformation models. This class includes as particular cases the well-known proportional hazards and proportional odds models. The transformation models for failure time data have given rise to an extremely rich literature so far; see, among others, [START_REF] Chen | Semiparametric analysis of transformation models with censored data[END_REF], Slud and Vonta (2004), [START_REF] Ma | Penalized log-likelihood estimation for partly linear transformation models with current status data[END_REF], [START_REF] Kosorok | Inference under right censoring for transformation models with a change-point based on a covariate threshold[END_REF]. See also [START_REF] Fleming | Survival analysis in clinical trials: past developments and future directions[END_REF], [START_REF] Bagdonavičius | Accelerated Life Models[END_REF], Martinussen and Scheike (2006), and [START_REF] Dupuy | Transformation models for failure time data: an overview of some recent developments[END_REF] for recent reviews and numerous references. In particular, [START_REF] Cheng | Analysis of transformation models with censored data[END_REF] have proposed simple estimating equations for estimating the regression parameter of interest in this class of models. The estimation proceeds from a sample of n independent observations (X 1 , ∆ 1 , Z 1 ), . . . , (X n , ∆ n , Z n ) of the vector (X, ∆, Z), where X denotes the (eventually right-censored) failure time, ∆ is the censoring indicator, and Z is a vector of explanatory variables. The proposed estimators are consistent and asymptotically normally distributed, with a covariance that can be consistently estimated. Using Cheng et al.'s approach, it is straightforward to carry out the statistical analysis of reliability data (estimation and testing). However, in reliability, cost constraints and unexpected technical issues arising during the course of the study often prevent engineers from observing the full vector (X, ∆, Z) on all the study items. In some situations, it may happen that the explanatory variables Z can be observed on all the items (at the beginning of the experiment, say) while X and ∆ can only be observed on a subset of the initial sample. The following two-stage procedure provides a simple way of making statistical inference with such data: i) remove from the analysis the observations i (i ∈ {1, . . . , n}) such that (X i , ∆ i ) is unobserved, ii) use Cheng et al.'s estimating equations on the resulting data set (the so-called complete data). As long as the missing-data mechanism is random (MAR, see Tsiatis (2006) for example), we may expect this complete-case analysis (CC therafter) to provide an unbiased estimator for the regression coefficient of interest. However, having discarded part of the sample, we may also expect an increase in the variance of this estimator, resulting in a loss of power of say, the Wald test of nullity of the regression parameter. This, in turn, may affect our interpretation of the true relationship between the covariates and the risk of failure. Therefore in this paper, we rely on an alternative estimation approach for the linear transformation model with missing values of the couple (X, ∆). This approach uses the full observed data that is, it also takes account of the Z i 's for those items with unobserved (X i , ∆ i ). We investigate the properties of this method (both theoretically and numerically), and we compare its performances with the CC approach. It appears from our simulation study that the proposed approach outperforms the CC analysis. The rest of the paper is organized as follows. In Section 2, we briefly describe the general class of linear transformation models, we recall the estimating equation proposed by [START_REF] Cheng | Analysis of transformation models with censored data[END_REF], and we describe our problem. In Section 3, we describe an inverse probability weighted estimation procedure adapted to this problem, and we investigate the asymptotic properties of the resulting estimator. Technical details are postponed to the Appendix. Section 4 describes a simulation study. A conclusion and a discussion are given in Section 5.

The linear transformation model and a problem of missing data

We provide a brief introduction to the linear transformation model for reliability data. We refer to [START_REF] Bagdonavičius | Accelerated Life Models[END_REF], Martinussen and Scheike (2006), and [START_REF] Dupuy | Transformation models for failure time data: an overview of some recent developments[END_REF] among others, for detailed treatments and reviews. Let T be some random failure time and Z be a p-dimensional vector of covariates. The class of linear transformation models relates T to Z via the following equation:

e(T ) = -β 0 Z + , ( 1
)
where e is an unknown strictly increasing transformation function, β 0 is a pdimensional unknown regression parameter of interest (β 0 reflects the amount of dependence between the covariates and the risk of failure), and is a random error variable with known distribution function F ( is assumed independent of Z). It is convenient to reparameterize the model (1) as H(T ) = e -β 0 Z e , where H(u) = exp(e(u)) is a strictly increasing positive function such that H(0) = 0 and lim u→∞ H(u) = ∞. Letting h be the derivative of H, some simple algebra shows that the hazard function λ(t) = lim δ↓0 δ -1 P(t ≤ T < t + δ|T ≥ t, Z) of T given Z can be expressed as

λ(t) = λ e (e β 0 Z H(t))e β 0 Z h(t), (2)
where λ e is the hazard function of exp( ). From this, we deduce the following well-known examples of transformation models (several other examples can be found in [START_REF] Kosorok | Inference under right censoring for transformation models with a change-point based on a covariate threshold[END_REF] and [START_REF] Ma | Penalized log-likelihood estimation for partly linear transformation models with current status data[END_REF]):

Exemple 1. Let have the extreme value distribution that is, F (u) = 1 -exp(-e u ). Then exp( ) is distributed as a standard exponential random variable and thus λ e (u) = 1. It follows that λ(t) = e β 0 Z h(t), and (2) reduces to the hazard function of a Cox proportional hazards model with baseline hazard rate h and cumulative baseline hazard function H.

Exemple 2. Let have the standard logistic distribution that is, F (u) = exp(u)/(1 + exp(u)). Then λ e (u) = (1 + u) -1 and (2) reduces to λ(t) = h(t)

H(t)+e -β 0
Z , which is known as the proportional odds model.

The usual objectives of the statistical inference in model ( 1) are to estimate the unknown β 0 and to examine various hypotheses about its value. Assume that we observe n independent copies (X 1 , ∆ 1 , Z 1 ), . . . , (X n , ∆ n , Z n ) of the random vector (X, ∆, Z), where X = T ∧ C (∧ denotes the minimum), T denotes the failure time of interest, C is a random right-censoring time, ∆ = 1(T ≤ C), and Z is a p-dimensional vector of explanatory variables. If s ≥ 0, let Y (s) = 1(X ≥ s) denote the at-risk indicator. Several procedures have been developed to estimate β 0 . We refer to Martinussen and Scheike (2006) for a detailed exposition. In particular, [START_REF] Cheng | Analysis of transformation models with censored data[END_REF] have proposed a fairly simple method, which estimates β 0 by the solution β of the estimating equation:

U (β) = n i=1 n j=1 ω(Z ij β)Z ij ∆ j Y i (X j ) G 2 (X j ) -ξ(Z ij β) = 0, (3)
where

Z ij = Z i -Z j , ω(•)
is a weight function [START_REF] Cheng | Analysis of transformation models with censored data[END_REF] proposed to use either ω

(•) = 1 or ω(•) = ξ(•)/{ξ(•)(1 -ξ(•))}, where ξ(s) =
As explained in the introduction, we consider the situation where the explanatory variables Z can be observed on all the items under study (at the beginning of the experiment, say) while the couple (X, ∆) can only be observed on a random subset of the initial sample. Let R be the random variable which equals 1 if (X, ∆) is observed and 0 otherwise. Then we observe n independent vectors (

X i R i , ∆ i R i , R i , Z i ), i = 1, . . . , n.
We consider the case where (X i , ∆ i ), if missing, is missing-at-random that is, R i is independent of (T i , C i ) given

Z i (we refer to Tsiatis (2006), for example, for a detailed description of the various missing data mechanisms).

The statistical problem is to estimate β 0 in model (1) from this incomplete data set. An obvious estimation procedure consists in solving the estimating equation ( 3), based on the subset S = {i ∈ {1, . . . , n} : R i = 1} that is, to remove from the analysis the items i such that (X i , ∆ i ) is missing. This however will cause a loss of information since the covariates {Z i : i / ∈ S} will be ignored. In the next section, we describe an alternative estimation procedure.

The proposed estimation method with missing data

We propose to estimate β 0 by adapting to our setting the general method of inverse weighting of complete cases. Let us denote by η i = P(R i = 1|Z i ) the probability of observing a complete case. The basic intuition underlying the method is as follows. Consider an item i, randomly sampled from a population with covariate value Z i . Then the probability that this item will have a complete observation (X i , ∆ i , Z i ) is η i . Therefore, any item with covariate Z i and complete observation can be thought of as representing 1 η i items from the population. This suggests that we may estimate β 0 by the solution β n of the following estimating equation:

U n (β) = n i=1 n j=1 ω(Z ij β)Z ij ∆ j Y i (X j )R i R j G 2 (X j )η i η j -ξ(Z ij β) := n i=1 n j=1 U ij (β) = 0 (4)
To motivate this equation, note that E[U ij (β 0 )] = 0. To see this, remark that

E ω(Z ij β 0 )Z ij ∆ j Y i (X j )R i R j G 2 (X j )η i η j Z i , Z j = E ω(Z ij β 0 )Z ij E 1(T j ≤ C j )1(T i ∧ C i ≥ T j )R i R j G 2 (T j )η i η j T j , Z i , Z j Z i , Z j = ω(Z ij β 0 )Z ij ξ(Z ij β 0 ). It follows that E[U ij (β 0 )|Z i , Z j ] = 0 and therefore E[U ij (β 0 )] = 0.
It is thus natural to estimate β 0 by the solution to the empirical counterpart of this latter equation, which is n i=1 n j=1 U ij (β) = 0. However, in practice, the probabilities η i and G 2 (T j ) are unknown, and thus have to be estimated. Following [START_REF] Cheng | Analysis of transformation models with censored data[END_REF], G can be estimated by the Kaplan-Meier estimator G (see [START_REF] Andersen | Statistical models based on counting processes[END_REF]), and we propose to estimate η i by fitting a regression model (such as the logistic regression model, for example) to the data (R 1 , Z 1 ), . . . , (R n , Z n ), which results in an estimate η i . Finally, we propose to estimate β 0 by the solution of the following approximated version of the equation (4):

n i=1 n j=1 ω(Z ij β)Z ij ∆ j Y i (X j )R i R j G 2 (X j ) η i η j -ξ(Z ij β) = 0. ( 5
)
This yields an approximation β n of the theoretical estimator β n . In the following, we establish the asymptotic properties of β n . If the censoring survival function G and the η i are reasonably estimated, we may expect the approximated estimator β n to inherit the large-sample properties of β n . This point will be investigated in the simulation study. Before turning to the asymptotic properties of β n , we introduce some additional notations and regularity conditions. If v is a p-dimensional column vector, we note v ⊗2 = vv . Then, define

e ij (β) = ∆ j Y i (X j )R i R j G 2 (X j ) η i η j -ξ(Z ij β) Λ -1 n = 1 n 2 n i=1 n j=1 ω(Z ij β n )Z ij Z ij ξ(Z ij β n ) Γ n = 1 n 3 n i=1 n j=1 k =j ω(Z ij β n ) e ij ( β n ) -ω(Z ji β n ) e ji ( β n ) × ω(Z ik β n ) e ik ( β n ) -ω(Z ki β n ) e ki ( β n ) Z ij Z ik - 4 n 3 n l=1 R l (1 -∆ l ) ( n k=1 R k Y k (X l )) 2    n i=1 n j=1 ω(Z ij β n )Z ij ∆ j Y i (X j )R i R j G 2 (X j ) η i η j Y j (X l )    ⊗2 .
We assume that the covariates are bounded that is, there exists a finite constant c such that |Z ij | ≤ c for every i = 1, . . . , n and j = 1, . . . , p. We also assume that the censoring is independent of Z. This condition is reasonable in many settings (note that this was also assumed in [START_REF] Cheng | Analysis of transformation models with censored data[END_REF]'s paper). Some ways to relax it are however discussed in Section 5. Finally, in order to identify β 0 , we assume that G(•) > 0 over the experiment time interval and that P(R = 1|Z = z) > 0 for every z. We are now in position to state the following theorem, whose proof is given in the appendix.

Theorem. Under the regularity conditions stated above, β n is a consistent estimator of β 0 , and n

We now investigate, via simulations, the numerical properties of the approximated estimator β n (bias, variance). We investigate the quality of the Gaussian approximation of its large-sample distribution. We also investigate the power and level of the Wald-type test based on β n (the Wald test is widely used to test the null hypothesis that a given covariate has no effect on the response variable).

A simulation study

The simulation setting is as follows. We simulate right-censored failure times from the linear transformation model e(T ) = -β 0 Z + , where e is the logarithm function, is distributed according to the extreme value distribution, and Z is normally distributed with mean 0 and variance 1. The values log(1.5) and 0 are considered for β 0 . An exponential distribution with parameter λ is used to simulate censoring (λ is successively chosen to yield 15% and 30% of censoring). We consider small (n = 75) and moderate (n = 150) sample sizes.

The missingness indicator R is simulated from a Bernoulli random variable with parameter

P(R = 1|Z) = exp(θ 0 + θ 1 Z)/(1 + exp(θ 0 + θ 1 Z))
, with θ 0 and θ 1 chosen to yield various missingness percentages (15% and 30%). The ideal case (referred to as FD for full-data in the sequel) where all the (X i , ∆ i ) are observed is also considered. In this case, β 0 is estimated using

i = exp( θ 0,n + θ 1,n Z i )/(1 + exp( θ 0,n + θ 1,n Z i )).
Based on the N replications, and for each of the combinations mentioned above, we obtain an averaged value N -1 N j=1 β (j) n of the estimates of β 0 , where β (j) n is the estimate obtained from the j-th simulated sample (formula (5) with ω(•) = 1). We also obtain the average value mean( s.e.) of the asymptotic standard error estimates, and the variance of the β (j) n . When β 0 = 0 (respectively β 0 = 0), we obtain the empirical power (respectively the empirical size) of the Wald test at the 5% level for testing β 0 = 0. The results are summarized in the Table 1. For comparison, the method which applies [START_REF] Cheng | Analysis of transformation models with censored data[END_REF]'s estimating equation (3) to the subset of complete cases (CC) only is also evaluated.

Table 1 about here

From these results, both the proposed estimator β n (referred to as IPW for Inverse Probability Weighted, in the sequel) and the CC estimator approximate reasonably well the unknown value β 0 . Also, as expected, the accuracy of both estimators degradates as the missingness percentage increases. The magnitude of the degradation is however larger for the CC than for the IPW estimator. This is particularly clear when the sample size is small. We note also that the power of the Wald-type tests based on the CC and IPW estimators decreases when the missingness percentage increases. But in all cases, the power of the IPW-based Wald test is higher than the power of the CC-based Wald test. The magnitude of the difference between powers is particularly large when the sample size is small. For moderate sample size, the IPW-based Wald test maintains a high power even when the missingness percentage is high. When β 0 = 0, both IPW-and CC-based tests globally satisfy the prescribed significance level α = 0.05, but again, the averaged standard error estimates reveal better performance for the proposed IPW approach, over the simpler CC method. For each simulation scenario, we plot the histograms of the N values β (j) n and of the N CC estimates, along with the associated Q-Q plots. Figures 1 to 8 display the results.

Figures 1 to 8 about here

From these figures, it appears that the normal approximation stated in the theorem is reasonably satisfied by the IPW estimator while the finite sample distribution of the CC estimator appears to be skewed when the sample size is small and/or when the percentage of missingness is high. These results indicate that a reliable statistical inference in the linear transformation model with missing durations and censoring indicators may be based on the IPW principle, provided that some reasonable conditions (such as n ≥ 75, and a moderately large missingness percentage) are met.

Conclusion and discussion

In this paper, we have considered the problem of estimation in the linear transformation model for failure time data, when the failure times and censoring indicators are missing at random for some units in the experiment sample. We have proposed an estimating approach for this problem, which relies on the principle of inverse probability weighting of complete cases. We have derived the asymptotic properties of the resulting estimator, and we have compared its numerical performances to the complete case estimator. It appears that the proposed IPW estimator outperforms the simpler CC estimator, under various realistic conditions of sample size, censoring, and missingness. The proposed procedure is valid under a set of reasonable conditions. The assumption that the censoring is independent of Z may not be satisfied in some experiments. In such a case, one may simply modify the proposed estimating equation (5) by using an appropriate estimator for the conditional survival function G(t|Z) = P(C ≥ t|Z) (a non-parametric estimator may be used if Z is continuous, while a stratified estimator G(•|Z = j) may be used when Z ∈ {1, . . . , J} is discrete). We have also assumed that the missing data mechanism is random. The case when the missingness is non-ignorable deserves its own attention, and constitutes a topic for future research. Finally, the IPW approach is useful if we are able to correctly specify the model for R given Z. The robustness of this approach to a misspecification of this model is also of interest, and constitutes a further topic for investigation.

Appendix

We give an outline of the proof of the theorem stated in Section 3. The arguments are similar to those given in [START_REF] Cheng | Analysis of transformation models with censored data[END_REF], but the technical details are different. Note first that the following holds in probability, as n → ∞:

1 n 2 U n (β)(β -β 0 ) → E ω(Z 12 β)Z 12 (β -β 0 ) ∆ 2 Y 1 (X 2 )R 1 R 2 G 2 (X 2 )η 1 η 2 -ξ(Z 12 β) .
Letting z 12 = z 1 -z 2 and H be the distribution function of Z, this limit can be written as

z 1 ,z 2 ω(Z 12 β)Z 12 (β -β 0 ) {ξ(Z 12 β 0 ) -ξ(Z 12 β)} dH(z 1 )dH(z 2 ),
which is zero only when β = β 0 and therefore, by classical arguments of Zestimation (see van der Vaart (1998) for example), β n converges in probability to β 0 .

To prove the asymptotic normality of β n , first write a Taylor series expansion of

U n ( β n ) around β 0 as 0 = U n ( β n ) = U n (β 0 ) + ( β n -β 0 ) Un (β * n )
, where β * n lies on the line segment between β n and β 0 , and Un denotes the derivative of U n with respect to the components of β. Next, some algebraic manipulations yield n

1 2 ( β n -β 0 ) = n -3 2 E ω(Z 12 β 0 )Z 12 Z 12 ξ(Z 12 β 0 ) U n (β 0 ) + n -3 2 U n (β 0 ) • o p (1),
where o p (1) denotes a sequence which converges to 0 in probability as n → ∞. By Slutsky's theorem, the asymptotic normality of n 1 2 ( β n -β 0 ) will be proved if we can prove that n -3 2 U n (β 0 ) is asymptotically normal. n -3 2 U n (β 0 ) is a Ustatistic, and it follows from the central limit theorem for U-statistics that it is asymptotically normally distributed with mean 0 (see [START_REF] Kowalski | Modern Applied U-Statistics[END_REF] for example). To calculate the asymptotic variance, we first use the martingale integral representation of ( G -G)/G given by [START_REF] Gill | Censoring and Stochastic Integrals[END_REF], and we re-write n -3 2 U n (β 0 ) as

n -3 2 n i=1 n j=1 ω(Z ij β 0 )Z ij e ij (β 0 ) + 2n -1 2 n k=1 ∞ 0 q(t) π(t) dM k (t) + o p (1) (6)
where

e ij (β) = ∆ j Y i (X j )R i R j G 2 (X j )η i η j -ξ(Z ij β), Λ G is the cumulative hazard function of C, M k (t) = 1(X k ≤ t, ∆ k = 0) -t 0 Y k (u)dΛ G (u)
denotes the martingale associated to the censoring counting process 1(X k ≤ t, ∆ k = 0), π(t) = P(X ≥ t), and

q(s) = lim n→∞ 1 n 2 n i=1 n j=1 ω(Z ij β 0 )Z ij ∆ j Y i (X j )R i R j G 2 (X j )η i η j Y j (s).
The first term in ( 6) is also a U-statistic, with limiting variance given by

E   n -3 n i=1 n j=1 k =j ω(Z ij β 0 )e ij (β 0 ) -ω(Z ji β 0 )e ji (β 0 ) × {ω(Z ik β 0 )e ik (β 0 ) -ω(Z ki β 0 )e ki (β 0 )} Z ij Z ik ] .
The second term in ( 6) is a martingale with limiting variance 4 ∞ 0 q(t)q (t) π(t) dΛ G (t). Finally, the covariance between the first two terms of (6) can be calculated as cov

   n -3 2 n i=1 n j=1 ω(Z ij β 0 )Z ij e ij (β 0 ), 2n -1 2 n k=1 ∞ 0 q(t) π(t) dM k (t) Z i , Z j    = 2 n 2 n i=1 n j=1 E ∞ 0 ω(Z ij β 0 )Z ij e ij (β 0 ) q (t) π(t) n k=1 dM k (t) Z i , Z j , which asymptotically, is -4 ∞ 0 q(t)q (t) π(t) dΛ G (t). Therefore, the asymptotic co- variance matrix of n -3 2 U n (β 0 ) is lim n→∞ n -3   n i=1 n j=1 k =j ω(Z ij β 0 )e ij (β 0 ) -ω(Z ji β 0 )e ji (β 0 ) × {ω(Z ik β 0 )e ik (β 0 ) -ω(Z ki β 0 )e ki (β 0 )} Z ij Z ik -4 ∞ 0 q(t)q (t) π(t) dΛ G (t) .
The expression above can be estimated by replacing β 0 , G, and η i by β n , G, and η i respectively, and by replacing Λ G by a standard Nelson-Aalen estimator (see [START_REF] Andersen | Statistical models based on counting processes[END_REF]), based on the subset S (note that under our regularity conditions, this estimator should be consistent for Λ G ). This yields the expression Γ n given in Section 3. Finally, the quantity

E ω(Z 12 β 0 )Z 12 Z 12 ξ(Z 12 β 0 ) -1
can be consistently estimated by its empirical counterpart Λ -1 n . This concludes the proof.

Martinussen, T., Scheike, T. H., 2006. Dynamic Regression Models for Survival

Data. New York: Springer. Meeker, W. Q., Escobar, L. A., 1998. Statistical 0 5 10 15 20 25 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 2 0. 4 0. 6 0. 8

IWP15 Theoretical Quantiles
Sampl e Quant i l es CC15 betatilde Fr equency 0.0 1.0 0 5 10 15 20 25 30 35 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 5 1. 0 1. 5 0 5 10 15 20 25 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 2 0. 4 0. 6 0. 8

IWP30 Theoretical Quantiles
Sampl e Quant i l es

CC30 betatilde
Fr equency -0.5 0.5 1.5 0 5 10 15 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 -0. 5 0. 0 0. 5 1. 0 1. 5

CC30 Theoretical Quantiles
Sampl e Quant i l es 0 5 10 15 20 25 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 2 0. 4 0. 6 0. 8

IWP15 Theoretical Quantiles
Sampl e Quant i l es CC15 betatilde Fr equency 0.0 1.0 0 5 10 15 20 25 30 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 5 1. 0 1. 5

CC15 Theoretical Quantiles
Sampl e Quant i l es

IWP30 betatilde
Fr equency 0.0 0.4 0.8 0 5 10 15 20 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 2 0. 4 0. 6 0. 8

IWP30 Theoretical Quantiles
Sampl e Quant i l es

CC30 betatilde
Fr equency -0.5 0.5 1.5 2.5 0 5 10 15 20 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 -0. 5 0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 2 0. 4 0. 6 0. 8 0 10 20 30 40 50 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 5 1. 0 0 5 10 15 20 25 30 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 2 0. 4 0. 6 0. 8

IWP15

CC15

IWP30 Theoretical Quantiles
Sampl e Quant i l es CC30 betatilde Fr equency 0.0 0.5 1.0 1.5 0 5 10 15 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 5 1. 0 1. 5 0 10 20 30 40 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 2 0. 4 0. 6 0. 8

IWP15 Theoretical Quantiles
Sampl e Quant i l es CC15 betatilde Fr equency 0.0 0.5 1.0 1.5 0 10 20 30 40 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 5 1. 0 1. 5 0 5 10 15 20 25 30 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 2 0. 4 0. 6 0. 8

IWP30 Theoretical Quantiles
Sampl e Quant i l es CC30 betatilde Fr equency 0.0 1.0 0 5 10 15 20 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 0. 0 0. 5 1. 0 1. 5

CC30 Theoretical Quantiles
Sampl e Quant i l es 0 5 10 15 20 25 30 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 -0. 4 -0. 2 0. 0 0. 2 0. 4

IWP15 Theoretical Quantiles
Sampl e Quant i l es CC15 betatilde Fr equency -0.5 0.5 0 10 20 30 40 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 -0. 5 0. 0 0. 5 0 5 10 15 20 25 30 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 -0. 4 -0. 2 0. 0 0. 2 0. 4

CC15

IWP30 Theoretical Quantiles
Sampl e Quant i l es CC30 betatilde Fr equency -1.0 0.0 1.0 0 5 10 15 20 25 30 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3

-1. 0 -0. 5 0. 0 0. 5 1. 0

CC30 Theoretical Quantiles
Sampl e Quant i l es 0 5 10 15 20 25 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 -0. 4 -0. 2 0. 0 0. 2 0. 4

IWP15 Theoretical Quantiles
Sampl e Quant i l es

CC15 betatilde
Fr equency -1.0 0.0 1.0 0 10 20 30 40 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 -0. 5 0. 0 0. 5

CC15 Theoretical Quantiles
Sampl e Quant i l es

IWP30 betatilde
Fr equency -0.5 0.5 0 10 20 30 40 50 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3 -0. 5 0. 0 0. 5

IWP30 Theoretical Quantiles
Sampl e Quant i l es CC30 betatilde Fr equency -2 0 1 2 0 10 20 30 40 50 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 -1 1 3

-2 -1 0 1 2

CC30 Theoretical Quantiles
Sampl e Quant i l es Fig. 6. Histograms and Q-Q plots for n = 75, β 0 = 0, and 30% of censoring.

IWP15 betatilde

Fr equency -0.3 0.0 0.3 0 5 10 15 20 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 0 2 -0. 3 -0. 1 0. 0 0. 1 0. 2 0. 3 0 5 10 15 20 25 30 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 0 2 -0. 4 -0. 2 0. 0 0. 2 0. 4

IWP15

CC15

Theoretical Quantiles

Sampl e Quant i l es IWP30 betatilde Fr equency -0.2 0.2 0 10 20 30 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 0 2 -0. 2 0. 0 0. 2 0. 4

IWP30

Theoretical Quantiles

Sampl e Quant i l es CC30 betatilde Fr equency -1.0 0.0 1.0 0 10 20 30 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 0 2 -1. 0 -0. 5 0. 0 0. 5 1. 0 0 10 20 30 40 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 0 2 -0. 4 -0. 2 0. 0 0. 2 0. 4

IWP15

Theoretical Quantiles

Sampl e Quant i l es CC15 betatilde Fr equency -0.6 0.0 0.6 0 5 10 15 20 25 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 0 2 -0. 6 -0. 2 0. 2 0. 4 0. 6 0 10 20 30 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 0 2 -0. 4 -0. 2 0. 0 0. 2 0. 4 0 10 20 30 40 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q -3 0 2 -1. 0 -0. 5 0. 0 0. 5 1. 0

Fig. 1 .

 1 Fig. 1. Histograms and Q-Q plots for n = 75, β 0 = log(1.5), and 15% of censoring.

Fig. 2 .

 2 Fig. 2. Histograms and Q-Q plots for n = 75, β 0 = log(1.5), and 30% of censoring.

Fig. 3 .

 3 Fig. 3. Histograms and Q-Q plots for n = 150, β 0 = log(1.5), and 15% of censoring.

Fig. 4 .

 4 Fig. 4. Histograms and Q-Q plots for n = 150, β 0 = log(1.5), and 30% of censoring.

Fig. 5 .

 5 Fig. 5. Histograms and Q-Q plots for n = 75, β 0 = 0, and 15% of censoring.

Fig. 7 .

 7 Fig. 7. Histograms and Q-Q plots for n = 150, β 0 = 0, and 15% of censoring.

Fig. 8 .

 8 Fig. 8. Histograms and Q-Q plots for n = 150, β 0 = 0, and 30% of censoring.

Table 1 .

 1 Methods for Reliability Data. New York: Wiley. Slud, E. V., Vonta, F., 2004. Consistency of the NPML estimator in the rightcensored transformation model. Scandinavian Journal of Statistics, 31, 21-41. Tsiatis, A. A., 2006. Semiparametric Theory and Missing Data. New York: Springer. van der Vaart, A. W., 1998. Asymptotic Statistics. Cambridge University Press. Simulation results.

	β 0	n	% censoring	% missing	0	15		30
				method	FD	IPW	CC	IPW	CC
	log(1.5) 75	15	mean( βn)	0.386	0.393	0.705	0.382	0.825
				mean( s.e.)	1.193	1.355	2.712	1.481	4.719
				var	0.019	0.027	0.152	0.031	0.259
				power	0.803	0.672	0.582	0.631	0.141
			30	mean( βn)	0.363	0.366	0.636	0.354	0.865
				mean( s.e.)	1.168	1.312	2.372	1.494	5.174
				var	0.021	0.032	0.135	0.041	0.358
				power	0.756	0.689	0.653	0.545	0.116
		150	15	mean( βn)	0.382	0.379	0.639	0.383	0.834
				mean( s.e.)	1.218	1.382	2.411	1.591	4.564
				var	0.011	0.016	0.062	0.026	0.129
				power	0.968	0.901	0.909	0.794	0.473
			30	mean( βn)	0.367	0.367	0.634	0.369	0.841
				mean( s.e.)	1.199	1.348	2.386	1.551	4.633
				var	0.011	0.018	0.079	0.028	0.150
				power	0.956	0.874	0.885	0.764	0.501
	0	75	15	mean( βn)	-0.003	0.002	0.003	0.010	0.015
				mean( s.e.)	1.125	1.351	1.697	1.583	2.476
				var	0.019	0.026	0.064	0.031	0.154
				level	0.028	0.038	0.038	0.037	0.015
			30	mean( βn)	0.002	0.003	0.002	-0.002	0.004
				mean( s.e.)	1.137	1.367	1.726	1.636	2.962
				var	0.022	0.030	0.077	0.047	0.293
				level	0.048	0.056	0.047	0.072	0.016
		150	15	mean( βn)	-0.002	0.000	0.000	0.000	-0.007
				mean( s.e.)	1.143	1.369	1.673	1.609	2.644
				var	0.009	0.012	0.030	0.017	0.105
				level	0.032	0.028	0.032	0.052	0.041
			30	mean( βn)	-0.002	-0.004 -0.007	-0.008 -0.028
				mean( s.e.)	1.138	1.382	1.704	1.649	2.737
				var	0.009	0.015	0.038	0.019	0.129
				level	0.024	0.040	0.044	0.052	0.028

Note: n denotes the sample size. FD stands for Full Data, IPW for Inverse Probability Weighted estimator, CC for Complete Case estimator.

( β n -β 0 ) is asymptotically distributed as a Gaussian vector with mean 0 and a covariance matrix that can be consistently estimated by Λ -1 n Γ n Λ -1 n .