E Brunel
email: ebrunel@math.univ-montp2.fr.

F Comte

V Genon-Catalot
email: valentine.genon-catalot@parisdescartes.fr.

NONPARAMETRIC DENSITY AND SURVIVAL FUNCTION ESTIMATION IN THE MULTIPLICATIVE CENSORING MODEL

Keywords: Adaptive procedure, Bandwidth selection, Kernel estimators, Multiplicative censoring model AMS 2000 subject classifications, 62G07 -62N01

Nonparametric density and survival function estimation in the multiplicative censoring model

F Comte, V Genon-Catalot, Elodie

Introduction

In this paper, we consider the model [START_REF] Abbaszadeh | Multiplicative censoring: estimation of a density and its derivatives under the Lp-risk[END_REF] Y i = X i U i , i = 1, . . . , n under the assumptions: the U i , i = 1, . . . , n are independent and identically distributed (i.i.d.) with uniform distribution on [0, 1]; the X i , i = 1, . . . , n are real valued, i.i.d., with unknown density f and cumulative distribution function (c.d.f.) F ; the sequences (U i ) 1≤i≤n and (X i ) 1≤i≤n are independent. We intend to propose estimation methods for f and F (or F = 1 -F ) when observing a sample (Y i ) 1≤i≤n only.

Model [START_REF] Abbaszadeh | Multiplicative censoring: estimation of a density and its derivatives under the Lp-risk[END_REF] has been widely investigated mostly when the random variables X i are nonnegative. In this case, Model (1) is usually called the multiplicative censoring model and was introduced in [START_REF] Vardi | Multiplicative censoring, renewal processes, deconvolution and decreasing density: nonparametric estimation[END_REF], studied in more details in [START_REF] Vardi | Large sample study of empirical distributions in a random-multiplicative censoring model[END_REF] and by [START_REF] Asgharian | Large-sample study of the kernel density estimators under multiplicative censoring[END_REF]. As explained in [START_REF] Vardi | Multiplicative censoring, renewal processes, deconvolution and decreasing density: nonparametric estimation[END_REF], the multicative censoring model unifies several important statistical problems (deconvolution of an exponential variable, estimation under decreasing density constraint or some estimation problems in renewal processes). However in the above quoted papers, authors assume that observations are composed of two independent samples, one of direct observations X with size m, in addition to the above Y n-sample. The statistical procedures for estimating the c.d.f. F rely on the fact that m tends to infinity and cannot be applied for m = 0. Let us mention that van Es et al. (2000) studied a survival analysis model involving both multiplicative censoring and length bias, in a parametric context.

The problem may be related to the moment problem: in Model (1), all moments of X can easily be estimated from the observations Y , so the question of distribution reconstruction from its moments as pointed out by Mnatsakanov (2006) can be addressed.

Another strategy is to take the logarithm of the squared model, as proposed in stochastic volatility models (see [START_REF] Van Es | Nonparametric volatility density estimation for discrete time models[END_REF], Comte and Genon-Catalot (2006)), and to apply deconvolution methods. In these papers, the U i 's are supposed to be Gaussian. But then, the estimated function is distorted and, in case of real random variables X i , information about their sign is lost, when proceeding so.

Series expansion methods have been proposed in [START_REF] Andersen | Multiplicative censoring: density estimation by a series expansion approach[END_REF]: they consider the problem as an inverse problem and apply Singular Value Decomposition in different bases to provide estimators. They obtain rates comparable to ours though on different regularity spaces, however their procedure is not adaptive and depends on the choice of a cutoff which is only empirically studied. Later on, wavelet methods have been applied by [START_REF] Abbaszadeh | Nonparametric estimation of density under bias and multiplicative censoring via wavelet methods[END_REF][START_REF] Abbaszadeh | Multiplicative censoring: estimation of a density and its derivatives under the Lp-risk[END_REF] to estimate the density and its derivatives, considering a general L p -risk, and in presence of additional bias. Their estimators are adaptive and reach the same rates as ours up to logarithmic terms (when p = 2). They do not provide lower bound, and consider neither global estimation of the density (wavelets are compactly supported) nor survival function estimation. Note that [START_REF] Chesneau | Wavelet estimation of a density in a GARCH-type model[END_REF] studies the multiplicative censoring model when the sequence (X i ) i∈N is α-mixing and the U i 's can be a product of independent uniform random variables. The dependence implies an important loss in the rate.

In this paper, we consider first the case where the X i 's are real-valued, and we investigate the pointwise nonparametric estimation on R of both f and the survival function F (x) = 1 -F (x). All nonparametric methods (likelihood, projection, kernel, . . . ) rely on relationships between the density f Y and survival function FY = 1 -F Y of Y i and those of X i , given by [START_REF] Abbaszadeh | Nonparametric estimation of density under bias and multiplicative censoring via wavelet methods[END_REF] ∀y ∈ R, f Y (y) = +∞ y f (x)

x dx 1 y≥0 + y -∞ f (x) |x| dx 1 y<0 , (3)
∀y ∈ R, FY (y) + yf Y (y) = F (y), which imply the following key property. Let t : R → R be bounded, derivable, with t belonging to L 2 (R), and assume that E|X| < +∞, then

E(t(Y ) + Y t (Y )) = Et(X). This relation allows us to propose adequate correction of the observation Y in order to get information about X, and yields simple kernel estimators of f and F (see Formulae [START_REF] Comte | Penalized projection estimator for volatility density[END_REF] and ( 5)). We first study their pointwise L 2 -risks properties. Under regularity assumptions, we can obtain rates of convergence for which lower bounds are provided. The study includes the classical case of nonnegative X i 's, for which pointwise kernel estimation of the density and the survival function is new.

Then we study the global risk for f on R or for F on R + when the variables are nonnegative. An adaptive choice of the bandwidths is proposed following the [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF] theory and proved to lead to automatic bias-variance tradeoff for the resulting adaptive density or survival function estimators.

Next, still considering nonnegative X i 's, we use convolution power kernel estimators fitted to nonparametric estimation of functions on R + , proposed in Comte and Genon-Catalot (2012) for standard density estimation. We introduce estimators of f, F (now on R + ), different from those based on classical kernels. The interest is to avoid boundary effects at 0. These kernels require the choice of an integer parameter m, for which a data driven procedure is proposed. The resulting estimator is proved to be adaptive.

The paper is organized as follows. Standard kernel estimators are described and studied in Section 2, and convolution power kernel method is explained in Section 3. Section 4 presents a simulation study that allows us to compare the two strategies. Lastly, proofs are gathered in Section 5.

2. Kernel estimation on the real line 2.1. Definition of kernel estimators. Let K : R → R be a kernel i.e. an integrable function with K(u)du = 1, which is also assumed to be square integrable. We set for h > 0, K h (u) = (1/h)K(u/h). Along the results hereafter, we possibly need additional conditions on K:

(A1) K is bounded. (A2) K is an even function with lim u→+∞ K(u) = 0, K is derivable and K is integrable. (A3) [K (u)] 2 du < +∞. (A4) |u|[K (u)] 2 du < +∞. (A5) [uK (u)] 2 du < +∞. The estimator of F (x) is defined by: Fh (x) = 1 nh n i=1 K u -x h 1 I Y i ≥u du + Y i K Y i -x h = K h FY (x) + 1 n n i=1 Y i K h (Y i -x) (5)
where s t(x) = s(x -u)t(u)du denotes the convolution product and ( 6)

FY (x) = 1 n n i=1 1 I Y i ≥x .
For K satisfying (A2), which implies (A1), we define the estimator of f (x) by: fh

(x) = 1 nh n i=1 Y i h K Y i -x h + K Y i -x h = 1 n n i=1 Y i K h (Y i -x) + K h (Y i -x) . (7)
With simple computations, we can prove:

Proposition 2.1. Under (A2), (i) fh (x)dx = 1, (ii) lim x→-∞ Fh (x) = 1, (iii) lim x→+∞ Fh (x) = 0, (iv) ( Fh ) (x) = -fh (x).
2.2. Pointwise risk. Consider the Hölder ball:

Σ I (β, R) = {f : I → R, f ( ) exists for = β , |f ( ) (x) -f ( ) (x )| ≤ R|x -x | β-, ∀x, x ∈ I}
where β is the largest integer strictly smaller than β. Recall that K is a kernel of order if:

|u| |K(u)|du < ∞ and u j K(u)du = 0 for j = 1, . . . , .

The following proposition shows that the risk at x 0 has a different rate for x 0 = 0 and for x 0 = 0.

E[( Fh (x 0 ) -F (x 0 )) 2 ] ≤ C 2 1 h 2(β+1) + C 2 nh + C 3 n , (9) E[( Fh (0) -F (0)) 2 ] ≤ C 2 1 h 2(β+1) + C 4 n , with C 1 = R |u| β+1 |K(u)|du/( + 1)!, C 2 = 2E(|X 1 |) K 2 , C 3 = 2 K 2 and C 4 = 2 K 2 + |u|K 2 (u)du. If K is of order with = β and |u| β |K(u)|du < +∞, then under (A2)-(A3) (8)
, for all h ∈ (0, 1), we have

(10) E[( fh (x 0 ) -f (x 0 )) 2 ] ≤ C 2 5 h 2β + C 6 nh 3 with C 5 = R |u| β |K(u)|du/ ! and C 6 = 2 E(|X 1 |) K 2 + K 2 ∞ . Under (A2)-(A4), for x 0 = 0, we have (11) E[( fh (0) -f (0)) 2 ] ≤ C 2 5 h 2β + C 6 nh 2 ,
where

C 6 = K ∞ + |u|[K (u)] 2 du. Under (A2)-(A3) and (A2), if E(1/|X|) = f Y ∞ < +∞, f ∞ < +∞, and x 0 = 0, we have (12) E[( fh (0) -f (0)) 2 ] ≤ C 5 2 h 2β + C 6 nh ,
where

C 6 = f ∞ K 2 + f Y ∞ u 2 [K (u)] 2 du.
For h of order n -1/(2β+3) , the estimator of F (x 0 ) has rate O(n -2(β+1)/(2β+3) ), except in 0, where choosing h = n 1/[2(β+1)] , gives the parametric rate. This is due to the fact that P(X > 0) = P(Y > 0) and thus F (0) = FY (0). For instance, n -1 n i=0 1 I Y i ≥0 is an estimator of F (0) with parametric rate. For h of order n -1/(2β+3) , the estimator of f (x 0 ) has rate O(n -2β/(2β+3) ) when x 0 = 0. The rate is better at x 0 = 0 and of order O(n -2β/(2β+2) ) or O(n -2β/(2β+1) ), provided that f Y ∞ < +∞.

The next theorem states that the rates obtained for points x 0 = 0 are optimal-minimax.

Theorem 2.1. Assume that x 0 = 0, x 0 ∈ I and let β > 0. There exists a constant c > 0 such that [START_REF] Hirsch | Elements of functional analysis[END_REF] liminf

n→+∞ n 2β/(2β+3) inf fn sup f ∈Σ I (β,R) E f ( fn (x 0 ) -f (x 0 )) 2 ≥ c
where inf fn denotes the infimum over all estimators of f based on (Y j ) 1≤j≤n . Moreover, for β ≥ 1, there exists a constant c > 0 such that

(14) liminf n→+∞ n 2β/(2β+1) inf Fn sup F ∈Σ I (β,R) E f ( Fn (x 0 ) -F (x 0 )) 2 ≥ c
where inf Fn denotes the infimum over all estimators of F based on (Y j ) 1≤j≤n .

2.3.

Global risk and bandwidth selection. We denote by ψ = ( ψ 2 (x)dx) 1/2 the L 2norm and by

ψ 1 = |ψ(x)|dx the L 1 -norm of a function ψ : R → R. Let f h (x) = f (u)K((x -u)/h)/hdu = f K h (x)
and Fh (x) = F K h (x). We can prove:

Proposition 2.3. Assume that E(X 2 1 ) < +∞. If f belongs to L 2 (R) and (A2)-(A3) hold, then (15) E( fh -f 2 ) ≤ f -f h 2 + K 2 nh + E(Y 2 1 ) K 2 nh 3 . If X is nonnegative, F belongs to L 2 (R + ) and K has compact support [-1, 1], then, for all h ≤ 1, ( 16
) E( R + ( Fh (x) -F (x)) 2 dx) ≤ R + ( Fh (x) -F (x)) 2 dx + 2E(Y 2 1 ) K 2 nh + 2E(Y 1 + 1) K 2 1 n .
By considering Nikols'ki classes of functions (see [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF]) instead of Hölder classes, we may evaluate the bias order and deduce rates of convergence. As the regularity is not known, we rather propose a bandwidth selection strategy inspired from Goldenshluger and Lepski (2011), which yields a nonasymptotic risk bound result. To that aim, let fh,h (x) = K h fh (x), and Fh,h = K h Fh (x).

Note that, as the kernel is even, fh,h = fh ,h and Fh,h = Fh ,h . Let H n be a finite set of bandwidths. Then set

A(h) = sup h ∈Hn fh -fh,h 2 -V (h ) + , B(h) = sup h ∈Hn Fh -Fh,h 2 R + -W (h ) + , with (17)
V (h) = κ 1 K 2 1 K 2 nh + E(Y 2 1 ) K 2 nh 3 , W (h) = κ 2 K 2 1 E(Y 2 1 ) K 2 nh ,
where κ 1 and κ 2 are numerical constants.

For each estimator, the term A(h) (resp. B(h)) approximates the square bias term while V (h) (resp. W (h)) is proportional to the variance term. Therefore, the data-driven bandwidths are defined by: (18) ĥ = arg min

h∈Hn (A(h) + V (h)) , h = arg min h∈Hn (B(h) + W (h)).
The above definitions depend on the unknown moment E(Y 2 1 ), which should be replaced by n -1 n i=1 Y 2 i . This substitution is possible both in theory and in practice. Note that K 1 ≥ 1 = K. The following holds:

Theorem 2.2. Assume that f belongs to L 2 (R), E(X 8 1 ) < +∞ and H n is such that (i) Card(H n ) ≤ n, (ii) ∀a > 0, ∃Σ(a) > 0 such that h∈Hn h -2 exp(-a/h) < Σ(a) < ∞, (iii) ∀h ∈ H n , 1/(nh 3 ) ≤ 1.
Then, under (A2)-(A3), there exists a numerical constant κ 1 in V (h) defined by [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF]such that

(19) E( fĥ -f 2 ) ≤ c inf h∈Hn K 2 1 f -f h 2 + V (h) + c n ,
where c is a numerical constant and c depends on K and f Y . If X is nonnegative, F belongs to L 2 (R + ), H n satisfies (i) and (ii) and K is chosen with compact support [-1, 1], then there exists a numerical constant κ 2 in W (h) defined by [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF] such that,

(20) E( R + ( Fh (x) -F (x)) 2 dx) ≤ c 1 inf h∈Hn K 2 1 R + ( Fh (x) -F (x)) 2 dx + W (h) + c 1 n ,
where c 1 is a numerical constant and c 1 depends on K and f Y .

The proof delivers numerical values for the constants κ 1 , κ 2 which are too large. Finding the minimal values is theoretically difficult. This is why it is standard to calibrate their value by preliminary simulations (see Section 4).

For instance

H n = {1/k, k = 1, . . . , n} or H n = {2 -k , k = 1, . . . , log(n)/ log 2} fulfill (i)-(ii)
. For (iii) the admissible values of k must be restricted to n 1/3 or log(n)/(3 log 2). Actually, (iii) can be replaced by 1/(nh 3 ) ≤ C for a constant C.

Convolution power kernel estimation on R +

Now, we assume that the X i 's are nonnegative. The properties of the kernel estimators of f and the survival function F of the previous section are still valid for x ≥ 0 by setting f (x) = 0 for x < 0. However, for estimating functions on R + , it is often better to use appropriate kernels so as to avoid the boundaries effects near 0. The convolution power kernels (Comte and Genon-Catalot (2012)) are well fitted to deal with this problem. where k m = k • • • k, m times and denotes the convolution product. For h integrable, we denote by h * (t) = e itu h(u)du, t ∈ R its Fourier transform. The Fourier transform of k m is given by

k * m (t) = (k * ( t m )) m , t ∈ R.
For α 1 , . . . , α L real numbers such that L j=1 α j = 1, k (1) , . . . , k (L) densities on R + with expectation 1, we define the convolution power kernel (CPK) by ( 22)

K m = L j=1 α j k (j) m .
The following assumptions are required on the densities k (j) , for j = 1, . . . , L.

(B1) For u ≥ 0, k (j) (u) ≥ 0, for u < 0, k (j) (u) = 0, +∞ 0 k (j) (u)du = 1, +∞ 0
(k (j) ) 2 (u)du < +∞, lim u→+∞ uk (j) (u) = 0, and

+∞ 0 uk (j) (u)du = 1, ∃γ ≥ 4, such that +∞ 0 |u -1| γ k (j) (u)du < +∞ (B2) For m large enough, +∞ 0 k (j) m (u) du u = 1 + O( 1 m ). (B3) There exists m 0 ≥ 1 such that the function t[(k (j) ) * (t)] m 0 belongs to L 1 (R) ∩ L 2 (R).
Note that assumptions (B2) is not stringent as k (j) m (u)du tends to δ 1 as m tends to infinity. Now, we define for x > 0, the estimator of F (x) by:

(23) Fm (x) = 1 x +∞ 0 K m u x FY (u)du + 1 nx n i=1 Y i K m Y i x .
Under (B3), K m is derivable, so we can define, for estimating f (x) at x > 0,

(24) fm (x) = 1 nx n i=1 K m Y i x + Y i x K m Y i x . Proposition 3.1. Under (B1), we have lim x→0 + Fm (x) = 1, lim x→+∞ Fm (x) = 0. Under (B1)-(B3), we have +∞ 0 fm (x)dx = 1 + O( 1
k m (x) = m (m -1)!2 m mx/2 i=0 (-1) i m i (mx -2i) m-1 1 I [0,2] (x)
Here, k is not continuous on (0, +∞) and successive convolutions increase the regularity. Thus, the exponent m plays clearly the role of regularity parameter. Assumptions (B1) and (B3) hold.

Example 2. Gamma kernels. For k the Gamma density G(a, a), a > 0, (B1)-(B3) hold and:

k m (u) = (am) am Γ(am) u am-1 e -mau 1 u>0 . For m > 1/a, +∞ 0 u -1 k m (u)du = (am)/(am -1) = 1 + O(1/m).
Example 3. Inverse Gaussian kernels. The inverse Gaussian distribution IG(a, θ) a > 0, θ > 0, is defined as the distribution of the hitting time T a = inf{t ≥ 0, θt + B t = a} where (B t ) is a standard Brownian motion. The density of an IG(a, θ) is (a/ √ 2πt 3 )e θa e -(1/2)(θ 2 t+a 2 /t) . For a = θ, the expectation is 1 and the variance is v = 1/a 2 . For k the inverse Gaussian density IG(a, a), (B1)-(B3) hold and k m is the density of the law

IG(a √ m, a √ m): k m (u) = a √ m √ 2πu 3 e ma 2 (1-1 2 ( 1 u +u)) 1 u>0 , +∞ 0 u -1 k m (u)du = 1 + 1/(a 2 m).

Pointwise risk.

To evaluate the order of the bias term, we need to define the notion of convolution power kernel of order .

Definition 3.1. We say that K m = L j=1 α j k (j)
m defines a convolution power kernel of order if, for j = 1, . . . , L, the density k (j) satisfies Assumptions (B1)-(B2), admits moments up to order and the coefficients α j , j = 1, . . . , L are such that L j=1 α j = 1 and for 1 ≤ k ≤ and all m (at least large enough)

+∞ 0 (u -1) k K m (u)du = L j=1 α j +∞ 0 (u -1) k k (j) m (u)du = 0.
These relations allow to compute the α j 's as functions of the moments of the k (j) 's (see Comte and Genon-Catalot (2012)). Note that a single convolution power kernel with L = 1 is of order one. Now, we can prove the following result. Proposition 3.2. Let x 0 > 0. Consider the estimator (23) built with a kernel ( 22) satisfying (B1). Set (25)

|α| 1 := L i=1 |α i |, v j = +∞ 0 (u -1) 2 k (j) (u)du, j = 1, . . . L.
If F belongs to Σ I (β, R) for I a neighborhood of x 0 , the kernel K m is order = β in the sense of Definition 3.1 and for j = 1, . . . , L,

+∞ 0 |u-1| β k (j) (u)du < +∞, then for m, n large enough, E[( Fm (x 0 ) -F (x 0 )) 2 ] ≤ C 1 (β) x 2β 0 m β + 4|α| 1 L j=1 |α j | 2πv j √ m n + 2 |α| 2 1 n
where C 1 (β) is a constant depending on R, β, the α j 's and the moments of the k (j) 's. Assume moreover that (B2) and(B3) hold and f is bounded. Then,

E[( fm (x 0 ) -f (x 0 )) 2 ] ≤ C 1 (β) x 2β 0 m β + C 2 f ∞ √ m nx 0 + C 2 ( √ m) 3 nx 3 0 where C 2 = 2 1≤i,j≤L |α i α j |/ 2π(v i + v j ), C 2 = 2 1≤i,j≤L |α i α j |/ 2π(v i + v j ) 3 ) and C 1 (β)
is the same as above.

3.4.

Global risk and adaptation. We prove a global result for Fm .

Proposition 3.3. Assume that (B1) holds and E(X 1 ) < +∞. Then the integrated risk satisfies:

E +∞ 0 Fm (x) -F (x) 2 dx ≤ +∞ 0 E F m (x) -F (x) 2 dx + C 2 10 √ m E(Y 1 ) n where C 2 is defined in Proposition 3.2.
Below, we do not search to link the bias term with the regularity property of the function F . We rather focus on finding a data-driven value of m without knowing the regularity of F on R + . From Propositions 3.2 and 3.3, √ m plays the role of the bandwidth. For two functions s and t on (0, +∞), let us define, each time it exists on (0, +∞), the function

u → s t(u) = +∞ 0 s(u/v)t(v)dv/v. If U 1 , U 2 are nonnegative independent random variables with densities k 1 , k 2 respectively, then the product U 1 U 2 has density k 1 k 2 (u). Now, we define (26) M n = {m = k 2 , log(n) ≤ k ≤ n/ log(n)}
as the set of possible indexes m and consider

K m = L j=1 α j k (j)
m , where the densities k (j) satisfy

(B1). Set (27) Fm,m (x) = 1 x +∞ 0 K m K m u x FY (u)du + 1 nx n i=1 Y i K m K m Y i x .
As K m K m = K m K m , we have Fm,m (x) = Fm ,m (x). For κ a numerical constant and (28)

C(K) = 2|α| 3 1 ( L i=1 |α i |/ √ 2πv i ), we set (29) Z(m) = κC(K)E(Y 1 ) √ m n , H(m) = sup m ∈Mn Fm -Fm,m 2 -Z(m ) + .
Note that, as

|α| 1 ≥ 1, the constant C 2 of Proposition 3.3 satisfies (30) C 2 ≤ 2|α| 1 L j=1 |α j | 2πv j ≤ C(K).
The adaptive estimator is then F m with

(31) m = arg min m∈Mn (H(m) + Z(m)) .
As noted above, we should replace the unknown moment E(Y 1 ) by its empirical counterpart. This is no difficulty in the proofs. We can prove the following result:

Theorem 3.1. Assume that F belongs to L 2 ((0, +∞)). Assume that (B1) holds and E(X 4 1 ) < +∞, then there exists a numerical constant κ such that

E +∞ 0 F m(x) -F (x) 2 dx ≤ C inf m∈Mn +∞ 0 E Fm (x)) -F (x) 2 dx + Z(m) + C n ,
where C is a numerical constant and C a constant depending on E(X 4 1 ) and on K.

Inverse Gaussian kernels (Example 3) are well fitted for practical implementation. Indeed if k is IG(1,1), then k m k m has the following explicit density

(32) k m k m (u) = √ mm πu 3/2 exp(m + m ) K0 m 2 + (m ) 2 + m m (u + 1 u ) 1/2
where K0 is the modified Bessel function of second kind with index 0 (available in R, library Bessel), see Proposition 3.6 in Comte and Genon-Catalot (2012).

Simulations

In this section, we implement our estimators on simulated data. We have selected the following distributions:

Model 1: a Gaussian density, X ∼ N (2. 4.1. Density estimation. We consider the estimator given by [START_REF] Comte | Penalized projection estimator for volatility density[END_REF] for Models 1 to 4, where K(x) is the standard Gaussian kernel. Bandwidths are selected between 0.1 and 1.5 in the set

H n = {0.1 + 0.05 k, k = 0, 1, . . . , 28}.
For each sample, we compute: -first, the oracle for = fhor where h or = arg min h∈Hn fh -f 2 , -second, the Goldenshluger and Lepski estimator fĥ with ĥ given by ( 18) and κ 1 = 1.2, -third, the estimator fh CV where h CV is selected by a cross validation criterion i.e. minimizing

CV (h) = f 2 h (x)dx - 2 n n i=1 [Y i f h,[i] (Y i ) + fh,[i] (Y i )],
where fh,[i] is the kernel estimator fh computed on the sample without Y i (leave-one-out),

-fourth, we compute the estimator of f based on the direct observations X 1 , . . . , X n , f (X)

h CV,X
where f (X)

h
is the standard kernel estimator of f and h CV,X is the bandwidth selected by usual density cross-validation criterion (see e.g. [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF], Section 1.5), -lastly, the oracle based on the direct observations X 1 , • • • , X n , also using the standard kernel estimator.

We investigate two sample sizes n = 200, 500, and 50 repetitions. Table 1 gives the estimated L 2 -risks of all estimators, together with medians and standard deviations except for the oracles. As expected, the oracle on direct observations performs better than the one with censored data. The comparison between the GL method and the oracle on censored data shows that the GL method is stable and that the loss with respect to the oracle is stable. We stress that the GL method gives smaller risks than the CV method, much smaller for means, and still smaller for medians. Indeed, looking at both medians and standard deviations for the CV method, we can see that it is very unstable. This is the reason why we also experimented the CV method on the direct sample, but in this case, it behaves smartlier. We conclude that the estimator fĥ proposed in our paper works well. As an illustration, in Figure 1, we plot the oracle, the GL estimator, the CV estimator on censored data and the CV estimator on direct data, for Model 3 with n = 200 (Figure 1 (a)-(b)), n = 1000 (Figure 1 (c)). When CV method works, it can be very competitive compared with GL method (see Figure 1 (a)), unfortunately, it sometimes completely fails as shown in Figure 1 (b). We can see on Figure 1 (c) that increasing n improves the estimators.

Survival function estimation.

For survival function estimation, we investigate for Models 3 to 6 two couples of estimators:

-the Goldenshluger and Lepski-type kernel estimator Fh given by ( 5) with h given by ( 18) with κ 2 = 1, and the associated oracle F (GL) hor , the bandwidths are selected among 30 equispaced values between 0.01 and 0.9.

-the convolution power estimator F m given by ( 23) with κ = 0.5 and (31) with the inverse Gaussian kernel IG(1,1) of Example 3 (see also Formula (32), function 'besselK(x, 0)' of the R-package Table 2 gives the L 2 -risks for sample size n = 100 and 50 repetitions. Comparing L 2 -risks of the GL and the CPK estimators, we find that the methods behave similarly and are stable over the four models. The difference between estimators and oracles is less important for survival function estimators (Table 2) than for density estimators (Table 1). For both methods, the loss between estimators and oracles is very small for Models 5, 6. The oracle of the GL method is much better than the estimator itself for Models 3, 4. This is less true for the CPK method.

For n = 500 and 100 repetitions, the L 2 -risks of oracles are comparable (Table 3). In Figure 2, ten estimators of both methods for n = 500 are plotted together with the true function (bold), corresponding to Model 5. The GL method is on the left and the CPK on the right. Both methods yield convincing results and monotonic estimators. Although the CPK method is computationally slower, it provides better estimators, especially near zero (Figure 2). One can prove that the risk of the monotonic version of any estimator on a bounded interval is smaller than the risk of the unmodified estimator. Finally, as the monotonic transformation leads to a step function, curves were smoothed using a method preserving monotonicity (Rfunction 'spline.fun' of the method "monoH.FC" from [START_REF] Fritsch | Monotone piecewise cubic interpolation[END_REF]). Clearly, the monotonic transformation improves the curves. The CPK method is better near zero, and the GL method seems better for large abscissa.

Proofs

We state a property used in proofs:

Lemma 5.1. Let ϕ belong to L 2 (R), E(Y 2 ϕ 2 (Y )) ≤ E|X| ϕ 2 .
5.1. Proof of Equations ( 2)-(4) and of Lemma 5.1. Equality (2) is elementary. For y ≥ 0,

FY (y) = +∞ y f Y (z)dz = +∞ y +∞ z f (x) x dxdz = ( x y dz) f (x) x 1 I y≤x dx = +∞ y (x -y) f (x) x dx = +∞ y f (x)dx -y +∞ y f (x) x dx = F (y) -yf Y (y).
For y ≤ 0,

F Y (y) = y -∞ f Y (z)dz = y -∞ dz z -∞ f (x) |x| dx = ( y x dz) f (x) |x| 1 I x≤y dx = y -∞ (y -x) f (x) |x| dx = yf Y (y) + F (y)
Thus, FY (y) = F (y) -yf Y (y), which is (3). For (4), by ( 2), yf Y (y) tends to 0 as both y tends to +∞ and -∞.

By (3), EY 2 (t (Y )) 2 is finite. Integrating by parts yields R f Y (y)(t(y) + yt (y))dy = - R yt(y)(f Y (y)) dy = -[ +∞ 0 yt(y)(- f (y) y )dy + 0 -∞ yt(y) f (y) |y| dy] = +∞ -∞ t(y)f (y)dy. Lemma 5.1 is immediate noting that EY 2 ϕ 2 (Y ) ≤ EX 2 ϕ 2 (U X). 2
5.2. Proof of proposition 2.1. For (i), we use K (u)du = 0 as K is integrable and K is even, and K(u)du = 1. For (ii) and (iii), we write K h (u-x)1 I Y i ≥u du = K(z)1 I z≤(Y i -x)/h dz for the first term and use that lim u→+∞ K(u) = 0 for the second term. Lastly (iv) is straightforward. 2 5.3. Proof of Proposition 2.2. First we study fh . Noting that

(yK( y -x h )) y = K( y -x h ) + y h K( y -x h ),
Equation ( 4) yields E( fh (x)) = f h (x) = f K h (x). Thus, for all x,

E[( fh (x) -f (x)) 2 ] = (f (x) -f h (x)) 2 + Var fh (x) .
As K is of order = β , the assumption on f gives, at point x 0 ,(f

(x 0 ) -f h (x 0 )) 2 ≤ C 2 2 h 2β with C 2 = R |u| β |K(u)|du/ ! (see Tsybakov (2009) Proposition 2.1). Next, we have Var( fh (x 0 )) ≤ 1 nh 2 E Y 1 h K Y 1 -x 0 h + K Y 1 -x 0 h 2 = 1 nh 2 E Y 2 1 h 2 K Y 1 -x 0 h 2 + E K 2 X 1 -x 0 h (33)
where the last equality follows from Equation (4). Then (34)

E K 2 X 1 -x 0 h ≤ min(h f ∞ K 2 , K 2 ∞ ). By Lemma 5.1 , E Y 2 1 K Y 1 -x 0 h 2 ≤ E(|X 1 |) (K ( u -x h )) 2 du = hE(|X 1 |) (K ) 2 .
Therefore, Var( fh (x))

≤ (nh 3 ) -1 E(|X 1 |) (K ) 2 + (nh 2 ) -1 K 2 ∞ .
This yields [START_REF] Van Es | Nonparametric volatility density estimation for discrete time models[END_REF].

The special value x 0 = 0 leads to other bounds. As,∀z

∈ R, |zf Y (z)| ≤ 1, E Y 2 1 K Y 1 -x 0 h 2 = (x 0 + uh) 2 [K (u)] 2 f Y (x 0 + uh)hdu ≤ |x 0 + uh|[K (u)] 2 hdu.
Therefore, for x 0 = 0, we have

(35) E Y 2 1 K Y 1 h 2 ≤ h 2 |u|[K (u)] 2 du.
Consequently,

Var( fh (0)) ≤ K ∞ + |u|[K (u)] 2 du nh 2 := C 6 nh 2 . If now f Y is bounded and u 2 [K (u)] 2 du < +∞, we get for x 0 = 0, (36) E Y 2 1 K Y 1 h 2 ≤ h 3 f Y ∞ u 2 [K (u)] 2 du.
Thus if moreover f ∞ < +∞, using (34),

Var( fh (0)) ≤ f ∞ K 2 + f Y ∞ u 2 [K (u)] 2 du nh := C 6 nh .
This gives inequalities ( 11) and ( 12). Now we study Fh to prove [START_REF] Comte | Convolution power kernels for density estimation[END_REF]. First we have E( Fh (x)) = F K h (x) so that the bias term can be studied using Proposition 2.1 in [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF]. Hence the bias order at x 0 . Next

Var( Fh (x)) ≤ 2 nh 2 E K u -x h 1 I Y 1 ≥u du 2 + E Y 2 1 K 2 Y 1 -x h . ( 37
)
We have

E K u -x h 1 I Y 1 ≥u du 2 ≤ K u -x h du 2 = h 2 |K(v)|dv 2 and E Y 2 1 K 2 Y 1 -x h ≤ hE|X 1 | K 2 .
Gathering terms gives [START_REF] Comte | Convolution power kernels for density estimation[END_REF]. Lastly, if x 0 = 0, inequality (35) applies with K replaced by K and gives the result (9) and thus ends the proof of Proposition 2.2. 2 5.4. Proof of Theorem 2.1. Proof of [START_REF] Hirsch | Elements of functional analysis[END_REF]. To obtain lower bounds, we follow the reduction scheme described in Tsybakov (2009), chapter 2. We have to find two densities

f 0,n , f 1,n such that (i) f j,n ∈ Σ I (β, R), j = 0, 1, (ii) (f 1,n (x 0 ) -f 0,n (x 0 )) 2 ≥ cγ 2 n
where γ 2 n is the desired rate, (iii) χ 2 = χ 2 (P f 1,n ,Y , P f 0,n ,Y ) ≤ c/n, where P f,Y is the law of Y when X has density f . We only prove the result for x 0 ∈ (0, 1) = I. Let h n be small enough to have [x 0 -h n , x 0 + h n ] (0, 1). We take f 0,n (x) = 1 I [0,1] (x) and

f 1,n (x) = f 0,n (x) + cγ n L( x -x 0 h n ) where L(v) = L(v)1 I [-1,1] (v), L ∈ Σ R (β, R), L(0) = 0 and 1 -1 L(v)dv = 0.
We set γ n = n -β/(2β+3) and h n = n -1/(2β+3) . We have f 1,n = f 0,n = 1 and we can choose c such that f 1,n (x) ≥ 0, ∀x ∈ [0, 1], so that f 1,n and f 0,n are [0, 1]-supported densities.

(i) The functions f j,n , j = 0, 1 are in Σ I (β, R) with I = (0, 1) as

γ n /h β n = 1. (ii) (f 1,n -f 0,n ) 2 (x 0 ) = c 2 γ 2 n L 2 (0) is of order n -2β/(2β+3) = γ 2
n , the expected rate.

(iii) Then we must prove that

χ 2 = 1 0 (g 1 -g 0 ) 2 (x) g 0 (x) dx ≤ c/n where g i (x) = 1
x (f i,n (u)/u)du, for i = 0, 1. We have

χ 2 = c 2 γ 2 n 1 0 x 1 L( u-x 0 hn ) u 1 I [x 0 -hn,x 0 +hn] du 2 | log(x)| dx := c 2 γ 2 n (I 1 + I 2 ),
with

I 1 = x 0 -hn 0 x 0 +hn x 0 -hn L( u-x 0 hn ) u du 2 | log(x)| dx, I 2 =
x 0 +hn x 0 -hn

x 0 +hn x L( u-x 0 hn ) u du 2 | log(x)| dx.
Using that 1 -1 L(v)dv = 0, we write

I 1 = x 0 -hn 0 1 -1 L(v) x 0 +vhn h n dv 2 | log(x)| dx = h 2 n x 0 -hn 0 1 -1 L(v) x 0 1 1+vhn/x 0 -1 dv 2 | log(x)| dx
and thus we get

I 1 ≤ h 4 x 2 0 | log(x 0 )| x 0 -h 0 1 -1 L(v) v/x 0 1 + vh/x 0 dv 2 dx ≤ h 4 x 2 0 (x 0 -h) 2 | log(x 0 )| x 0 -h 0 1 -1 |L(v)|dv 2 dx = 1 -1 |L(v)|dv 2 x 2 0 (x 0 -h)| log(x 0 )| h 4 .
Next

I 2 = h 2 x 0 +h x 0 -h 1 | log(x)| 1 (x-x 0 )/h L(v) x 0 + vh dv 2 dx ≤ 2 1 -1 |L(v)|dv 2 (x 0 -h) 2 | log(x 0 + h)| h 3 . Therefore χ 2 ≤ c(x 0 )γ 2 n h 3 = c(x 0 )/n which is the desired result. 2
Proof of ( 14). We seek a rate τ 2 n = n -2β/(2β+1) . We build S 0,n (x) = (1 -x) for x ∈ [0, 1], the survival function associated to f 0,n and

S 1,n (x) = S 0,n (x) + cτ n L x -x 0 h n for x ∈ [0, 1], with L = -L, L(x) = 1
x L(v)dv and L as above and L(0) = 0. We take here τ n = n -β/(2β+1) and h n = n -1/(2β+1) . Note that S 1,n is the survival function associated to f1,n (x) = f 0,n (x) + c(τ n /h n )L((x -x 0 )/h n ). Indeed τ n /h n = n -(β-1)/(2β+1) is O(1) for β ≥ 1 so that c can be chosen small enough to have f1,n ≥ 0.

(i) The functions S 0,n and S 1,n are survival functions belonging to Σ I (β, R) with I = (0, 1) as

τ n /h β n = 1. (ii) (S 0,n (x 0 ) -S 1,n (x 0 )) 2 = c 2 τ 2 n L 2 (0).
(iii) For the χ 2 distance between the observations laws, it follows the same lines as previously and yields an order (τ

2 n /h 2 n ) × h 3 n , i.e. τ 2 n h n = n -2β/(2β+1) × n -1/(2β+1)
= n -1 . 5.5. Proof of Proposition 2.3. The integrated mean-square risk is decomposed as the integrate of the squared bias plus the integrate of the variance. We integrate equation ( 33) and easily obtain bound [START_REF] Mnatsakanov | Hausdorff moment problem: reconstruction of distributions[END_REF]. Now we turn to [START_REF] Rényi | Probability Theory[END_REF]. We start from (37) and get

R + Var( Fh (x)dx ≤ 2 n R + E (K h (u -x) 1 I Y 1 ≥u du 2 + 2 nh K 2 E Y 2 1 .

Now we write

R + E (K h (u -x) 1 I Y 1 ≥u du 2 dx = E R + (K h (u -x) 1 I Y 1 ≥u 1 I u≥-1 du 2 dx
by interchanging expectation and integral and using that as

K has support [-1, 1], u ∈ [x - h, x + h] ⊂ [-1, +∞) for x ≥ 0 and h ≤ 1. Therefore R + E (K h (u -x) 1 I Y 1 ≥u du 2 dx = E K h g Y 1 2
where g Y 1 (u) = 1 I Y 1 ≥u 1 I u≥-1 . Applying the Young Inequality (55) for p = 1, r = q = 2, yields

K h g Y 1 2 ≤ K h 2 1 g Y 1 2 = K 2 1 (Y 1 + 1). This implies R + Var( Fh (x)dx ≤ 2 n K 2 1 E(Y 1 + 1) + 2 nh K 2 E Y 2 1 ,
and thus Inequality (16). 2 5.6. Proof of Theorem 2.2. We start by proving [START_REF] Vardi | Large sample study of empirical distributions in a random-multiplicative censoring model[END_REF]. By using the definitions of A(h), V (h) and ĥ, we note that

∀h, h ∈ H n , fh,h -f h 2 ≤ A(h) + V (h ),
and ∀h ∈ H n , A( ĥ) + V ( ĥ) ≤ A(h) + V (h). Therefore, for all h ∈ H n , fĥ -f 2 ≤ 3 fĥfh, ĥ 2 + 3 fh, ĥ -fh

2 + 3 fh -f 2 ≤ 3(A(h) + V ( ĥ)) + 3(A( ĥ) + V (h)) + 3 fh -f 2 ≤ 6A(h) + 6V (h) + 3 fh -f 2 .
The term E( fh -f 2 ) is ruled by Inequality [START_REF] Mnatsakanov | Hausdorff moment problem: reconstruction of distributions[END_REF] and we only need to study E(A(h)). Recall that fh,h = K h * fh , and denote

f h (x) = E[ fh (x)], f h,h (x) = E[ fh,h (x)]. We split fh := f (1) h + f (2) h , f h := f (1) h + f (2) h with f (1) h (x) = 1 nh n i=1 [Y i K h (Y i -x) + K h (Y i -x)]1 I |Y i |≤cn , f (1) h (x) = E[ f (1) h (x)],
and analogously for fh,h and f h,h . Then using the definition of A(h) we get

A(h) ≤ 5 sup h ∈Hn f (1) h -f (1) h 2 -V (h )/10 + + 5 sup h ∈Hn f (1) h,h -f (1) h,h 2 -V (h )/10 + + 5 sup h ∈Hn f (2) h -f (2) h 2 + 5 sup h ∈Hn f (2) h,h -f (2) h,h 2 + 5 sup h ∈Hn f h -f h,h 2 := 5(T 1 + T 2 + T 3 + T 4 + T 5 ).
Using (55) with p = 1, q = r = 2, and K h 1 = K 1 , we obtain

T 5 = f h -f h,h 2 = K h (f -K h f ) 2 ≤ ( K 1 ) 2 f -K h f 2 .
For T 1 , we write

T 1 = sup h ∈Hn f (1) h -f (1) h 2 -V (h )/10 + ≤ h∈Hn f (1) h -f (1) h 2 -V (h)/10 + ,
and note that

(38) f (1) h -f (1) h 2 = sup t∈L 2 (R), t =1 f (1) h -f (1)
h , t 2 = sup t∈B( 1)

f (1) h -f (1) h , t 2 ,
where B(1) denotes a countable dense subset of {t ∈ L 2 (R), t = 1}. Now we introduce the centered empirical process

ν n,h (ψ t ) = f (1) h -f (1)
h , t = 1 n n i=1 [ψ t (Y i ) -E(ψ t (Y i ))], ψ t (y) := y h 2 K y -x h + 1 h K y -x h 1 I |y|≤cn t(x)dx = yK h t(y) + K h t(y) 1 I |y|≤cn .
Therefore,

E[T 1 ] ≤ h∈Hn E sup t∈B(1) ν 2 n,h (ψ t ) -V (h)/10 + .
We bound the above expectation using the Talagrand inequality (see Appendix). To apply it, we compute H, M and v. Clearly, H 2 = V (h)/κ 1 suits. Next, we get sup t∈B( 1)

sup u∈R |ψ t (u)| ≤ √ 2 h sup u∈R c 2 n h 2 (K ) 2 u -x h + K 2 u -x h dx 1/2 ≤ √ 2 h c 2 n h K 2 + h K 2 1/2 ≤ C(K) c n h 3/2 := M.
Lastly, we search for v.

sup t∈B( 1)

Var (ψ t (Y 1 )) ≤ sup t∈B(1) E ψ 2 t (Y 1 ) .
Remark that

ψ 2 t (y) = y 2 (K h t) 2 (y) + y(K h t) 2 (y) 1 I |y|≤cn .
Thus by Equation ( 4),

E(ψ 2 t (Y 1 )) ≤ E Y 2 1 (K h t) 2 (Y 1 ) + E (K h t) 2 (X 1 ) := S 1 + S 2 .
Next, by Lemma 5.1, Young's Inequality (55) and as t = 1, we get

S 1 ≤ E(|X 1 |) K h t 2 ≤ E(|X 1 |) K h 2 1 t 2 = E(|X 1 |) K 2 1 h 2 .
For S 2 , we write, applying twice the Young Inequality for r = +∞, p = q = 2 and p = 1, q = r = 2

S 2 = E[(K h t) 2 (X 1 )] = (K h t) 2 (x)f (x)dx ≤ K h t ∞ K h t f ≤ K h t K 1 t f = K K 1 √ h f . Thus we get v = c(K, f )/h 2 where c(K, f ) = K 2 1 E(|X 1 |)+ K 1 K f . Then, for κ 1 /10 = 3 ( = 1/2), we get E sup t∈B(1) ν 2 n,h (ψ t ) -V (h)/10 + ≤ C 1 n 1 h 2 exp (-C 2 /h) + c 2 n nh 3 exp -C 3 √ n c n .
By the definition of H n , we have 1/(nh

3 ) ≤ 1, h∈Hn h -2 exp(-C 2 /h) < Σ(C 2 ) < ∞, and Card(H n ) ≤ n. So, choosing c n = C 3 √ n/(4 log(n)), we obtain E[T 1 ] ≤ c/n.
The term T 2 is studied analogously, with additional factors K 2 1 due to an additional application of Young's Inequality.

For the terms T 3 , T 4 , rough bounds are used together with the definition of H n , in particular 1/(nh 3 ) ≤ 1 to get

T 3 ≤ C(K)nE(|Y 1 | 2+p /c p n
) for all p > 0, where C(K) is a constant depending on the kernel. Thus, with the definition of c n we obtain an order 1/n by choosing p = 6 with constraint E(|Y 1 | 8 ) < +∞. Hence we get [START_REF] Vardi | Large sample study of empirical distributions in a random-multiplicative censoring model[END_REF]. Now we turn to the proof of (20). The study follows the same line as previously, so we mainly give a sketch of proof. Here we can split in three parts Fh = F (1)

h + F (2) h + F (3) h with F (1) h (x) = 1 nh n i=1 Y i 1 I Y i <cn K Y i -x h , F (2) h (x)
= 1 nh n i=1 Y i 1 I Y i ≥cn K Y i -x h , F (3) h
(x) = K h (u -x) FY (u)du, with F (i) h = E[ F (i) h ] for i = 1, 2, 3
, and analogously for

F (i) h,h , i = 1, 2, 3.
The first two terms are studied as previously T 1 , T 2 , T 3 , T 4 . There is also a term analogous to

T 5 . Let G Y (u) = ( FY (u) -FY (u))1 I -1≤u .
The additional new terms are

T 6 := E sup h ∈Hn +∞ 0 [ F (3) h (x) - F (3) h (x)] 2 dx = E sup h ∈Hn R + [K h G Y (x)] 2 dx
and its twin in h, h . Thus using Inequality (55) as previously, we get

T 6 ≤ E sup h ∈Hn K h 2 1 G Y 2 ≤ K 2 1 E ( FY (u) -FY (u)) 2 1 I -1≤u du = K 2 1 n Var(1 I Y 1 ≥u )1 I u≥-1 )du ≤ K 2 1 E(Y 1 + 1) n .
This ends the proof. 2 5.7. Proof of Proposition 3.1. For sake of simplicity, we assume that L = 1 and k (1) = k. By (B1),

1 x +∞ 0 k m ( u x )1 I Y i ≥u du = Y i /x 0 k m (v)dv ≤ +∞ 0 k m (v)dv = 1,
so that Fm is well defined. Moreover, it is obvious from the formula above that

lim x→+∞ 1 x +∞ 0 k m ( u x ) FY (u)du = 0, lim x→0 + 1 x +∞ 0 k m ( u x ) FY (u)du = 1.
From Young's Inequality (see (55) with r = +∞, p = q = 2) and

(B1), k k ∞ ≤ k 2 so that for all m ≥ 2, k m ∞ < ∞. Consequently, lim x→+∞ 1 x k m ( Y i x ) = 0.
As uk(u) → 0 when u → +∞, by induction we easily prove that uk m (u) → 0 when u → +∞. Therefore,

lim x→0 1 x k m ( Y i x ) = 0.
In summary, we proved that, under (B1), lim x→+∞ Fm (x) = 0 and lim x→0 + Fm (x) = 1. Without loss of generality, we assume that (B3) holds for m 0 = 1 and write that

k(x) = 1 2π R e -itx k * (t)dt, k (x) = -i 2π R e -itx tk * (t)dt.
This implies that k and k are continuous, tend to zero at +∞, and k(0

) = k (0) = 0. As k m (y) = y 0 k(y -z)k m-1 (z)dz for m > 1, we have k m (0) = 0, lim y→+∞ k m (y) = 0 and k m is continuously derivable, with (39) (k m ) * (t) = -itk * m (t).
Using (B2) and

+∞ 0 k m (v)dv = [k m (v)] +∞ 0 = 0 yields +∞ 0 fm (x)dx = +∞ 0 k m (v) dv v + +∞ 0 k m (v)dv = 1 + O( 1 m ).
5.8. Proof of Proposition 3.2.

Lemma 5.2. Let k, k (1) , k (2)

satisfying (B1), then for v, v 1 , v 2 their variances(i.e. v = +∞ 0 (u- 1) 2 k(u)du), we have (40) k m 2 = +∞ 0 k 2 m (u)du = √ m(1/2 √ πv(1 + o(1)), k m ∞ ≤ √ m(1/ √ 2πv(1 + o(1)), (41) k
(1) m , k (2) m = +∞ 0 k (1) m (u)k (2) m (u)du = √ m(1/ 2π(v 1 + v 2 )(1 + o(1)).
Let k, k (1) , k (2) satisfying (B1) and (B3), then

k m 2 = 1 4 √ π m v 3/2 (1 + o(1)), (k (1) m ) , (k (2) m ) = 1 √ 2π m v 1 + v 2 3/2 (1 + o(1)). (42)
Proof of Lemma 5.2. Equalities (40) and (41) are proved in Lemma A.1. of Comte and Genon-Catalot (2012).

Thus we turn to (42). Under (B1) and (B3)

with m 0 = 1, as (k m ) * (t) = -itk * m (t), k m 2 = +∞ 0 (k m (y)) 2 dy = 1 2π R t 2 |k * m (t)| 2 dt = ( √ m) 3 2π R s 2 |k * ( s √ m )| 2m ds.
Under the assumption t 2 |k * (t)| 2 dt < +∞ (see (B3)), we can mimick the proof of Lemma A.1.

(Comte and Genon-Catalot (2012)) to obtain:

1 ( √ m) 3 R t 2 |k * m (t) 2 dt → s 2 e -vs 2 ds = √ π 2v 3/2 .
And for the case of two densities,

(k (1) m ) , (k (2) m ) = +∞ 0 (k (1) m ) (y)(k (2) m ) (y)dy ∼ ( √ m) 3 2π R s 2 e -(v 1 +v 2 )s 2 /2 ds.
Hence (42). 2

Now we turn to the proof of Proposition 3.2. For the bias order, we have

E( Fm (x)) = 1 x +∞ 0 K m ( u x ) FY (u)du + 1 x E(Y 1 K m ( Y 1 x )) = 1 x +∞ 0 K m ( u x ) FY (u) + uf Y (u) du = 1 x +∞ 0 K m ( u x ) F (u)du = +∞ 0 K m (v) F (vx)dv.
The order of this term on Σ I (β, C), is given in Proposition 3.2 in Comte and Genon-Catalot (2012). Now, we bound the variance term.

Var( Fm (x)) = 1 nx 2 Var +∞ 0 K m ( u x )1 I Y 1 ≥u du + Y 1 K m ( Y 1 x ) ≤ 2 nx 2 E[( +∞ 0 K m ( u x )1 I Y 1 ≥u du) 2 ] + E[Y 2 1 K 2 m ( Y 1 x )] := T 1 (x) + T 2 (x) (43) T 1 (x) ≤ 2 nx 2 +∞ 0 |K m ( u x )|du 2 ≤ 2 nx 2   L j=1 |α j | +∞ 0 k (j) m (v)xdv   2 = 2 n |α| 2 1 , T 2 (x) = 2 nx 2 ((uv) 2 K 2 m ( uv x )1 I [0,1] (u)f (v)1 I R + (v)dudv = 2 nx 2 +∞ 0 v 2 f (v) 1 0 u 2 K 2 m ( uv x )du dv = 2 nx 2 +∞ 0 f (v) v x 3 v/x 0 z 2 K 2 m (z)dz dv.
Thus using Lemma 5.2 namely

+∞ 0 zk (j) m (z)dz = 1 and k (j) m ∞ ≤ 2 √ m/ 2πv j ), we get T 2 (x) ≤ 2 n +∞ 0 f (v) v/x 0 zK 2 m (z)dz dv ≤ 2 n +∞ 0 zK 2 m (z)dz ≤ 2 n K m ∞ +∞ 0 z|K m (z)|dz ≤ 2 n L j=1 2|α j | √ m 2πv j L j=1 |α j | zk (j) m (z)dz ≤ 4|α| 1 √ m n L j=1 |α j | 2πv j .
Consequently we have

Var( Fm (x)) ≤ |α| 1 2 n   |α| 1 + 2 √ m L j=1 |α j | 2πv j   .
Next, we study the estimator of f . As (∂/∂y)(yK m ( y x )) = K m ( y x ) + (y/x)K m ( y x ), we have

E fm (x) = 1 x EK m ( X 1 x ) = +∞ 0 K m (v)f (xv)dv.
As for the bias term of Fm (x), the study of the bias term of fm (x) is a direct application of Proposition 3.2 of Comte and Genon-Catalot (2012). For the variance term, we use that

Var fm (x) ≤ 1 nx 2 E K m ( Y 1 x ) + Y 1 x K m ( Y 1 x ) 2 = 1 nx 2 EK 2 m ( X 1 x ) + E Y 1 x K m ( Y 1 x ) 2
We have

EK 2 m ( X 1 x ) = x +∞ 0 K 2 m (v)f (xv)dv ≤ x f ∞ K m 2
where, by Lemma 5.2, the L 2 -norm of K m satisfies, using (30), (44)

K m 2 ≤ √ m 1≤i,j≤L 2 |α i α j | 2π(v i + v j ) = C 2 √ m ≤ C(K) √ m.
For the other term, we have

E Y 1 x K m ( Y i x ) 2 = v≥0,0≤u≤1 (uv) 2 x 2 (K m ( uv x )) 2 f (v)dudv = x +∞ 0 f (v) v dv v x 0 t 2 (K m (t)) 2 dt ≤ 1 x E(X 1 ) +∞ 0 (K m (t)) 2 dt.
Now, we use (42) of Lemma 5.2

K m 2 ≤ m 3/2 √ 2π 1≤i,j≤L 2|α i α j | (v i + v j ) 3/2 .
The result follows. This ends the proof of Proposition 3.2. 2 5.9. Proof of Proposition 3.3. Inequality (43) for Var( Fm (x)) must be integrated over R + . The second term is the easiest:

(45) +∞ 0 T 2 (x) dx ≤ 2 n E(Y 2 1 +∞ 0 K 2 m ( Y 1 x ) dx x 2 ) = 2 n E(Y 1 ) K m 2 .
For the term T 1 (x), we apply the generalized Minkowski inequality (see (54) in appendix):

+∞ 0 T 1 (x) dx ≤ 2 n E 1 I x≥0 dx x 2 K m ( u x )1 I Y 1 ≥u 1 I u≥0 du 2 ≤ 2 n E 1 I Y 1 ≥u 1 I u≥0 du K 2 m ( u x )1 I x≥0 dx x 2 1/2 2 ≤ 2 n E 1 I Y 1 ≥u≥0 1 √ u du K 2 m (v)1 I v≥0 dv 1/2 2 = 2 n E(2 Y 1 ) 2 ) +∞ 0 K 2 m (v)dv = 8 n EY 1 K m 2 . (46) Finally, +∞ 0 Var( Fm (x))dx ≤ 10 n EY 1 K m 2
which is the announced result using (44). We also state a result with useful bounds concerning the convolution power kernels.

Lemma 5.4. Recall notations (25). Under assumptions (B1)-(B2), we have

(o) ||K m K m || ∞ ≤ |α| 1 L j=1 |α j | √ 2πv j √ m ∧ m (1 + o(1)). (i) K m K m 2 ≤ C(K) √ m ∧ m where C(K) is defined by (28). (ii) +∞ 0 (|K m (z)|/ √ z)dz ≤ 3|α| 1 . (iii) +∞ 0 (|K m K m (z)|/ √ z)dz ≤ 3|α| 2 1 .
Proof of Lemma 5.4. For (o), see Lemma A4 of Comte and Genon-Catalot (2012). For (i), we write

(48) (K m K m ) 2 (u)du ≤ K m K m ∞ +∞ 0 |K m K m |(u)du. Now we know from (o) that K m K m ∞ ≤ 2|α| 1 L j=1 |α j | √ 2πv j √ m ∧ m . Moreover +∞ 0 |K m K m |(u)du ≤ +∞ 0 +∞ 0 |K m ( u v )||K m (v)| dv v du = +∞ 0 |K m (v)|dv +∞ 0 |K m (z)|dz ≤ |α| 2 1 .
Plugging these two bounds in (48) gives the first result. For (ii), we simply split the integral

+∞ 0 |K m (z)| √ z dz = 1 0 |K m (z)| √ z dz + +∞ 1 |K m (z)| √ z dz ≤ 1 0 |K m (z)| z dz + +∞ 1 |K m (z)|dz. By (B2), +∞ 0 z -1 |K m (z)|dz ≤ 2|α| 1 and +∞ 0 |K m (z)|dz ≤ |α| 1 .
This ends the proof of (ii). For (iii), we write

+∞ 0 K m K m (s) √ s ds = +∞ 0 +∞ 0 K m ( u s ) ds s 1/2 K m (u) du u
and this term is simply equal to

+∞ 0 K m (u)/ √ udu K m (v)
/vdv, so that the result (iii) follows by applying (ii). Hence Lemma 5.4. 2

Recall that C 2 ≤ C(K). Thus, a straightforward consequence of Lemma 5.4 (i) and the bounds in Proposition 3.3 is the following Lemma. where the constant C(K) (see (28)) does not depend on the density f . 5.10.2. Proof of Theorem 3.1. First note that the definition of m implies that H( m) + Z( m) ≤ H(m) + Z(m) for all m ∈ M n . From now on, we extend all functions by setting them equal to 0 on (-∞, 0) so that . is the L 2 -norm on R + . Hence, for m any element of M n , we can write the decomposition

F m -F 2 ≤ 3( F m -Fm, m 2 + Fm, m -Fm 2 + Fm -F 2 ) ≤ 3(H(m) + Z( m)) + 3(H( m) + Z(m)) + 3 Fm -F 2 ≤ 6(H(m) + Z(m)) + 3 Fm -F 2 . Therefore, E( F m -F 2 ) ≤ 3E( Fm -F 2 ) + 6Z(m) + 6E(H(m)). Let us study H(m) (see (29)). Let E( Fm (x)) = Fm (x) and E( Fm,m (x)) = Fm,m (x). Then Fm -Fm,m 2 ≤ 3 Fm -Fm 2 + 3 Fm,m -Fm,m 2 + 3 Fm -Fm,m 2 . By Lemma 5.3, for all m, m ∈ M n , Fm -Fm,m 2 = +∞ 0 +∞ 0 B m F (xu)K m (u)du 2 dx.
Therefore, using that each k

(i)
m is a density, we obtain:

Fm -Fm,m 2 ≤ |α| 1 +∞ 0 L i=1 |α i | +∞ 0 (B m F )(xu)k (i) m (u)du 2 dx ≤ |α| 1 L i=1 |α i | (B m F ) 2 (xu)k (i) m (u)dudx ≤ |α| 1 +∞ 0 (B m F ) 2 (v)dv L i=1 |α i | +∞ 0 k (i) m (u) u du,
having used Fubini and the change of variable v = xu. Now, k

i) m (u)/u du = 1 + O(1/m ) ≤ 2. Therefore H(m) ≤ 3 sup m Fm -Fm 2 - Z(m ) 6 + + 3 sup m Fm,m -Fm,m 2 - Z(m ) 6 + +3 sup m Fm -Fm,m 2 ≤ 3 sup m Fm -Fm 2 - Z(m ) 6 + + 3 sup m Fm,m -Fm,m 2 - Z(m ) 6 + +6|α| 2 1 +∞ 0 (B m F ) 2 (v)dv. (
Now, we can prove the following Lemmas:

Lemma 5.6. Under the assumptions of Theorem 3.1, we have

E sup m Fm -Fm 2 - Z(m ) 6 + ≤ C n .
Lemma 5.7. Under the assumptions of Theorem 3.1, we have

E sup m Fm,m -Fm,m 2 - Z(m ) 6 + ≤ C n .
This yields that, ∀m ∈ M n ,

E( F m -F 2 ) ≤ 3E( Fm -F 2 ) + 6Z(m) + 6|α| 2 1 (B m F ) 2 (v)dv + 6C n . As E( Fm -F 2 ) ≤ C(Z(m) + (B m F ) 2 (v)dv),
E sup m Fm -Fm 2 - Z(m ) 6 + ≤ m∈Mn E Fm -Fm 2 - Z(m) (49)
+ .

Next, we split the estimator and its expectation in two parts,

Fm -Fm = ( F (1) m -F (1) m ) + ( F (2) m -F (2) m ) where F (1) m (x) = 1 n n i=1 1 x +∞ 0 K m u x 1 I Y i ≥u 1 I Y i ≤cn (u)du + 1 nx Y i 1 I Y i ≤cn K m Y i x , F (2) m = Fm - F (1) m , F (k) m = E( F (k) m ) for k = 1, 2 and for a a numerical constant (50) c n = nE(Y 1 )
a log 2 (n) .

We get

E Fm -Fm 2 - Z(m) 6 + ≤ 2E F (1) m -F (1) m 2 - Z(m) 12 + + 2E F (2) m -F (2) m 2
and from the variance bound, for

√ m ≤ n, E F (2) m -F (2) m 2 ≤ CE(Y 1 1 I Y 1 >cn ) √ m n ≤ C E(Y p+1 1 ) c p n .
With c n given by (50), p = 3 and cardM n ≤ n/ log(n), we get

m∈Mn E F (2) m -F (2) m 2 ≤ Ca 3 E(Y 4 1 ) (EY 1 ) 3 log 5 (n) n 2 ≤ C n ,
provided that E(Y 4 1 ) < +∞, which makes this term negligible. Next, we note that

F (1) m - F (1) m 2 = sup t, t =1 F (1) m - F (1
) m , t 2 , and the supremum can be taken over a dense countable family of functions t such that t = 1; we denote by B(1) this set.

Thus, setting

θ t (y) = +∞ 0 1 x +∞ 0 K m ( u x )(1 I y≥u 1 I y≤cn du + yK m ( y x )1 I y≤cn t(x)dx := θ (1)
t (y) + θ

with obvious splitting into two terms, we introduce the centered empirical process

(51) ν n (θ t ) = F (1) m -F (1) m , t = 1 n n i=1 [θ t (Y i ) -Eθ t (Y i )] .
We can apply the Talagrand inequality (see Appendix). For this, we search for H, v, M such that:

E( sup t∈B(1) ν 2 n (θ t )) ≤ H 2 , sup t∈B (1)
Var (θ t (Y 1 )) ≤ v and sup

ν 2 n (θ t )) ≤ E( Fm -Fm 2 ) ≤ 10C(K)E(Y 1 ) √ m/n := H 2
where C(K) is defined in (28). Next, for t = 1, we have by ( 44) Thus, we can take v = 3|α| 1 C(K) is the analogous of θ t with K m K m instead of K m . We have to find the three quantities H, v, M. This reduces to using Lemma 5.5 for H and inequalities given in (i) and (iii) of Lemma 5.4. The bounds being the same as for Lemma 5.6, the conclusion is also analogous. 2 6. Appendix 6.1. Auxiliary result. We recall the generalized Minkowski inequality. The proof of the following inequality can be found in e.g. Tsybakov (2004, p. 161). For all Borel function g on R × R, we have The Young inequality. (see [START_REF] Hirsch | Elements of functional analysis[END_REF]). Let f be a function belonging to L p (R) and g belonging to L q (R), let p, q, r be real numbers in [1, +∞] and such that 1

θ (1) t (y) = y 0 +∞ 0 1 x K m ( u x )t(x)dx du1 I y≤cn ≤ cn 0 +∞ 0 1 x 2 K 2 m ( u x )dx 1/2 du = cn 0 1 √ u +∞ 0 K 2 m (v)dv 1/2 du = 2 K m √ c n ≤ 2 C(K)m 1/4 √ c n ,
p + 1 q = 1 r + 1. Then (55) f g r ≤ f p g q .
where f g is the convolution product and f p p = |f (x)| p dx. In particular, for p = 1, r = q = 2, we have f g 2 ≤ f 1 g 2. By standard density arguments, this result can be extended to the case where F is a unit ball of a linear normed space, after checking that f → νn(f ) is continuous and F contains a countable dense family.

3. 1 .

 1 Definition of convolution power kernel estimators. For k a density on R + with expectation 1, we denote by k m the density of (E1 + • • • + E m )/m with E i i.i.d. with density k, i.e.

  k m (u) = mk m (mu), u ≥ 0

Figure 1 .

 1 Figure 1. True density (solid black), oracle for (green) and estimators fĥ (blue dash-dotted), fh CV (red dashed), f (X) h CV,X (magenta long-dashed). n = 200 in (a) and (b), n = 1000 in (c).

Figure 2 .

 2 Figure 2. True survival function (solid black) and 10 estimators in dotted blue, Fh left (GL method), and F m right (CPK method), for Model 5 and n = 500.

Figure 3 concerns

 3 Model 6, with still 10 estimators and n = 500. On top left and right, the GL and CPK estimators. As they are not always monotonic, we have used (bottom left and right) the monotonic transformation of estimators defined by (see[START_REF] Chernozhukov | Improving point and interval estimators of monotone functions by rearrangement[END_REF], R-package 'quantreg'): Ḡ → Ǧ(y) = inf{z; 1 Ḡ(u)≥z du ≤ y}.

Figure 3 .

 3 Figure 3. True survival function (solid black) and 10 estimators in dotted blue for Model 6 and n = 500. Top left: Fh (GL method). Top right F m (CPK method). Bottom left: GL with monotonic transformation and smoothing. Bottom right: CPK with monotonic transformation and smoothing.

2 5. 10 . 5 . 10 . 1 .Lemma 5 . 3 .

 210510153 Proof of Theorem 3.1. Some preliminary Lemmas. Let us set, for m, m > 0, (47) B m F (x) = E F m (x) -F (x), B m,m F (x) = E Fm,m (x) -F (x). Similarly to Lemma A.3 of Comte and Genon-Catalot (2012), the following relation between bias terms holds. We have B m F (x) = +∞ 0 K m (u) F (xu)du -F (x) and B m,m F (x) = B m F (x) + +∞ 0 K m (u)B m F (xu)du.

Lemma 5 . 5 .

 55 Under assumptions (B1)-(B2), we have +∞ 0 Var Fm,m (x) dx ≤ 10 E(Y 1 )

2 5. 10 . 3 .

 2103 the proof of Theorem 3.1 is complete. Proof of Lemma 5.6. First we write,

0 (

 0 K m (u)) 2 ydu 1/2 ≤ √ c n K m ≤ C(K)m 1/4 √ c nHence, we can take M = 3 C(K) √ c n m 1/4

3 fu 2 x 2 x 2 y 3 f 2 Y

 32222 Y (y)(y/x)K m (y/x)t(x)(y/z)K m (y/z)t(z)dxdydz.First, (y2 /xz)f Y (y)K m (y/x)K m (y/z)dy = (x 2 /z) u 2 K m (u)K m (xu/z)f Y (xu)du. Hence, |K m (u)| +∞ 0 |t(x)|f Y (xu)( +∞ 0 |t(z)K m (xu/z)| dz z )dx du.Next, with v = xu/z, we get+∞ 0 |t(z)K m (xu/z)| dz z ≤ +∞ 0 (K m (xu/z)(1/z)) 2 dz |t(x)|f Y (xu) 1 √ x dx C(K) 1/2 m 1/4 .Then, as yf Y (y) ≤ 1, we get +∞ 0 (y)dy ≤ E(

The 2 - 2 - 1 ,

 221 Talagrand inequality. The result below follows from the Talagrand concentration inequality given in Klein and Rio (2005) and arguments in Birgé and Massart (1998) (see the proof of their Corollary 2 page 354). Lemma 6.1. (Talagrand Inequality) Let Y1, . . . , Yn be independent random variables, let νn,Y(f ) = (1/n) n i=1 [f (Yi)-E(f (Yi))] and let F be a countable class of uniformly bounded measurable functions. Then for 2 > 0E sup f ∈F |νn,Y (f )| 2 -2(1 + 2 2 )H 2 + K1n 2 C 2 ( 2 )e K1 = 1/6, andsup f ∈F f ∞ ≤ M, E sup f ∈F |νn,Y (f )| ≤ H, sup (Y k )) ≤ v.

  Proposition 2.2. Assume that E(|X 1 |) < +∞. Let x 0 ∈ R. Assume that f belongs to Σ I (β, R) for I a neighborhood of x 0 . If the kernel K is of order + 1 with = β and |u| β+1 |K(u)|du < +∞, then under (A1),

Table 1 .

 1 Table of risks for density estimators and oracles.

	5, 0.75),

Model 3: a Gamma distribution, X ∼ Γ(8, 4), Model 4: a rescaled Beta distribution, X = 5X , X ∼ β(3, 3), Model 5: an Exponential distribution X ∼ exp(2), Model 6: a mixture of Gamma distributions X ∼ 0.4Γ(1, 10) + 0.6Γ(40, 30).

Table 2 .

 2 Table of risks, n = 100 for survival function estimators and oracles.

			Model 3 Model 4 Model 5 Model 6
		Mean	0.025	0.043	0.006	0.013
	GL	(std)	(0.028)	(0.038)	(0.004)	(0.012)
		Med.	0.015	0.027	0.005	0.012
		Mean	0.010	0.014	0.004	0.009
	Oracle GL	(std)	(0.009)	(0.012)	(0.009)	(0.007)
		Med.	0.007	0.012	0.004	0.007
		Mean	0.021	0.034	0.005	0.017
	CPK	(std)	(0.015)	(0.023)	(0.005)	(0.009)
		Med.	0.018	0.028	0.004	0.014
		Mean	0.013	0.021	0.004	0.011
	Oracle CPK	(std)	(0.011)	(0.016)	(0.003)	(0.008)
		Med.	0.009	0.016	0.003	0.009

Table 3 .

 3 Table of risks (n = 500)Bessel), and its associated oracle Fmor . The values of m are chosen among {5+3k, k = 0, . . . , 10}. This is not exactly consistent with the theoretical constraint but computationally more tractable, with good results.

			Model 3 Model 4 Model 5 Model 6
		Mean	0.004	0.005	0.001	0.003
	Oracle GL	(std)	(0.003)	(0.004)	(0.001)	(0.002)
		Med.	0.003	0.004	0.001	0.002
		Mean	0.005	0.008	0.001	0.004
	Oracle CPK	(std)	(0.003)	(0.005)	(0.001)	(0.002)
		Med.	0.004	0.007	0.001	0.004

  . Now, we studyK m (s)1 I y≥sx t(x)K m (w)1 I y≥wz t(z)f Y (y)dsdwdxdydz (Y 1 )) is bounded by a constant independent of m, n. Next we consider Var(θ

	Var(θ	(1) t (Y 1 )) ≤ E θ	t (Y 1 ) (1)	2
				≤	[0,+∞[ 5	1 x	K m (	u x	)1 I y≥u t(x)	1 z	K m (	v z	)1 I y≥v t(z)f Y (y)dudvdxdydz
				≤
					[0,+∞[ 5
	Writing that	+∞ 0	t(x)1 I y≥sx dx ≤ t (	+∞ 0	1 I y≥sx dx) 1/2 = y/s, we get
		Var(θ	(1) t (Y 1 )) ≤		[0,+∞[ 3	|K m (s)|	y s	|K m (w)|	y w	f Y (y)dsdwdy
					≤ E(Y 1 )		0	+∞	|K m (s)| √ s	ds	2	= 9|α| 2 1 E(Y 1 ),
	by (ii) of Lemma 5.4. Therefore, Var(θ	(1)
				(2) t (Y 1 )).
	Var(θ

t

  1/2 E(Y 2 1 )m1/4 . Lastly, This yields, choosing 2 = 1/2, using (50) and taking a such that 2K 1 C( 2 ) √ 10 a/(21 √ 2) = 2, and using m ≤ n 2 for any m in M n we get This ends the proof of Lemma 5.6. 2. 5.10.4. Proof of Lemma 5.7. The proof of Lemma (5.7) follows the same line as previously with K m replaced by K m K m , where m is fixed and the sum is now taken over m in M n .The truncation of the Y i 's by c n is done as previously, and the bound given in Lemma 5.5 leads to the same result. Therefore, we can work as if the Y i 's were bounded by c n .Thus, we apply the Talagrand inequality to the empirical process

		E sup t∈B(1) (ν 2 n (t) -4H 2 ) + ≤ C 1	m 1/4 n	e -Bm 1/4 +	1 log 2 n	e -2 log(n) .
	Now, reminding of (49) and Card(M n ) ≤ n, we get
		E sup m	Fm -Fm	2 -	Z(m ) 6	+	≤	C 1 n	m∈Mn	m 1/4 e -Bm 1/4 +	C 2 n	≤	C n
	(53)
	E sup m	Fm,m -Fm,m	2 -	Z(m ) 6	+	≤	m ∈Mn	E	Fm,m -Fm,m	2 -	Z(m ) 6
							ν * n (θ	(m,m ) t
		nH M	=	√	10 a 3	log(n),	nH 2 v	= B m 1/4 , B =	10 C(K)E(Y 1 3|α| 1 E(Y 2 1 )

+ . ) = Fm,m -Fm,m , t 2

where θ (m,m ) t