INTRODUCTION

The undrained condition is defined theoretically as a condition in which there is no change in the fluid mass of the porous material. For performing an undrained test in the laboratory, this condition cannot be achieved just by closing the valves of the drainage system as it is done classically in a conventional triaxial system (Figure ( 1)). In a triaxial cell, the tested sample is connected to the drainage system of the cell and also to the pore pressure transducer. As the drainage system has a non-zero volume filled with water, it experiences volume changes due to its compressibility and its thermal expansion. The variations of the volume of the drainage system and of the fluid filling the drainage system induce a fluid flow into or out of the sample to achieve pressure equilibrium between the sample and the drainage system. This fluid mass exchanged between the sample and the drainage system modifies the measured pore pressure and consequently the measured strains during the test. [START_REF] Wissa | Pore pressure measurement in saturated stiff soils[END_REF] and [START_REF] Bishop | Influence of system compressibility on observed pore pressure response to an undrained change in stress in saturated rock[END_REF] were the first who studied this problem for a mechanical undrained loading and presented a method for correction of the measured pore pressure. [START_REF] Ghabezloo | Comportement thermo-poro-mécanique d'un ciment pétrolier[END_REF]Sulem (2009, 2010) presented an extension to the work of [START_REF] Bishop | Influence of system compressibility on observed pore pressure response to an undrained change in stress in saturated rock[END_REF] to correct the pore pressure and the volumetric strain measured during undrained heating and cooling tests, as well as undrained compression tests, by taking into account the compressibility and the thermal expansion of the drainage system, the inhomogeneous temperature distribution in the drainage system and also the compressibility and the thermal expansion of the fluid filling the drainage system. The correction method depends also on the porosity, the compressibility and the thermal expansion of the tested porous material. The proposed method was applied to the results of isotropic compression and undrained heating tests performed on Rothbach sandstone [START_REF] Ghabezloo | Comportement thermo-poro-mécanique d'un ciment pétrolier[END_REF]Sulem 2009, 2010) and on a hardened cement paste (Ghabezloo et al., 2009). The proposed correction method is presented briefly in this paper.

Experimental artefacts in undrained triaxial testing S. Ghabezloo & J. Sulem Université Paris-Est, UR Navier, ERMES, Ecole des Ponts ParisTech, Marne la Vallée, France ABSTRACT: For evaluation of the undrained thermo-poro-elastic properties of saturated porous materials in conventional triaxial cells, it is important to take into account the effect of the dead volume of the drainage system. The compressibility and the thermal expansion of the drainage system along with the dead volume of the fluid filling this system, influence the measured pore pressure and volumetric strain during undrained thermal or mechanical loading in a triaxial cell. A correction method is presented in this paper to correct these effects during an undrained isotropic compression test or an undrained heating test. A parametric study has demonstrated that the porosity and the drained compressibility of the tested material and the ratio of the volume of the drainage system to the one of the tested sample are the key parameters which influence the most the error induced on the measurements by the drainage system.

Please cite this paper as : [START_REF] Ghabezloo | Effect of the volume of the drainage system on the measurement of undrained thermo-poroelastic parameters[END_REF] Experimental artefacts in undrained triaxial testing, Eurock 2010, Rock Mechanics in Civil and Environmental Engineering: European Rock Mechanics Symposium EUROCK 2010, Lausanne, Switzerland, 15-18 June 2010. We consider a fluid-saturated porous material with a porosity φ . The variations of the total volume V and of the pore volume V φ are given as a function of variations of Terzaghi effective stress d σ , pore pressure f p and the temperature T :

d d s f d dV V c d c dp dT σ α = - - + (1) p d f dV V c d c dp dT φ φ φ φ σ α = - - + (2)
where d c , s c , p c and c φ are four elastic compressibility coefficients, and d α and φ α are two thermal expansion coefficients defined below:

( ) ( ) , , 1 , 1 f f d d p d p T p T c V V c V V φ φ σ σ = - ∂ ∂ = - ∂ ∂ (3) ( ) ( ) , , 1 , 1 d d s f f T T c V V p c V V p φ φ φ σ σ = - ∂ ∂ = - ∂ ∂ (4) ( ) ( ) , , 1 , 1 f d f d d p p V V T V V T φ φ φ σ σ α α = ∂ ∂ = ∂ ∂ (5)
A detailed description of these parameters is presented in [START_REF] Ghabezloo | Poromechanical behaviour of hardened cement paste under isotropic loading[END_REF]Ghabezloo et al. ( ,2009)). In the undrained condition in which the mass of the fluid phase is constant ( 0 f dm = ), we can define four new parameters to describe the response of the porous material in undrained condition:

( ) ( ) , , , 1
f f f u m T m T B p c V V σ σ = ∂ ∂ = - ∂ ∂ (6) ( ) ( ) , , , 1
f f f u m m p T V V T σ σ α Λ = ∂ ∂ = ∂ ∂ (7)
The parameter u c is the undrained bulk compressibility, B is the so-called Skempton coefficient, u α is the undrained volumetric thermal expansion coefficient and Λ is the thermal pressurization coefficient. Writing the fluid mass conservation under undrained condition ( 0 f dm = ) leads to the following expressions for the undrained parameters:

( ) ( ) ( ) d s d s f B c c c c c c φ φ = - -+ - (8) ( ) ( ) ( ) f d s f c c c c φ φ φ α α φ Λ = - -+ - (9)
( )

u d d s c c B c c = - - (10)
( )

u d d s c c α α = + Λ - (11)
where f c and f α are respectively the pore-fluid compressibility and thermal expansion coefficient.

CORRECTION OF THE EFFECT OF DRAINAGE SYSTEM

In a triaxial cell the tested sample is connected to the drainage system and the undrained condition is achieved by closing the valves of this system (Figure ( 1)). Consequently, the condition 0 f dm = is applied to the total volume of the fluid which fills the pore volume of the sample and also the drainage system

( f f L fL m V V φ ρ ρ = +
), where fL ρ is the density of the fluid in the drainage system and L V is its volume. As the sample and the drainage system may have different temperatures, temperature-dependent fluid densities are considered. The variation of volume of the drainage system is written as:

L L L f L L L dV V c dp dT d α κ σ = + - (12)
where L dT is the equivalent temperature change in the drainage system, L c and L κ are isothermal compressibilities and L α is the thermal expansion coefficient of the drainage system defined as:

( ) , 1 L L L L f T c V V p σ = ∂ ∂ (13) ( ) , 1 f L L L L p V V T σ α = ∂ ∂ (14) ( ) , 1 f L L L L p T V V κ σ = - ∂ ∂ (15)
In most triaxial devices, the drainage system can be separated into two parts: one situated inside the triaxial cell and the other one situated outside the cell. In the part inside the cell, the temperature change dT is identical to the one of the sample; in the part situated outside the cell, the temperature change is smaller than dT and varies along the drainage lines. We define an equivalent homogeneous temperature change 12) the following expressions are obtained for the correction of the measured undrained thermo-poro-elastic parameters:

( ) ( )
1 mes cor L fL mes L fL L f d s B B V B c c V c c ρ κ ρ =   + - +   - (17) ( ) ( )
1 mes cor d u u d mes L fL d u L fL L f d s d s c c c c V c c c c V c c c c ρ κ ρ - = -   - + - +   - -   (18) ( ) ( ) ( )
1 mes cor L fL mes fL L fL L f f V c c V φ ρ β α α ρ φ α α Λ Λ =   + - -Λ +   - (19) ( ) (
) ( )

1 mes cor u d u d L fL fL L mes fL L u d d s f f V c c c c V φ α α α α ρ β α α α α ρ φ α α - = + +   + - - -   - -   (20)

CALIBRATION OF THE CORRECTION PARAMETERS

The triaxial cell used in this study can sustain a confining pressure up to 60MPa. The axial and radial strains are measured directly on the sample with two axial transducers and four radial ones of LVDT type. The confining pressure is applied by a servo controlled high pressure generator. The pore pressure is applied by another servo-controlled pressure generator. The heating system consists of a heating belt around the cell which can apply a temperature change with a given rate and regulate the temperature, and a thermocouple which measures the temperature of the sample. More details about this triax-ial cell and a schematic view of the system are presented in [START_REF] Ghabezloo | Comportement thermo-poro-mécanique d'un ciment pétrolier[END_REF].

The drainage system is composed of all the parts of the system which are connected to the pore volume of the sample and filled with the fluid, including pipes, pore pressure transducers, porous stones. The volume of fluid in the drainage system L V , can be measured directly or evaluated by using the geometrical dimensions of the drainage system. For the triaxial cell used in the present study, the volume of the drainage system was measured directly using a pressure/volume controller equal to 3 2300 mm

L V = .
The compressibility of the drainage and pressure measurement systems L c is evaluated by applying a fluid pressure and by measuring the corresponding volume change in the pressure/volume controller. A metallic sample is installed inside the cell to prevent the fluid to go out from the drainage system. Fluid mass conservation is written in the following equation which is used to calculate the compressibility L c of the drainage system:

( )

L L L fL f dV V c c dp = + ( 21
)
where f dp and L dV are respectively the applied pore pressure and the volume change measured by the pressure/volume controller. For a single measurement, the volume change L dV accounts also for the compressibility of the pressure/volume controller and of the lines used to connect the pressure/volume controller to the main drainage system. This effect is corrected by performing a second measurement only on the pressure/volume controller and the connecting lines. The estimated value is ) ( )

mes fL L fL L c c β α α Λ = - + (22)
The physical properties of water fL α and fL c are known as functions of temperature and fluid pressure. As these variations are highly non-linear, the parameters β and L α cannot be evaluated directly but are back analysed from the calibration test results using equation ( 22). The parameters β and L α are back-calculated by minimizing the error between the measurements and the computed results using a least-square algorithm. β is found equal to 0.46 and the thermal expansion coefficient of the drainage system L α is found equal to 1.57×10 -4 (°C) -1 . The evaluation of the compressibility L κ which represents the effect of the confining pressure on the volume of the drainage system is performed using an analytical method. As can be seen in Figure ( 1), only a part of the drainage system which is the pipe connected to the top of the sample, is influenced by the confining pressure. The effect of the confining pressure on the variations of the volume of this pipe can be evaluated using the elastic solution of the radial displacement of a hollow cylinder and the following expression is obtained:

( ) ( ) ( ) 2 2 2 2 2 4 1 L L a b L b a V E κ π ν = - - (23)
where a and b are the inner and the outer radius and L is the length of the drainage pipe. E and ν are the elastic parameters of the pipe material. For the dimensions of the triaxial system used in this study we obtain . This is due to the fact that only a small part of the drainage system, less than 8% of its volume, is influenced by the confining pressure.

PARAMETRIC STUDY

Examples of the application of the proposed correction method on the results of the undrained isotropic compression test and undrained heating tests performed on a granular rock and a hardened cement paste are presented in Ghabezloo and Sulem (2009), Ghabezloo et al. (2009) and [START_REF] Ghabezloo | Effect of the volume of the drainage system on the measurement of undrained thermo-poroelastic parameters[END_REF]. In this section, a parametric study on the error made on the measurement of different undrained thermo-poro-elastic parameters is presented. The error on a measured quantity Q is evaluated as

( ) measured real real Q Q Q -
and takes positive or negative values with indicates if the measurement overestimates or underestimates the considered quantity.

Among the different parameters appearing in equations ( 17) to (20), the porosity φ of the tested material, its drained compressibility d c and the ratio of the volume of the drainage system to the one of the tested sample, L V V are the most influent parameters. For this parametric study the parameters of the drainage system are taken equal to the ones of the triaxial system used in this study. We take also V V . Three different values of the drained compressibility are considered, respectively equal to 0.03GPa -1 , 0.1GPa -1 and 0.5GPa -1 , which covers a range from a rock with a low compressibility to a relatively highly compressible rock. The porosity is varied from 0.05 to 0.35. The ratio L V V is taken equal to 0.025 which corresponds to the conditions of the triaxial system used in this study. We analyze also the effect of a greater volume of the drainage system on the measurement errors by choosing another value twice bigger, equal to 0.05. We can see in Figure (2) that the error on the measurement of B is always negative (the measurement underestimates the real value) and covers an important range between 2% and 50%. The measurement error is more significant for low-porosity rocks with low-compressibility. We can also see the significant effect of the volume of the drainage system on the measurement error. The measurement error for the undrained compressibility u c is presented in Figure (3) where we can observe that it is more important for low-porosity rocks and for greater volume of the drainage system. As opposite to what we observe for B, the error of the measurement of u c is more important when the tested material is more compressible. Figures (4) and ( 5) show the errors corresponding to the measurements of the thermal pressurization coefficient Λ and of the undrained thermal expansion coefficient u α respectively. The error of the measurement for Λ varies between -40% and +10%, which shows that the measured value may be smaller or greater than the real one. As for the isothermal undrained parameters, the error is more important for low-porosity materials and for a greater volume of the drainage system. The error for the undrained thermal expansion coefficient u α varies between -6% and +4%, which is a narrower range, as compared to the other undrained parameters.

CONCLUSIONS

A simple method is presented for analysis of the error induced by the dead volume of the drainage system of a triaxial cell on the measurement of undrained thermo-poro-elastic parameters. A parametric study demonstrated that the porosity φ of the tested material, its drained compressibility and the ratio of the volume of the drainage system to the one of the tested sample, L V V are the key parameters which influence the most the error induced on the measurements by the drainage system. It was also shown that the Skempton coefficient, the thermal pressurization coefficient and the undrained compressibility measurements are much more affected than the measurement of the undrained thermal expansion coefficient.

Figure 1 .

 1 Figure 1. Schematic view of a conventional triaxial cell.

L

  dT such that the volume change of the entire drainage system caused by L dT is equal to the volume change induced by the true non-homogeneous temperature field. The temperature ratio β is thus defined as:

  and L α are evaluated using the results of an undrained heating test performed using a metallic sample with the measurement of the fluid pressure change in the drainage system. For the metallic

(

  values. Figure (2) presents the error on the measurement of the Skempton coefficient as a function of the sample porosity, for three different values of drained compressibility and two different values of the ratio L

Figure 2 .

 2 Figure 2. Error on Skempton coefficient B

Figure 4 .

 4 Figure 4. Error on the thermal pressurization coefficient Λ


Piercesare Secchi
email: piercesare.secchi@polimi.it

Aymeric Stamm
email: aymeric.stamm@irisa.fr

Simone Vantini
email: simone.vantini@polimi.it

Large p Small n: Inference for the Mean

Keywords:

We present a new result that enables inference for the mean vector of a multivariate normal random variable when the number p of its components is far larger than the number n of sample units and the covariance structure is completely unknown. The result turns out to be a useful tool for the inferential analysis (e.i. confidence region and hypothesis testing) of data up to now mostly studied only within an explorative perspective, like functional data. To this purpose, an application to the analysis of brain vascular vessel geometry is developed and shown.

Introduction

The advent and development of high precision data acquisition technologies in active fields of research (e.g., medicine, engineering, climatology, economics), that are able to capture real-time and/or spatially-referenced measures, have provided the scientific community with large amount of data that challenge the classical approach to data analysis.

Modern data sets (large p small n data sets, i.e. data sets characterized by a number of random variables that is much larger than the number of sample units) contrast traditional data sets (small p large n data sets, i.e. data sets characterized by a number of sample units that is much larger than the number of random variables) that drove the evolution of statistics and data analysis during the last century. This makes all classical inferential tools nearly useless in many fields at the forefront of scientific research and raises the demand for new modern inferential tools that suit this new kind of data. The aim of this paper is to provide statistical tools for the inferential analysis of large p small n data sets.

An active area of statistical research moving in this direction is functional data analysis (FDA). Indeed, in FDA each sample unit is represented by means of a function (e.g. [START_REF] Ramsay | Functional Data Analysis[END_REF][START_REF] Ferraty | Nonparametric functional data analysis[END_REF]. Nowadays, the typical inferential approach of FDA is the projection of the n functions under investigation -virtually belonging to an ∞-dimensional functional space -onto a finite p-dimensional functional subspace with p smaller than n. Roughly speaking, the original FDA is replaced by a classical multivariate analysis that is expected to well approximate the former one. Technically, performing this replacement means implicitly assuming that the image of the random function under investigation (i.e. the space which the realizations of the random function under analysis belong to) coincides with a specific finite p-dimensional functional subspace. This paper is a first attempt to provide inferential tools for the analysis of large p small n data (e.g. functional data) in a basis-free framework (for functional data this means that there are no assumptions on the spaces which the mean function and the auto-covariance function belong to).

The work of [START_REF] Srivastava | Multivariate theory for analyzing high dimensional data[END_REF] moves in the same direction. In this work, some inferential results non depending on strong assumptions on the covariance structure are presented. Unfortunately, these results are asymptotic in both p and n (i.e. large p large n data). This makes them non suitable to perform inferential statistical analysis of large p small n data.

For clarity of exposition, in Section 2 we recall a few well known results about inference for the mean of a multivariate normal random variable; in Section 3, our new results about inference for the mean when the number p of random variables is far larger than the number n of sample units are presented, while in Section 4 an application of the previous results to the inferential analysis of the local radius of the internal carotid artery is reported.

Inference for the Mean: State of the Art

The classical approach to inference for the mean µ p of a p-variate normal random variable with unknown covariance matrix Σ p relies on a famous corollary of the Hotelling's Theorem that holds when the number n of sample units is larger than the number p of random variables.

Theorem 1 (Hotelling's Theorem). Assume that (i) X ∼ N p (µ p , Σ p ), (ii) W ∼ Wishart p ( 1 m Σ p , m), (iii) X and W are independent, then for m ≥ p:

m -p + 1 mp (X -µ p ) W -1 (X -µ p ) ∼ F(p, m -p + 1) .
Corollary 1 (Hotelling's Corollary). Assume that (i') {X i } i=1,...,n ∼ iid N p (µ p , Σ p ), then, for n > p:

(n -p)n (n -1)p (X -µ p ) S -1 (X -µ p ) ∼ F(p, n -p) ,
where X and S -1 are the sample mean and the inverse of the sample covariance matrix, respectively.

Corollary 1 makes possible the development of inferential tools for the estimate of the mean value of a p-variate normal random variable (e.g. confidence ellipsoidal regions or hypothesis testing) when the number n of sample units is larger than the number p of random variables; there are no assumptions on the covariance matrix Σ p that is only required to be positively definite. Proofs of the previous results can be found, for instance, in [START_REF] Anderson | An introduction to multivariate statistical analysis[END_REF].

In more and more applications, the number p of random variables is far larger than the number n of sample units and the covariance matrix is unknown. Thus, Corollary 1 cannot be used to make inference for the mean in these cases. In the following section we provide a theorem and a corollary that can be used to make inference for the mean when n is finite, p goes to infinity, and the covariance matrix is unknown.

3 Inference for the Mean of Large p Small n Data Given a real positively semi-definite p × p matrix A with {λ i } i=1,...,p and {e i } i=1,...,p being its eigenvalues and eigenvectors respectively, A + = ∑ i:λ i =0 1 λ i e i e i is called the Moore-Penrose inverse of A (e.g [START_REF] Rao | Generalized inverse of matrices and its applications[END_REF]. Moreover, we indicate with χ 2 1-α (m) the (1α)-quantile of a random variable with distribution χ 2 (m).

Theorem 2 (Generalized Hotelling's Theorem). Assume that:

(i) X ∼ N p (µ p , Σ p ); (ii) W ∼ Wishart p (Σ p , m) ;
(iii) X and W are independent;

then, for m ≥ 1:

lim p→+∞ P (trΣ p ) 2 trΣ 2 p (X -µ p ) W + (X -µ p ) ≤ χ 2 1-α (m) = 1 -α .
Corollary 2 (Generalized Hotelling's Corollary). Assume that:

(i') {X i } i=1,...,n ∼ iid N p (µ p , Σ p );
then, for n ≥ 2:

lim p→+∞ P n(trΣ p ) 2 (n -1) trΣ 2 p (X -µ p ) S + (X -µ p ) ≤ χ 2 1-α (n -1) = 1 -α ,
where X and S + are the sample mean and the Moore-Penrose inverse of the sample covariance matrix, respectively.

The proof of Theorem 2 and of Corollary 2 can be found in [START_REF] Secchi | Large p Small n: Inference for the Mean[END_REF]. This proof is quite technical and -at the moment -it requires some specific regularity assumptions about the asymptotic behavior of Σ p .

Note that Corollary 2 is based on the univariate statistics n(Xµ p ) S + (Xµ p ). We named this statistics Generalized Hotelling's T 2 since it can be proven [START_REF] Secchi | Large p Small n: Inference for the Mean[END_REF] that it generalizes Hotelling's T 2 = n(Xµ p ) S -1 (Xµ p ) that appears in Corollary 1. Indeed, Hotelling's T 2 is defined only for n > p ≥ 1 while the Generalized Hotelling's T 2 is defined for any n and p such that n ≥ 2 and p ≥ 1, and it coincides with the former when n > p. Strong connections with the univariate t statistic are discussed in [START_REF] Secchi | Large p Small n: Inference for the Mean[END_REF].

Corollary 2 turns out to be a useful tool for the construction of confidence regions and hypothesis tests for the mean in all practical situations where the number p of random variables is far larger than the number n of sample units (e.g. genetics) or even virtually infinite (e.g. functional data).

A Confidence Region for the mean µ p can be defined as follows:

CR γ (µ p ) := m p : n(trΣ p ) 2 (n -1) trΣ 2 p (m p -X) S + (m p -X) ≤ χ 2 γ (n -1) , (1)
with γ being the asymptotic confidence level.

Equivalently, an Hypothesis Test for H 0 : µ p = µ 0p versus H 1 : µ p = µ 0p with asymptotic significance level α has the following rejection region:

Reject H 0 in favor of H 1 if: n(trΣ p ) 2 (n -1) trΣ 2 p (X -µ 0p ) S + (X -µ 0p ) > χ 2 1-α (n -1) . (2)
In practice, the use of Corollary 2, requires the computation of the coefficient (trΣ p ) 2 /trΣ 2 p . Two different scenarios may occur:

• The coefficient (trΣ p ) 2 /trΣ 2 p is known even if Σ p is not completely known. This occurs, for instance, in any situation where Σ p is known up to a multiplying constant (e.g. if Σ p = σ 2 I p with unknown σ 2 , (trΣ p ) 2 /trΣ 2 p turns out to be equal to p, or if [Σ p ] i j = σ 2 min(i, j) with unknown σ 2 , i.e. a discrete time brownian motion, (trΣ p ) 2 /trΣ 2 p turns out to be equal to 3/2). • The coefficient (trΣ p ) 2 /trΣ 2 p is unknown. In this case we proceed by replacing it with an estimate while the confidence level in (1) and the significance level in (2) become approximate; for instance, in Section 4 the estimator (n-2)(n+1)

(n-1) 2 (trS) 2 trS 2 -1 n-1 (trS) 2
suggested in Secchi et al. ( 2010) is used.

The case Σ p = σ 2 I p with unknown σ 2 is widely discussed in [START_REF] Srivastava | Multivariate theory for analyzing high dimensional data[END_REF]; in that work the distribution of the statistic (p-n+2)n n-1 (Xµ p ) S + (Xµ p ) is detected for any p > n under the assumption Σ p = σ 2 I p . The results presented in Srivastava ( 2007) are consistent with Corollary 2. Indeed, when Σ p = σ 2 I p , for p → +∞, the difference between the statistic (p-n+2)n n-1 (Xµ p ) S + (Xµ p ) and the statistic n(trΣ p ) 2 (n-1) trΣ 2 p (Xµ p ) S + (Xµ p ) converges a.s. to 0, and the limit distribution of the former statistic is a χ 2 (n -1), as expected by Corollary 2.

Before showing an application of our results within the context of functional data analysis (e.g. the analysis of the local radius of 65 Internal Carotid Arteries), we want to point out some peculiar features of confidence region (1) and of test (2).

Because S + is positive semi-definite, the confidence region CR γ (µ p ) -which for n > p is a p-dimensional ellipsoid in a p-dimensional space -turns out to be a cylinder belonging to a p-dimensional space generated by an n -1-dimensional ellipsoid. Its graphical visualization is of course non trivial. Nevertheless, knowing if a given vector m p of interest belongs to CR γ (µ p ) is always trivial. Moreover it can be shown that CR γ (µ p ) is bounded in all directions belonging to the random space Im(S) and thus all projections onto these directions can be used to partially visualize CR γ (µ p ). For instance, one may use the n -1 sample principal components (PCs) as natural directions onto which project and visualize CR γ (µ p ).

Due to the non-null dimension of the random space ker(S) and to the orthogonality between ker(S) and Im(S), we have that the statistics

n(trΣ p ) 2 (n-1) trΣ 2 p (X-µ 0p ) S + (X- µ 0p ) in the hypothesis test (2) does not change if µ 0p is replaced by µ 0p + µ ker(S)
with µ ker(S) being any vector belonging to ker(S). This means that it might happen that H 0 is not rejected even for values of the sample mean X that are "really very far" from µ 0p in some direction within ker(S). This is not surprising, because the use of S + implies an exclusive focus on the space Im(S), neglecting all pn + 1 directions associated to ker(S).

An Application to the Radius of Brain Vascular Vessels

We present here an application of the results introduced in Section 3 to the analysis of a functional data set: the local radius of 65 internal carotid arteries (ICA). Details about the origin and elicitation of these data can be found in Sangalli et al. (2009a) and Sangalli et al. (2009b). The 65 patients are divided into two groups according to the presence and location of a cerebral aneurysm in their brain vascular system: the Lower group (made of 32 patients having an aneurysm along the ICA or healthy) and the Upper group (made of 33 patients having an aneurysm downstream of the ICA). For both groups, a 95% confidence region for the mean radius function and a test for the constancy of the mean radius function are computed.

From a numerical point of view, functional data have been discretized on a sufficiently fine grid and the values of the 65 functions in correspondence of the grid points have been used as realizations of the random variables. The grid size (in this case made of 258 points) has been chosen large enough to make the value of the terms on the left of inequalities ( 1) and (2) stabilize towards their limit values. In practice, we are dealing with two data sets characterized by n = 32 and p = 258, and n = 33 and p = 258, respectively, and the p-asymptotic approximation is going to be used.

In Figure 1, for the two data sets (first row of Figure 1), projections of the radius functions (colored curves) and of the 95% confidence region for the mean function (black dotted curves) along the 1st and 2nd PCs are reported (second and third row of Figure 1, respectively); in the language of multivariate statistics, we would refer to these bounds as the extremities of the T 2 -simultaneous confidence intervals along the direction indicated by the 1st and 2nd PCs. Similarly to the multivariate case, this representation is not exhaustive, indeed an infinite number of other directions can be explored by means of similar graphics to help the visualization of the confidence region. Note that the fact that a given function lyes within the confidence bounds shown in Figure 1 does not imply that this function belongs to the confidence region (the correct procedure to know if a given function belongs to the confidence region remains using equation (1) directly).

Moreover, note that the bounds in Figure 1 appear to be extremely large compared to the variability presented by the data; this is not surprising in a framework where p >> n. To this purpose, in the fourth row of Figure 1, the projection of the 95% confidence region (black dotted ellipse) onto the subspace generated by the 1st and 2nd PCs is compared with the confidence ellipse based on the classical Hotelling's T 2 with p = 2 (black full ellipse); this is the confidence region that one would have used if he had projected functional data onto the subspace generated by the 1st and 2nd PCs and had performed -therein -a classical bi-variate statistical analysis neglecting the randomness of the bi-dimensional space onto which the data are projected (i.e. assuming it as deterministically chosen instead of data dependent). It is evident that in the framework p >> n, ignoring this randomness can make the actual confidence (or test significance) drift away from its nominal value.

Finally, we perform for the two populations a test for checking the constancy of the mean radius, i.e. test is equivalent to the test H 0 : E[R (s)] = 0 versus H 1 : E[R (s)] = 0 that can be performed by applying test (2) to the first derivatives of the radius functions. In particular, the 65 first derivatives have been estimated by means of free-knot regression splines (Sangalli et al., 2009b) and the same grid used to build the confidence regions for the radius functions have been used to perform the test. At the significance level 5%, the null hypothesis of constancy of the mean radius is rejected for the Upper group (p-value < 0.001) and it is not for the Lower group (p-value = 0.076). Thus, a strong statistical evidence for the non-constancy of the mean radius within the last 30 mm of the ICA of patients with an aneurysm downstream of the ICA is found. This conclusion is coherent with the results illustrated in Sangalli et al. (2009a). But -differently from the latter work, where this conclusion is heuristically derived by means of subjective interpretations of functional principal components -in the present work the same conclusion is reached from an inferential perspective where the hypotheses of constancy and non-constancy of radius are formally defined and quantified by means of an hypothesis test procedure.

H 0 : E[R(s)] = constant versus H 1 : E[R(s)] = constant. This

Fig. 1

 1 Fig.1First row: radius functions (colored curves). Second and third rows: projections of the radius functions (colored curves) and of the 95% confidence region (black dotted curves) along the 1st and 2nd principal components respectively. Fourth row: projections (by means of PC scores) onto the subspace generated by the 1st and 2nd principal components of the radius functions (colored points) and of the 95% confidence region (black dotted ellipse); for comparison the confidence ellipse based on the classical Hotelling's T 2 with p = 2 is reported (black full ellipse). Lower group on the left and Upper group on the right.


Marc Metian

Dr Michel Warnau
email: warnaumichel@yahoo.com

The Tropical Brown Alga Lobophora variegata (Lamouroux) Womersley: A Prospective Bioindicator for Ag Contamination in Tropical Coastal Waters

Keywords: Silver, Bioconcentration, Coral reef ecosystems, Biomonitoring

published or not. The documents may come   L'archive ouverte pluridisciplinaire

One of the unique characteristics of silver relates to the diversity of its applications [START_REF] Eisler | Silver hazards to fish, wildlife and invertebrates: a synoptic review[END_REF]Silver Institute 2003). On a global scale, 27,000 tons of silver are used every year (Silver Institute 2003), of which a minimum of 150 tons year -1 enters the aquatic environment from, for example, mine tailings, electroplating, or wastewater treatment plants (Silver Institute 2003).

Due to the Ag enrichment in sewage sludge from coastal cities, this metal is considered to be a good proxy for anthropogenic inputs in coastal waters (e.g., [START_REF] Sanudo-Wilhelmy | Anthropogenic silver in the southern California bight: a new tracer of sewage in coastal waters[END_REF]. In addition, it has been shown both in the field and in the laboratory that several marine organisms can bioconcentrate Ag up to very high levels (see [START_REF] Ratte | Bioaccumulation and toxicity of silver compounds: a review[END_REF][START_REF] Metian | Accumulation of trace elements in the tropical scallop Comptopallium radula from coral reefs in New Caledonia[END_REF]. Hence, the use of Ag as a proxi for bioindicator species has been proposed as an efficient tool to survey and monitor urban contamination in the temperate coastal zones [START_REF] Sanudo-Wilhelmy | Anthropogenic silver in the southern California bight: a new tracer of sewage in coastal waters[END_REF][START_REF] Langston | Bioindicators of Ag availability in UK estuaries In[END_REF]. According to [START_REF] Rainbow | Cosmopolitan biomonitors of trace metals[END_REF], extending this concept to the tropical zones, using local bioindicators, would be most useful. However, although Ag + is one of the most toxic ions to marine biota (e.g. [START_REF] Ratte | Bioaccumulation and toxicity of silver compounds: a review[END_REF]Warnau et al. 1996b), studies on Ag dynamics in marine biota are relatively scarce and this holds particularly true for tropical environments (see e.g. Gorsuch et al. 2003).

The present work investigates Ag uptake and depuration biokinetics in the tropical brown alga Lobophora variegata, using radiotracer techniques. This species is widely distributed in the tropical zone [START_REF] Coen | Morphological variation and differential susceptibility to herbivory in the tropical brown alga Lobophora variegata[END_REF] and is known to display quite elevated metal concentrations in the field [START_REF] Metian | Accumulation of trace elements in the tropical scallop Comptopallium radula from coral reefs in New Caledonia[END_REF]; it is therefore a promising bioindicator species.

Materials and Methods

Specimens of L. variegata were collected by SCUBA diving in Maa Bay, SW lagoon of New Caledonia in October 2003. Algae were then shipped to IAEA-MEL, Principality of Monaco and acclimated to laboratory conditions (constantly aerated open circuit aquarium, 20% seawater renewal h-1; salinity 37 p.s.u.; temperature 25.0 ± 0.5°C; 12-h photoperiod with a light irradiance of 300 lmol photons m -2 s -1 ).

Twelve L. variegata thallia (average wet weight = 1.7 ± 0.7 g) were placed in a closed-circuit aquarium containing 70 L of natural seawater (salinity, temperature and light conditions as described above). In order to allow for individual recognition, each alga was placed in a cylindrical 200-mL PST container covered above and below with a 300-lm mesh net. Algae were then exposed for 28 days in seawater spiked with 0.5 kBq L -1 of 110m Ag (as 110m AgNO 3 ; CERCA, France). Due to the specific activity of the radiotracer, this spike corresponds to an addition of 21 ng Ag L -1 , viz. a concentration that is actually found in contaminated environments [START_REF] Smith | Silver in San Francisco Bay estuarine waters[END_REF]. The seawater was changed and the radio-tracer spike was renewed daily to maintain 110m Ag activity constant. Radioactivity in the water was measured before and after each seawater renewal in order to determine the time-integrated radiotracer activity (Rodriguez y [START_REF] Rodriguez Y Baena | Experimental evidence for 234 Th bioaccumulation in three Antarctic crustaceans: potential implications in particle flux studies[END_REF]. L. variegata thallia were regularly ccounted to determine the 110m Ag uptake kinetics. Radioactivity was measured using a high-resolution c-spectrometry system consisting of four coaxial Germanium (N-or P-type) detectors (EGNC 33-195-R, Canberra ® and Eurysis ® ) connected to a multi-channel analyzer and a computer equipped with a spectra analysis software (Interwinner ® 6). Activities of the samples were determined by comparison with standards of known activities and appropriate geometry and were corrected for background and physical decay. Counting times were adapted to obtain counting rates with relative propagated errors < 5%.

Following the exposure period, all algae were transferred to new meshsided PST containers and placed in clean flowing seawater (open circuit; flux: 50 L h -1 ; constantly aerated; salinity, temperature and light conditions as described above) for 62 days. These were then radioanalysed at different times (as described above) to determine 110m Ag depuration biokinetics.

Uptake of 110m Ag was expressed in terms of concen-tration factors (CF; viz. ratio between 110m Ag activity in alga-Bq g -1 wet wt-and time-integrated activity in seawater-Bq g -1 ) over time (Warnau et al. 1996a). Uptake kinetics were best fitted using the following saturation exponential equation: CF = CF ss (1-e -k e t ) where CF and CF ss are concentration factors at time t (d) and at steady state, respectively, and k e is the depuration rate constant (d -1 ).

Depuration of 110m Ag was expressed as decrease in percentage of remaining activity (viz., radioactivity at time t divided by the initial radioactivity measured in the alga at the beginning of the depuration period * 100) over time (Warnau et al. 1996a). Depuration kinetics were best fitted using the following exponential equation: A t = A 0 e -k e t where A t and A 0 are the remaining activities (%) at time t (d) and 0, respectively, and k e is the depuration rate constant (d -1 ). The biological half-life (T b½ ) of 110m Ag in L. variegata was calculated according to the relation T b½ = ln2/k e (Warnau et al. 1996a).

Constants of the models and their statistics were estimated by iterative adjustments of the model and Hessian matrix computation using the non-linear curve-fitting routines in the Statistica ® 5.1 software. The level of significance for statistical analyses was always set at α = 0.05.

Results and Discussion

Lobophora variegata concentrated readily 110m Ag, reaching a mean concentration factor, CF, higher than 7,000 after 28 days of exposure. Uptake biokinetics were best described by a saturation model (Fig. 1), which was characterised by an estimated steady-state CF, CF ss , of 12,900 ± 2,350 (p < 0.0001).

Subsequent depuration of incorporated 110m Ag was followed over a 62-day period. Depuration kinetics were best fitted using a single exponential equation (Fig. 2), characterised by a relatively slow depuration rate constant (k e = 0.0097 ± 0.0005; p < 0.0001), indicating that Ag was efficiently retained within the alga tissues, with a biological half-life of 72 ± 4 days.

Information on Ag bioaccumulation kinetics in macroflora representatives is scarce in the scientific literature, whether in the field or in the laboratory [START_REF] Langston | Bioindicators of Ag availability in UK estuaries In[END_REF]Warnau et al. 1996a;[START_REF] Boisson | Influence of temperature on the accumulation and retention of 11 radionuclides by the marine alga Fucus vesiculosus (L.)[END_REF][START_REF] Ratte | Bioaccumulation and toxicity of silver compounds: a review[END_REF]). Nevertheless, comparison with the data available from other marine macroalgae and macrophytes clearly shows the intensity of Ag bioaccumulation in L. variegata (Table 1). The high CFs observed in L. variegata could result from (1) biosorption processes, where the metal in solution binds to the cell walls of the macroalgae [START_REF] Schiewer | Ionic strength effects in biosorption of metals by marine algae[END_REF] and/or (2) bioconcentration and sequestration processes, where the metal enters within the cell and strongly binds to macromolecules such as polyphenols, phytochelatins and metallothioneins [START_REF] Morris | Identification and characterization of a recombinant metallothionein protein from a marine alga, Fucus vesiculosus[END_REF][START_REF] Cobbett | Phytochelatins and their roles in heavy metal detoxification[END_REF].

According to the literature related to Ag marine geochemistry and algal bioaccumulation, the high Ag CF observed in L. variegata is likely driven by both adsorption and cell concentration processes. It is indeed documented that charged or polar Ag compounds can be adsorbed onto algae cell surfaces, and that lipophilic, non-polar Ag chloride complexes are directly incorporated into the cells (see e.g. Ratte 1999). The high CF observed in L. variegata suggests that bioconcentration should be particularly intense compared to adsorption in this species. This is also supported by the fact that L. variegata has a high content of polyphenolic compounds (up to 13.4% on a dry weight basis; [START_REF] Targett | Biogeographic comparisons of marine algal polyphenolics: evidence against a latitudinal trend[END_REF], for which Ag has a high affinity. This suggests that metal bioaccumulation and chelation with phlorotannins (viz. the brown algae polyphenols) could play a major role in Ag bioaccumulation in L. variegata, as well as in the strong retention capacity shown by the alga for Ag.

Overall, our results indicate that L. variegata could be considered as a relevant bioindicator species for Ag contamination in tropical waters, as it shows a rapid response in metal uptake and strong retention capacity. Therefore it has the potential to provide valuable information on the contamination levels occurring in its environment. Further research is needed to verify whether phlorotannins in L. variegata could also be proposed as a sensitive biomarker of Ag contamination in seawater. A t = 98 * e -0.0097 t R 2 = 0.88

Fig. 1 Fig. 2

 12 Fig. 1 Uptake kinetics of 110m Ag in Lobophora variegata (mean concentration factor ± SD, n = 12)

Table 1

 1 Experimental 110m Ag concentration factors (CF) in different organisms.

	Organisms	CF	References
	Lobophora variegata (brown alga)	1.29 10 4a	present study
	Fucus vesiculosus (brown alga)	10 3b	Boisson et al. 1997
	Caulerpa taxifolia (green alga)	2.5 10 2c	Warnau et al. 1996
	Posidonia oceanica (seagrass)	1.4 10 2a	Warnau et al. 1996

a CF at steady state b CF calculated after 12 days of exposure c CF calculated after 15 days of exposure

Acknowledgments

The IAEA is grateful for the support provided to its Marine Environment Laboratories by the Government of the Principality of Monaco. Authors thank P. Bustamante (LIENSs) for constructive comments on the ms and J.L. Teyssié (IAEA-MEL) for technical assistance. MW is an Honorary Senior Research Associate of the National Fund for Scientific Research (NFSR, Belgium) and holds a 2008 Invited Expert position at LIENSs, supported by the Conseil Régional de Poitou-Charentes. This work was supported by the IAEA and the French PNEC Programme (Chantier Nouvelle-Calédonie).


Hansj Örg Albrecher
email: hansjoerg.albrecher@unil.ch

Corina Constantinescu
email: corina.constantinescu@unil.ch

Stephane Loisel
email: stephane.loisel@univ-lyon1.fr

Explicit ruin formulas for models with dependence among risks

Keywords: Ruin probability, frailty models, mixing, Archimedean copulas, completely monotone distributions

We show that a simple mixing idea allows to establish a number of explicit formulas for ruin probabilities and related quantities in collective risk models with dependence among claim sizes and among claim inter-occurrence times. Examples include compound Poisson risk models with completely monotone marginal claim size distributions that are dependent according to Archimedean survival copulas as well as renewal risk models with dependent inter-occurrence times.

Introduction

Assume that the free surplus of an insurance company at time t is modeled by

R(t) = u + ct - N(t) k=1 X k , (1)
where u is the initial surplus, the stochastic process N(t) denotes the number of claims up to time t and the random variables X k refer to the corresponding claim amounts. Here c > 0 is a constant premium intensity and u ≥ 0 is the initial surplus in the portfolio. The classical Cramér-Lundberg risk model assumes that N(t) is a homogeneous Poisson process with intensity λ, which is independent of the claim sizes and the claim sizes are independent and identically distributed. Under this assumption, various quantities can be calculated explicitly for certain classes of claim size distributions, including the probability of ruin ψ(u) = P(R(t) < 0 for some t > 0). However, the independence assumption can be too restrictive in practical applications and it is natural to look for explicit formulas for ψ(u) and related quantities in the presence of dependence among the risks. Over the last decades a number of dependence structures have been identified that allow for analytical formulas (see e.g. Chapter XIII of Asmussen and Albrecher (2010) for a recent survey), but an exhaustive identification of this set of models is far from complete.

The purpose of this paper is to provide an additional class of dependence models for which explicit expressions for ψ(u) can be obtained. To that end, we start with simpler models for which explicit solutions are available and subsequently mix over involved parameters. This changes the marginal distributions and introduces dependence among the risks. One can then balance the marginal distribution of the risks with their dependence structure in such a way that properties like level crossing probabilities can be studied without direct treatment of the dynamics of the process. In other words, the mixing of the parameters can be carried over to the mixing of the final quantities under study. This results in a new set of dependence models for which explicit results can be obtained and may serve as a skeleton within larger model classes. For transparency of exposition, the analysis of the paper will be in terms of the ruin probability ψ(u), but the principle can be applied to many other quantities and also to applications beyond risk theory. We would like to note that mixing over parameter values of the risk process is a classical tool (see for instance B ühlmann (1972), who considered this in the context of credibilitybased dynamic premium rules). In this paper we want to suggest such mixing procedures as a general tool of dependence modeling in collective risk theory, as its potential does not seem to be sufficiently explored yet. The rest of the paper is organized as follows. Sections 2 and 3 work out the ideas in detail and show how the Archimedean dependence structure naturally enters in this approach. Examples for explicit ruin probability formulas are given. Section 4 then highlights a number of further possible extensions of the method and Section 5 concludes.

Compound Poisson models with completely monotone claim sizes and Archimedean dependence

Let Θ be a positive random variable with cdf F Θ and consider the classical compound Poisson risk model [START_REF]  | On ruin probability and aggregate claim representations for Pareto claim size distributions[END_REF] with exponential claim sizes that fulfill, for each n,

P (X 1 > x 1 , . . . , X n > x n | Θ = θ) = n k=1 e -θx k . ( 2
)
That is, given Θ = θ, the X k (k ≥ 1) are conditionally independent and distributed as Exp(θ). However, the resulting marginal distributions of the X k 's will now in general not be exponential any more and the claim sizes will be dependent. Let ψ θ (u) denote the ruin probability of the classical compound Poisson risk model with independent Exp(θ) claim amounts given by

ψ θ (u) = min λ θc exp{-(θ - λ c )u}, 1 , u ≥ 0. (3)
Then, for the dependence model ( 2), the ruin probability is given by

ψ(u) = ∞ 0 ψ θ (u)dF Θ (θ). (4)
Since for θ ≤ θ 0 = λ/c the net profit condition is violated and consequently ψ θ (u) = 1 for all u ≥ 0, this can be rewritten as

ψ(u) = F Θ (θ 0 ) + ∞ θ 0 ψ θ (u)dF Θ (θ). ( 5
)
An immediate consequence is that in this dependence model

lim u→∞ ψ(u) = F Θ (θ 0 ), (6)
which is positive whenever the random variable Θ has probability mass at or below θ 0 = λ/c (and the latter is useful for the construction of dependence models as will be seen below).

Proposition 2.1. The dependence model characterized by (2) can equivalently be described by having marginal claim sizes X 1 , X 2 , . . . that are completely monotone, with a dependence structure due to an Archimedean survival copula with generator φ = F Θ -1 for each subset (X j 1 , . . . , X j n ) (for j 1 , . . . , j n pairwise different), where F Θ denotes the Laplace-Stieltjes transform of F Θ .

Proof. The proof is essentially due to Oakes (1989), who considered the case n = 2, see also Denuit et al. (2005, p.229). In order to keep the paper self-contained, we give here a rather direct line of reasoning. Due to (2), for each n, the joint distribution of the tail of X 1 , . . . , X n can be written as

P (X 1 > x 1 , . . . , X n > x n ) = ∞ 0 e -θ(x 1 +•••+x n ) dF Θ (θ) = F Θ (x 1 + • • • + x n ). ( 7
)
At the same time, the representation with survival copula C is given by

P (X 1 > x 1 , . . . , X n > x n ) = C(F X (x 1 ), . . . , F X (x n )),
where

F X (x i ) = 1 -F X (x i
) is the tail of the marginal claim size X i (note that the X i 's are all identically distributed). If the survival copula is Archimedean with generator φ, then it admits the representation C(F X (x 1 ), . . . ,

F(x n )) = φ -1 φ(F X (x 1 )) + • • • + φ(F X (x n )) , which due to F X (x i ) = ∞ 0 e -θx i dF Θ (θ) = F Θ (x i ), i = 1, . . . , n, (8)
exactly matches with [START_REF] D | Actuarial Theory for Dependent Risks[END_REF] when the generator is chosen to be φ(t) = F Θ -1 (t). Note that as the inverse of a Laplace-Stieltjes transform of a cdf, φ is a continuous strictly decreasing function from [0,1] to [0, ∞] with φ(0) = ∞ and φ(1) = 0 and φ -1 is completely monotone, so that the Archimedean copula is well-defined for all n (see e.g. Nelsen (1999, Th.4.6.2). At the same time, from (8) one sees that the marginal random variables X i are necessarily completely monotone.

Remark 2.2. The above mixing construction may be thought of as sampling a realisation θ of Θ according to F Θ and then following a trajectory of the usual independent risk model with parameter θ, so that the dependence is introduced through the common realized value of θ among all possible values from the positive halfline. Correspondingly, the resulting dependence will be stronger the more spread out the distribution of Θ is. However, this mixing construction is just the tool to establish the explicit formula (3), but need not be the causal reason for the dependence model. Having established this result, one may now equivalently start in two ways: For any risk model with completely monotone marginal claim size random variable X i , there is an expression of the form (8) for some positive random variable Θ and then formula (3) holds for a dependence structure with Archimedean survival copula and generator φ = F Θ -1 . Alternatively, one may start with specifying the Archimedean survival copula through its generator and then the above relations give a corresponding marginal distribution for which the explicit formula (3) holds. Note that for each choice of the generator φ, Kendall's tau for each bivariate pair is given by

τ 0 = 1 + 4 1 0 φ(t)/φ (t) dt.
Let us now look at some particular examples.

Example 2.3 (Pareto claims with Clayton copula dependence).

If Θ is Gamma(α, β) distributed with density

f Θ (θ) = β α Γ(α) θ α-1 e -βθ , θ > 0,
the resulting mixing distribution for the marginal claim size X k is

F X (x) = ∞ 0 e -θx f Θ (θ)dθ = 1 + x β -α , x ≥ 0
(see e.g. Klugman et al. ( 2008)). That is, the X k 's are Pareto(α, β) distributed and due to Proposition 2.1 they follow a dependence structure according to an Archimedean survival copula with generator

φ(t) = t -1/α -1,
which is the Clayton copula with parameter α (see also Yeh (2007)). Consequently, the upper tail dependence index between two claim amounts is

λ U = 2 -2 -1/α
(the asymptotic behavior of P(X 1 + X 2 > x) in this situation is for instance studied in Alink et al. ( 2004)). From ( 5) it now follows that for this model

ψ(u) = 1 - Γ(α, βθ 0 ) Γ(α) + θ 0 e θ 0 u β 1 + u β -(α-1) Γ(α -1, (β + u)θ 0 ) Γ(α) ,
where Γ(α, x) = ∞ x w α-1 e -w dw is the incomplete Gamma function and θ 0 = λ/c. In particular (also from ( 6)),

lim u→∞ ψ(u) = 1 - Γ(α, βλ c ) Γ(α) . (9)
Using lim x→∞ Γ(s,x)

x s-1 e -x = 1, one can further deduce that the convergence towards this constant is of asymptotic order u -1 . Finally, for u = 0 we obtain the simple formula

ψ(0) = 1 - Γ(α, βθ 0 ) Γ(α) + βθ 0 Γ(α -1, βθ 0 ) Γ(α) .

Example 2.4 (Weibull claims with Gumbel copula dependence).

If Θ is stable (1/2) distributed (also called Lévy distributed) with density

f Θ (θ) = α 2 √ πθ 3 e -α 2 /4θ , θ > 0,
then the resulting marginal distribution tail of the claim size random variable X is

F X (x) = ∞ 0 e -θx f Θ (θ)dθ = exp{-αx 1/2 }, x ≥ 0,
so that the claim sizes are Weibull distributed with shape parameter 1/2. Since F Θ (s) = e -α √ s , one obtains that the generator of the Archimedean copula is in this case given by φ(t) = (-ln t) 2 (for all values of α). Hence the underlying survival copula is a particular Gumbel copula, and consequently the claim sizes are asymptotically independent. Note that the marginal distribution varies according to the choice of α, whereas the copula stays invariant. From (5) we now get

ψ(u) = F Θ (λ/c) + ∞ λ/c λ θc e -θu e λc/u α 2 √ πθ 3 e -α 2 /4θ dθ,
which can be expressed through the error function Erfc

(z) = 1-Erf(z) = 2 √ π ∞ z e -w 2 dw = 2Φ(-z √ 2) as ψ(u) = Erfc α 2 √ λ/c + λ α 2 c e -α 2 c/4λ - 2α √ πλ/c +e (cα-2 √ uλ) 2 /4cλ (1+α √ u) Erfc uλ/c - α 2 √ λ/c + e (cα+2 √ uλ) 2 /4cλ (-1 + α √ u) Erfc uλ/c + α 2 √ λ/c . Note that for z > 0, Erfc(z) = Γ(1/2, z 2 )/ √ π. For u = 0 we have ψ(0) = Erfc α 2 √ λ/c - 2 √ λ/c α √ π e -cα 2 /4λ + 2λ cα 2 Erf α 2 √ λ/c and lim u→∞ ψ(u) = Erfc α 2 √ λ/c . ( 10
)
Whenever F Θ (a) = 0 for some a > 0 (i.e. there is no probability mass in the neighborhood of the origin), the resulting marginal claim size distribution will be light-tailed.

The following example illustrates this for a resulting marginal (completely monotone) Gamma distribution.

Example 2.5 (Dependent Gamma claims).

Since the Gamma(α, β) distribution with shape parameter α ≤ 1 is completely monotone, it can be obtained as a marginal claim distribution with the above method. Correspondingly, if

F X (x i ) = Γ(α, βx i )/Γ(α) with α < 1, (11)
then due to [START_REF] D | Probabilité de ruine lorsque le paramétre de Poisson est adjusté à posteriori[END_REF] we need to identify the mixing distribution by inverse Laplace transformation. This can be done by complex analysis techniques in an analogous way as in Albrecher and Kortschak (2009) and one arrives at the mixing density

f Θ (θ) = sin(απ) πθ θ β -1 -α , θ > β,
i.e. the mixing distribution has a Pareto-type tail. Formula (4) now gives the ruin probability for a risk model with marginal claim size distribution [START_REF] G | Aspects of Risk Theory[END_REF] and Archimedean dependence structure among claims with generator φ(t), which is the inverse with respect to x of the normalized incomplete Gamma function Γ(α, βx)/Γ(α). Although this generator can not be determined explicitly, it is clear that it does not depend on β (since the generator is invariant with respect to multiplication by a constant). Correspondingly, varying over β, (4) gives an explicit representation of ψ(u) for a whole class of marginal Gamma distributions with the same dependence structure. The result can be compared with the exact ruin probability for the same marginal claim distribution, but with independent claims (see e.g. Grandell (1992, p.14)).

Note also that if the support of Θ is the interval [θ 1 , +∞) for a θ 1 > θ 0 , then the probability of ruin ψ(u) decays exponentially fast with u and is upper-bounded by ψ θ 1 (u), where ψ θ 1 (u) = λ θ 1 c exp{-(θ 1 -λ c )u} is the ruin probability of the independent Cramér-Lundberg model with parameter θ 1 .

Remark 2.6. From the general construction of Archimedean copulas (see e.g. Joe (1997, p.86)), it is clear that the principle outlined in this section can be extended to conditionally independent tails beyond the used exponential tails in [START_REF] A | Diversification of aggregate dependent risks[END_REF], still preserving the Archimedean dependence structure. In particular, whenever the conditional tail of the marginals can be written in the power form P(

X i > x i |Θ = θ) = (G(x i )) θ for some distribution function G(x) and P (X 1 > x 1 , . . . , X n > x n | Θ = θ) = n k=1 (G(x k )) θ
for all n, i.e. θ is the common mixing parameter, then

P(X 1 > x 1 , . . . , X n > x n ) = ∞ 0 [G(x 1 )] θ • • • [G(x n )] θ dF Θ (θ) = F Θ -log G(x 1 ) -• • • -log G(x n ) = F Θ F -1 Θ (F X (x 1 )) + • • • + F -1 Θ (F X (x n )) .
Hence, the dependence structure is again Archimedean with generator

φ(t) = ( F Θ ) -1 (t), where now F X (x i ) = F Θ (-log G(x i )) (in

Renewal risk models with completely monotone inter-occurrence time distributions and Archimedean dependence

In much the same way as in Section 2, one can also use ruin probability formulas of the Cramér-Lundberg risk model (which are explicit for certain classes of claim size distributions, see e.g. Asmussen and Albrecher (2010)) and mix over the Poisson parameter λ. If the mixing cdf is denoted by F Λ , then the resulting ruin probability in the new dependence model is

ψ(u) = ∞ 0 ψ λ (u)dF Λ (λ), u ≥ 0. ( 12
)
Note that this formula was already given by B ühlmann (1972, Eq. ( 3)), where the mixing procedure was used in the context of dynamic credibility-based premiums for the risk process (see also Dubey (1977), B ühlmann and Gerber (1978) and Gerber (1979, Section 5 of Chapter 9)). Proposition 2.1 now still holds by replacing θ with λ, i.e. the resulting dependence structure between the inter-occurrence times T 1 , T 2 , . . . (and each subset of them) will be described by an Archimedean copula with generator φ = F Λ -1 , where F Λ is the Laplace-Stieltjes transform of F Λ . Note that the resulting (dependent) interoccurrence times are marginally not exponential any more, but are completely monotone with distribution tail P(T i > t) = F Λ (t). Now the net profit condition will be violated whenever the realisation of Λ is larger than the threshold value λ 0 = c/E(X i ). Hence, a refined version of ( 12) is

ψ(u) = λ 0 0 ψ λ (u)dF Λ (λ) + F Λ (λ 0 ), u ≥ 0. Correspondingly, lim u→∞ ψ(u) = F Λ (λ 0 ),
which is positive whenever Λ has a positive probability to be larger or equal to λ 0 .

Example 3.1 (Pareto inter-occurrence times with Clayton copula dependence).

As in Example 2.3, if Λ is Gamma(α, β) distributed, the resulting mixing distribution for the marginal inter-occurrence times T k is Pareto distributed with tail

F T (t) = ∞ 0 e -λt f Λ (λ)dλ = 1 + t β -α , t ≥ 0.
and their dependence structure is described by a Clayton survival copula with generator

φ(t) = t -1/α -1.
Note that at the same time it is a classical result that the number N(t) of claims up to a fixed time point t under the Gamma mixing distribution follows a negative binomial distribution.

If one considers the special case of exponentially distributed claim sizes with (now fixed!) parameter θ, then clearly

ψ λ (u) = min λ θc exp{-(θ - λ c )u}, 1 , u ≥ 0.
With [START_REF] J | Multivariate Models and Dependence Concepts[END_REF] and λ 0 = cθ, this leads to the explicit formula

ψ(u) = β α e -θu θc (β -u/c) -1-α α - Γ(α + 1, cθβ -θu) Γ(α) + Γ(α, βcθ) Γ(α) , u ≥ 0. ( 13
)
In particular, we have

ψ(0) = 1 βcθ α - Γ(α + 1, cθβ) Γ(α) + Γ(α, βcθ) Γ(α) and lim u→∞ ψ(u) = Γ(α, βcθ) Γ(α) .
In view of lim x→∞ Γ(s, x)/(x s-1 e -x ) = 1, the convergence towards this constant is again of asymptotic order u -1 .

Example 3.2 (Weibull inter-occurrence times with Gumbel copula dependence).

As in Example 2.3, if Λ is stable (1/2) distributed with density

f Λ (λ) = α 2 √ πλ 3 e -α 2 /4λ , λ > 0,
then the resulting marginal distribution tail of the inter-occurrence times T k is

F T (t) = ∞ 0 e -λt f Λ (λ)dλ = exp{-αt 1/2 }, t ≥ 0.
That is, the inter-occurrence times are Weibull distributed with shape parameter 1/2 and their dependence structure is described by a Gumbel survival copula with generator φ(t) = (-ln t) 2 . From ( 12), the ruin probability for such a model is given by

ψ(u) = αi e -iα √ u/c-uθ 4θ √ cu -1 + Erf α 2 √ cθ -i √ uθ + e 2i √ u/cα Erfc α 2 √ cθ + i √ uθ + Erfc α 2 √ cθ , (14)
where i = √ -1. It can be shown that the resulting imaginary part of the right-hand side of ( 14) is zero, so the expression is indeed real. For u = 0 we obtain the limit

ψ(0) = 1 - α 2 2cθ Erfc α 2 √ cθ + α √ θπc e -α 2 /4cθ
and for large u we have lim u→∞ ψ(u) = Erfc α 2 √ cθ .

Remark 3.3. Since for heavy-tailed inter-occurrence times the usual techniques such as exploiting certain Markovian conditioning arguments do not work, there does not seem to be any explicit formula for ψ(u) available in the literature. Consequently, formulas (13) and ( 14) may be the first instances of an explicit ruin probability formula for heavy-tailed identically distributed interoccurrence times, albeit dependent according to a Clayton and Gumbel copula, respectively.

Further extensions of the method

The mixing idea outlined in this paper can be developed further in many directions. We list here a couple of examples.

Example 4.1 (Independent parallel mixing). One can mix both inter-occurence times and claim sizes independently at the same time, leading to the above-described Archimedean copula structure for each. The ruin probability is then

ψ(u) = ∞ 0 ∞ 0 ψ (θ,λ) (u) dF Θ (θ)dF Λ (λ),
where ψ (θ,λ) is the conditional probability of ruin given that Θ = θ and Λ = λ. Whenever there is an explicit expression for ψ (θ,λ) , this leads to an explicit expression for ψ(u) in renewal models with both dependent inter-occurrence times and dependent claim sizes.

Example 4.2 (Comonotonic mixing). One can also consider Archimedean dependence between inter-occurence times and claim sizes and at the same time dependence among claim sizes and among inter-occurence times. One way to do this is comonotonic mixing, where the realization λ of Λ is a deterministic function of the realisation θ of Θ in the form

λ(θ) = F -1 Λ (F Θ (θ)
) . The ruin probability in this model then is

ψ(u) = ∞ 0 ψ (θ,λ(θ)) (u) dF Θ (θ),
where ψ (θ,λ) is the conditional probability of ruin given that Θ = θ and Λ = λ.

Example 4.3 (Independent light-tailed and dependent heavy-tailed claims). Recall that in Example 2.3, in order to obtain a ruin probability formula for a certain (completely monotone) marginal claim distribution, the dependence structure needed to be fixed. There is in fact a way to vary the dependence structure while leaving the asymptotic tail of the marginal claim distribution unchanged and still receive explicit formulas. Let the surplus process be given by

R(t) = u + ct - N(t) k=1 X k - N (t) l=1 Y l ,
where there are two independent Poisson processes: N (t) with intensity λ generates (independent) Exp(ν) claims Y 1 , Y 2 , . . ., where ν is a fixed constant. On the other hand, N(t) with fixed intensity λ generates a dependent claim stream X 1 , X 2 , . . . as in [START_REF] A | Diversification of aggregate dependent risks[END_REF], where Θ is a positive random variable. If the distribution of Θ is for instance as in Examples 2.3 and 2.4, then these dependent claims X 1 , X 2 , . . . are heavy-tailed. Equivalently, we may view the resulting risk process as

R(t) = u + ct - N (t) k=1 Z k ,
where N (t) is a Poisson process with intensity λ + λ and the marginal density of the claim sizes Z 1 , Z 2 , . . . is given by

f Z (x) = λ λ + λ θe -θx + λ λ + λ νe -νx , x ≥ 0. ( 15
)
That is, given the realisation of θ, the claim size distribution is a mixture of two exponential distributions, but mixed over θ, the marginal tail of X 1 , X 2 , . . . will determine the tail behavior if it is heavy-tailed. Since for fixed θ the ruin probability ψ θ (u) in the classical risk model with a claim size density of the form ( 15) has an explicit form as a weighted sum of two exponential terms in u (see for instance Gerber (1979)), one again obtains an explicit expression for ψ(u) by virtue of (4). Note that mixing the claim stream X 1 , X 2 , . . . with independent (in this case exponential) claims reduces the resulting dependence among the claims. In particular, if Kendall's tau of two arbitrary claims X i , X j is τ 0 , the value of Kendall's tau of two arbitrary claims Z i , Z j in the new model is τ 1 = ( λ λ+λ ) 2 τ 0 , because the only way to get positive correlation between two randomly selected claim amounts is to pick two claims coming from the thinned process N(t), and because the probability that a claim generated by N (t) comes from N(t) is λ λ+λ . Hence we identified a computational vehicle to determine an explicit ruin probability formula, if the dependence among claim sizes should be weakened, but the marginal tail behavior should stay in the way described by N(t), and by choosing the parameter λ accordingly, one can generate explicit formulas for models with any value of τ 1 between 0 and τ 0 . Note that this is done at the expense of losing the explicit Archimedean type dependence structure that was available for λ = 0. Instead, the joint distribution tail of Z 1 , . . . , Z n is obtained as follows. Each of the occurred claims Z i is of the type X i with probability λ/(λ + λ ) and of type Y i else. The joint distribution is then the corresponding mixture of the independent Y i -types with the Archimedean dependence structure of the X i -types.

In complete analogy, one can vary the dependence structure of the inter-occurrence times in the above fashion, still leading to explicit ruin probability formulas.

Remark 4.4. The resulting risk model in the above example eventually comprised two types of claims: independent light-tailed claims, and dependent heavy-tailed claim amounts (under suitable assumptions on θ). In terms of practical interpretation, this may indeed describe a realistic portfolio situation: the dependence between heavy-tailed claim amounts may come from parameter uncertainty, or from other sources of correlation like environmental risk, climate change and legal risk. In fact, a number of internal models for Solvency II work with regularly varying random losses that are aggregated by a Clayton survival copula. Remark 4.5. Inspired by Example 4.3, it is clear that one can generate many more examples of explicit formulas for risk models with dependence by replacing the exponential distribution (given Θ in (2)) by a more general distribution for which ψ θ (u) can be determined explicitly (for instance phase-type random variables). The resulting dependence structure and marginal claim size distribution will be a result of the interplay between this choice and the distribution of Θ (see Remark 2.6 when to still expect an Archimedean dependence structure).

Conclusion and Outlook

In this paper we utilized a simple mixing idea over values of involved parameter to enlarge the class of collective risk models for which explicit representations of the ruin probability are possible. In that way we developed a recipe how to identify explicit solutions for certain combinations of dependence structures and marginal distributions of the involved risks, both for claim sizes and the times of their occurrence. In addition, the method can also be used to derive explicit formulas for dependence between claim sizes and their inter-occurrence times. It should be noted that the computational vehicle to obtain these explicit formulas need not be the causal reason for the dependence in the model. One may as well find different interpretations of the resulting dependence model and then just use the statistical equivalence to identify the explicit formula. This fact was illustrated in the Examples of Section 2 and 3, where the dependence was introduced through the common factor Θ (Λ, respectively), but the resulting dependence structure was an Archimedean copula, which itself may be motivated as an appropriate model by other means. Apart from dependence modeling, the mixing can also be motivated by parameter uncertainty, so that the results can alternatively be interpreted as ruin probabilities when we only have a distribution of an involved parameter available (which for the uncertainty about the Poisson parameter λ was already exploited for credibility-based dynamic premium rules in B ühlmann (1972) and Dubey (1977)). The approach proposed in this paper can be pushed forward to more general risk models including Markov-additive processes and those Lévy processes for which explicit expressions for the ruin probability exist (see Chapter I.4b of Asmussen and Albrecher (2010) for a list of risk models for which this is the case).

As outlined in the introduction, this versatile recipe can easily be extended beyond the study of ruin probabilities, including Gerber-Shiu functions, the maximum severity of ruin and the expected time-integrated negative part of the risk process (see e.g. Loisel (2005)). Whenever in an independence model an explicit formula for a quantity related to the risk process is available, one can mix over involved parameters and obtain explicit formulas for related models that exhibit a certain degree of dependence among the risks. The resulting skeleton of models for which exact expressions of such quantities are available may help to increase the understanding of the effects of dependence for risk management purposes in general.

  the context of survival analysis, θ is often interpreted as the frailty parameter). One concrete example would be to choose the claim size distribution in the independent risk model to be Pareto(α, β) distributed, where α would now be the mixing parameter

θ (so that G(x) would be the tail of a Pareto(1, β) random variable). Then (4) applies and one still has the freedom to choose the mixing distribution of Θ to identify a number of formulas for different dependence structures and marginals. However, this is only of limited usefulness, as for Pareto distributed claims there is no fully explicit formula for ψ θ (u) available (but see Ramsay

(2003)

and

Albrecher and Kortschak (2009)

for integral representations of ψ θ (u)).

Remark 2.7. In fact, the construction of dependence through a common factor is not as restrictive as it may appear at first sight: due to a generalization of De Finetti's Theorem by B ühlmann

(1960)

, an exchangable family of random variables (and exchangability may be seen as a natural assumption in the risk model context) can in great generality be generated as a mixture over a common parameter of an i.i.d. sequence (see also

Feller (1970, p.229)

).

Acknowledgements

. The authors would like to thank Hans-Ulrich Gerber and Dominik Kortschak for helpful remarks. This work has been partially supported by the research chair Actuariat Durable sponsored by Milliman Paris, the French Research National Agency (ANR) under the reference ANR-08-BLAN-0314-01, the Model Management research chair funded by BNP Paribas Assurance, and the Swiss National Science Foundation Project 200021-124635/1.