3 PUBLICATIONS AND EDITING ACTIVITIES

The Institute publishes a Discussion Papers series and a Reprint series. The papers in both series are the output from the statistical research activities. Many collaborations (national and international) are going on with researchers from abroad. The following Discussion Papers and Reprints were issued during the period concerned by this report.

3.1 Discussion Papers

0101.

DELAIGLE, A. and I. GIJBELS, Estimation of integrated squared density derivatives from a contaminated sample.
 

In this paper we propose a kernel estimator of integrated squared density derivatives, from a sample that has been contaminated by random noise. We derive asymptotic expressions for the bias and the variance of the estimator and show that the squared bias term dominates the variance term. This coincides with results available in the case of non­contaminated observations. We then discuss the selection of the bandwidth parameter when estimating integrated squared density derivatives based on contaminated data. We propose a data­driven bandwidth selection procedure of the plug­in type and investigate its finite sample performance via a simulation study.
 

0102.

VANDENHENDE, F. and Ph. LAMBERT, Modeling longitudinal data with non-random drop-outs using copulas.
 

This paper develops a class of parametric models for longitudinal data with non­random drop­outs. Marginal regression models are fitted to the repeated measurements and the drop­out profiles to account for covariate and time effects. The dependence between successive responses and between drop­out and response is also modeled using particular dependence functions, called copulas. Copulas are used to create a joint distribution with given marginal distributions. In our case, two copulas are used to obtain a parametric form for the joint density of the repeated responses and of the drop­out indicators. Parameters of the two marginal and copula models are jointly estimated using maximum likelihood. The method evenly applies to continuous or non­continuous ordinal responses and illustrations are provided for the two cases. A log­normal response (heart rate) is analysed in a toxicology study and a discrete ordinal variable (pain severity) is considered in a migraine trial. Three copula families are used: the product copula, the copula of Frank (1979) and a copula generated from the standard multivariate normal distribution. The use of copulas allows a consistent and flexible modeling of the dependence structure for continuous and non­continuous repeated measurements. Moreover, it permits a valid likelihood­based inference on marginal models for both the response and the drop­out process when the pattern of drop­out is not missing completely at random. In the two examples, serial dependence was detected in the data and the drop­out hazard was moderately depending on previous responses. Parameter estimates from the marginal models were found insensitive to the choice of the dependence model between drop­out and response.
 

0103.

CEBRIAN, A. C., DENUIT, M. and Ph. LAMBERT, Determination of the PML : a case study in group medical insurance.
 

This paper aims to analyze the 1991­92 group medical large claims database of the Society of Actuaries. The framework of extreme values, namely the Generalized Pareto distribution, provides the adequate tool for the evaluation of the probable maximum loss. Key words and phrases: large claims, probable maximum loss, extreme value theory, Generalized Pareto distribution, Generalized Extreme Value distribution, POT method
 

0104.

BERMUDEZ, L., DENUIT, M. and J. DHAENE, Exponential bonus-malus systems integrating a priori risk classification.
 

This paper examines an integrated ratemaking scheme including a priori risk classification and a posteriori experience rating. In order to avoid the high penalties implied by the quadratic loss function, the symmetry between the overcharges and the undercharges is bro­ ken by introducing parametric loss functions of exponential type.
 

0105.

DENUIT, M., DHAENE, J., LE BAILLY de TILLEGHEM, C. and S. TEGHEM, Measuring the impact of a dependence among insured lifelenghts.
 

Actuaries usually compute multiple life premiums based on the unrealistic assumption of independence of the lifelengths of insured persons. Many clinical studies, however, have demonstrated dependence of the lifetimes of paired lives such as husband and wife. In this respect, the present article tries to give an answer to the following question: does this simplifying hypothesis constitute a real financial danger for the insurance company? The answer turns out to be affirmative: this dependence materially affects the values of annuities and insurances involving multiple lives. In order to quantify the impact of a possible dependence on the amount of premium charged for annuities, insurances and widow's pension, we resort here on the Frechet bounds, Markov processes and some copula models. These techniques are applied to classical insurance contracts issued to married couples and illustrated on NIS data as well as on observations from Brussels city.
 

0106.

AKRITAS, M.G. and I. VAN KEILEGOM, Estimation of the bivariate and marginal distributions with censored data.
 

New estimators for the bivariate and marginal distributions when both variables are subject to censoring are proposed under two censoring schemes. The estimators of the marginal distributions are not the marginals of the corresponding estimator of the bivariate distribution. Under the usual scheme which assumes that the censoring variables are independent from the variables of interest, the proposed estimator of the marginal distribution outperforms the Kaplan­Meier estimator. Under the assumption that one of the censoring variables is conditionally independent of the survival time it censors given the other survival time, the proposed estimator of the marginal distribution generalizes the estimator of Cheng (1989). All estimators require estimation of the conditional distribution when the conditioning variable is subject to censoring. Such a method of estimation is proposed. The weak convergence of the proposed estimators is obtained. Simulation results suggest that they perform well relative to existing estimators.
 

0107.

DENUIT, M., LEFEVRE, Cl. and Ph. PICARD, Polynomial structures in order statistics distributions.
 

This paper is concerned with the exact joint tail distributions of order statistics for i.i.d. random variables with arbitrary common distribution function. It is pointed out that the left tail distributions can be expressed in terms of Abel­Gontcharoff polynomials, while the right tail distributions can be written using Appell polynomials. This property is exploited to obtain, in a simple and unified way, closed forms and recursions for the evaluation of the corresponding probabilities. The Kolmogorov­Smirnov goodness­of­fit test for discrete models is discussed as a special case.
 

0108.

BOUEZMARNI, T. and J.M. ROLIN, Consistency of Beta kernel density function estimator.
 

We consider the Beta Kernel estimator for an unknown probability density function f with a compact support [0,1] :

$${\textstyle\frac{1}{n}} \sum_{i=1}^{n}$$ (1)

where the kernel K is the beta density function with parameters x/b and (1 - x)/b , b is the smoothing bandwidth and X₁, $\cdots$ ,X_n is a random sample from a distribution F with an unknown probability density function f .
For such estimator, we obtain, for densities f in C ²([0,1]) , the exact asymptotic behaviour of $\mathbb {E}$(J_bn) , where

J_bn = $$\int_{0}^{1}$$ | f_b(x) - f (x) | dx

Further, we show that there exist a B(f ) depending only upon f such that

$$\inf_{b}^{}$$n ^2/5$$\mathbb {E}$$(J_bn) $$\leq$$ C ^* B(f )+ o(1),

and C ^* = 1.3768102 $\cdots$ is a universal constant.
The uniform weak consistency on [0,1] is proved when f is continuous on [0,1] . For f $\in$ C((0,1)) and unbounded in 0 (or 1), we prove that the Beta kernel estimator converges in probability to infinity in 0 (or 1). By simulations, comparison in L₁[0,1] of the Beta Kernel estimator with Bernstein estimator and another mixture of beta's will be shown.
 

0109.

DU, Y., AKRITAS, M.G. and I. VAN KEILEGOM, Nonparametric analysis of covariance for censored data.
 

The fully nonparametric model for nonlinear analysis of covariance, proposed in Akritas, Arnold & Du (2000), is considered in the context of censored observations. Under this model, the distributions for each factor level combination and covariate value are not restricted to comply to any parametric or semiparametric model. The data can be continuous or ordinal categorical. The possibility of different shapes of covariate effect in different factor level combinations is also allowed. This generality is useful whenever modeling assumptions such as additive risks, proportional hazards or proportional odds appear suspect. Test statistics for the nonparametric hypotheses of no main effects, and no interaction effects which adjust for the presence of a covariate are obtained. They are quadratic forms based on averages over the covariate values of Beran estimators of the conditional distribution of the survival time given each covariate value. The derivation of the asymptotic $\chi^{2}_{}$ -distribution of the test statistics uses a recently obtained asymptotic representation of the Beran estimator as average of independent random variables. A real data set is analyzed and results of simulation studies are reported.
 

0110.

DENUIT, M., Laplace transform ordering of actuarial quantities.
 

This paper aims to study both univariate and multivariate versions of the Laplace transform order. The importance of this stochastic order relation in actuarial sciences is enhanced by outlining several possible applications.
 

0111.

DENUIT, M. and O. SCAILLET, Nonparametric tests for positive quadrant dependence.
 

We consider distributional free inference to test for positive quadrant dependence, i.e. for the probability that two variables are simultaneously small (or large) being at least as great as it would be were they dependent. Tests for its generalisation in higher dimensions, namely positive orthant dependences, are also analysed. We propose two types of testing procedures. The first procedure is based on the specification of the dependence concepts in terms of distribution functions, while the second procedure exploits the copula representation. For each specification a distance test and an intersection­union test for inequality constraints are developed depending on the definition of null and alternative hypotheses. An empirical illustration is given for US and Danish insurance claim data. Practical implications for the design of reinsurance treaties are also discussed.
 

0112.

DENUIT, M., PITREBOIS, S. and J.F. WALHIN, Méthodes de construction de systèmes bonus-malus en RC Auto.
 

In casualty ratemaking the a priori risk classification and the a posteriori corrections to the amount of premium are closely related. These two actuarial techniques form an integrated risk evaluation process for each policy of the portfolio. Consequently, the evolution of the premium amount according to the past claims history depends on the a priori risk classification strategy adopted by the company. This paper aims to enlighten the complementarity of those different techniques in third party liability automobile insurance, relating the discounts and penalties produces by a bonus-malus system to the amount of information taken into account in the price list.

In a deregulated environment, it is essential to distinguish between technical and commercial reevaluations of the premium (viewed as a measure of the risk associated to a given policy). The latter must be a simple as possible to be implemented in practice whereas the former have to be accurate enough to allow for a dynamic treatment of the portfolio (policy cancellations, for instance). Both levels require different actuarial techniques (credibility theory and bonus-malus systems).

Even if it is difficult to include the claim severities in third party liability ratemaking, the average cost per claim considerably vary between accidents with or without bodily injuries. The last part of this work allows for a distinction between these two types of claims.
 

0113.

DONOHO, D., MALLAT, S., von SACHS, R. and Y. SAMUELIDES, Signal and covariance estimation with macrotiles.
 

A macrotile estimation algorithm is introduced to remove additive noise from signals and to estimate the covariance of non­stationary processes. A macrotile algorithm uses a penalized method to optimize the partition of the space in orthogonal subspaces, and the estimation is computed with a projection operator. It is implemented by searching for a best basis among a dictionary of orthogonal bases, and by constructing an adaptive segmentation of this basis. Macrotile models are studied with local cosine bases to remove noise from sounds and to estimate the covariance matrices of locally stationary processes. The model selection and the estimation are implemented with a fast algorithm.
 

0114.

GIJBELS, I. and U. GURLER, Estimation in change point models for hazard function with censored data.
 

The hazard function plays an important role in reliability or survival studies since it describes the instantaneous risk of failure of items at a time point, given that they have not failed before. In some real life applications, abrupt changes in the hazard function are observed due to overhauls, major operations or specific maintenance activities. In such situations it is of interest to detect the location where such a change occurs and estimate the size of the change. In this paper we consider the problem of estimating the change point in a hazard function when the observed variables are subject to random censoring. When the hazard function is piecewise constant and there is a single change point, we suggest an estimation procedure that is based on certain structural properties and on least squares ideas. A simulation study is carried out to compare the performance of this estimator with two estimators available in the literature: an estimator based on a functional of the Nelson­Aalen estimator and a maximum likelihood estimator. The proposed least squares estimator turns out to be less biased than the other two estimators, but has a larger variance. We illustrate the estimation method on some real data sets.
 

0115.

ROLIN, J.M., Nonparametric competing risks models : identification and strong consistency.
 

In this paper, it is shown, in the general case, that a multiple causes death model is equivalent to a competing independent risks model. Uniqueness of the representation and identification conditions are discussed. Martingales arguments generalizing results of Stute and Wang (1993) are used to show the almost sure convergence of simple functionals of the predictable hazard measures and of the distributions of the latent or ''fictitious'' independent risks. These results entails the almost sure uniform convergence on the real line of the distributions of these independent risks.
 

0116.

DELAIGLE, A. and I. GIJBELS, Bootstrap bandwidth selection in kernel density estimation from a contaminated sample.
 

In this paper we consider kernel estimation of a density when the data are contaminated by random noise. More specifically we deal with the problem of how to choose the bandwidth parameter in practice. A theoretical optimal bandwidth is defined as the minimizer of the mean integrated squared error. We propose a bootstrap procedure to estimate this optimal bandwidth, and show its consistency. These results remain valid for the case of no measurement error, and hence also summarize part of the theory of bootstrap bandwidth selection in ordinary kernel density estimation. The finite sample performance of the proposed bootstrap selection procedure is demonstrated with a simulation study. Some applications to real data examples illustrate the use of the method.
 

0117.

DELOUILLE, V., SIMOENS, J. and R. von SACHS, Smooth design-adapted wavelets for nonparametric stochastic regression.
 

In the setting of nonparametric stochastic regression, we introduce a new way to build smooth design­adapted wavelets. Starting from the Unbalanced Haar basis, we use the lifting scheme framework to build improved biorthogonal filters. A weighted average interpolation scheme allows us to construct wavelets with a higher number of vanishing analysing moments. We include a step which stabilizes the transform by local semi­ orthogonalisation. The achievement of this article is to provide a uniform solution to the usual criticisms of wavelet estimators. Indeed, our transform automatically adapts to the nature of the regression problem, that is, to the irregularity of the design, to data on the interval, and to an arbitrary sample size (which does not need to be a power of two). We propose a wavelet thresholding algorithm and show its numerical performance both on real data and simulations including white, correlated and heteroscedastic noise.
 

0118.

TAJAR, A., DENUIT, M. and P. LAMBERT, Copula-type representation for random couples with Bernoulli margins.
 

We propose a copula­type representation for random couples with Bernoulli margins. Some dependence measures for binary data are re­examined. It is stressed that satisfactory dependence measure should only depend on the discrete copula, and not on the margins.
 

0119.

LAMBERT, P. and S. LAURENT, Modelling skewness dynamics in series of financial data using skewed location-scale distributions.
 

We show how the ARMA-Power GARCH model for the conditional mean and variance can be adapted to analyze times series data showing asymmetry. Dynamics is introduced in the location and the dispersion parameters of the skewed Student and of the skewed stable distributions using the same type of structure found in the conditional mean and in the conditional variance in the ARMA-APARCH model. We also propose a general dynamic model for skewness as measured by the odds ratio of having the next observation greater than the conditional mode. This general tool is illustrated by the analysis of the DEM-USD exchange rate over the 1980-1996 period.
 

0120.

VANNUCCI, M. and I. GIJBELS, Wavelet-based testing for multimodality of a density function.
 

This paper aims at exploring wavelet-based testing for multimodality of a density function. A linear wavelet estimator of the density is considered and the test statistic is based on the resolution level, the discrete smoothing parameter of the estimate. A "critical resolution" level is defined and the distribution of the test statistic under the null hypothesis is investigated via a smoothed boostrap method. Thresholding techniques are employed as a tool to calibrate the test to improve its level. The performance of the testing procedure as comparison with the "critical bandwidth" kernel method of Silverman (1981). The procedure is also demonstrated on the Chondrite dataset. While investigating such a wavelet-based testing procedure some interesting issues of wavelet-based estimators had to be explored. This led to some discussions on deriving density estimates which are positive and integrate to one, and to exploring results on wavelet-based estimators of the derivates of a density. Moreover, the impact of having a test statistic based on a discrete rather than on a continuous smoothing parameter became more transparent with this study.
 

0121.

KAAS, R., DHAENE, J., VYNCKE, D., GOOVAERTS, M.J. and M. DENUIT, A simple geometric proof that comonotonic risks have the convex-largest sum.
 

In the recent actuarial literature, several proofs have been given for the fact that if a random vector (X₁,X₂,...,X_n) with given marginals has a comonotonic joint distribution, the sum X₁ + X₂ +...+ X_n is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.
 

0122.

DENUIT, M., LEFEVRE, C. and S. UTEV, Measuring the impact of dependence between claims occurrences.
 

The purpose of this paper is to provide a quantitative measure of the impact of a possible dependence between the occurrences of claims in an individual risk model. Firstly, probabilistic distances, of stop­loss or total variation types, specific to arithmetic random variables are introduced and studied, especially in connection with related probabilistic orderings. Then, these results are applied to derive effective bounds for the distance between the total numbers of claims in the original model and under a standard independence assumption.
 

0123.

CLIMOV, D., DELECROIX, M. and L. SIMAR, Semiparametric estimation in single index Poisson regression : a practical approach.
 

In a single index Poisson regression model with unknown link function, the index parameter can be root­n consistently estimated by the method of pseudo maximumum likelihood. In this paper, we study, by simulation arguments, the practical validity of the asymptotic behavior of the pseudo maximum likelihood index estimator and of some associated cross­validation bandwidths. A robust practical rule for implementing the pseudo maximum likelihood estimation method is suggested, which uses the bootstrap for estimating the variance of the index estimator and a variant of bagging for numerically stabilizing its variance. Our method gives reasonable results even for moderate sized samples thus it can be used for doing statistical inference in practical situations. The procedure is illustrated through a real data example.
 

0124.

DE MACQ, I. and L. SIMAR, Hyper-rectangular space partitioning classification trees : a few insights.
 

The process of computation of classification trees can be characterized as involving three basic choices: the type of splits considered in the growing process,the criterion to be optimized on the successively constructed partitions, and the way to get right-sized trees. Most implementations are ordinary binary trees, i.e. whose successive cuts are made by hyper-planes perpendicular to the axes, while most of the literature concerns the various possible criteria and pruning methods. L. Devroye, L. Györfy and G. Lugosi (1996) define and consider the remarkable theoretical properties of a binary tree classifier whose prominent feature is the particular type of splits used in its construction: at a given node, partitioning is made by hyper-rectangles rather than hyper-planes. We propose a simple algorithm for the optimization problem involved. The we compare the performance of two different implementations of our algorithm with two leading algorithms for ordinary binary trees, namely CART and C4.5 as implemented in the Splus "tree" procedure in SAS's Enterprise Miner respectively. For this purpose, data sets which traditionally enhance the weaknesses of classification trees are used.
 

0125.

LAMBERT, Ph. and S. LAURENT, Modelling financial time series using GARCH-type models with a skewed student distribution for the innovations.
 

Fernandez and Steel (1998) propose a 4-parameter skewed Student distribution where the parameters specifying the location, the dispersion, the asymmetry and the tail thickness have a meaningful interpretation. We first reparametrize their density as a function of the conditional mean and of the conditional variance and derive its cumulative density function and quantile function. We also proceed to a Monte Carlo simulation to assess its practical applicability in a MLE estimation procedure in the GARCH framework. Finally, this general tool is illustrated by the analysis of the NASDAQ on the period 1985-1996. Using both in- and out-of-sample density forecast tests, we validate the choice of this density and reject the normal and Student densities.
 

0126.

MOUCHART, M. and A. OULHAJ, A note on partial sufficiency with connection to the identification problem.
 

Let (R_X,X,P ^$\Theta$ = P ^$\theta$:$\theta$ $\in$ $\Theta$) be a parametrized statistical model and g : $\Theta$ $\rightarrow$ G be a non injective function characterising a parameter of interest. The basic idea of partial sufficiency is to find a (minimal) statistic containing all the relevant information for the inference on the parameter of interest. Following Fraser (1956), Barndorff­Nielsen (1978) has defined a concept of S­sufficiency. This definition is essentially based on two fundamental concepts: The first one concerns the notion of sufficient parameter, and the second one is related to the notion of a variation free reparametrization. Our contribution to this area is to give some properties of the S ­sufficiency concept. In particular, we establish the connection between S ­sufficiency and the identification concept. We show that S ­sufficiency preserves the identification property, i.e. if we reduce the complete model by marginalizing on an S ­sufficient statistic, we do not lose the identification of the parameter of interest. Furthermore, we review the properties of the standard concept of sufficiency and we check wich ones also hold for S ­sufficiency.
 

0127.

GOVAERTS, B., BECK, B., LECOUTRE, E., LE BAILLY, C. and P. VANDEN EECKAUT, From monitoring data to regional distributions : a practical methodology applied to water risk assessment.
 

During the last decade, the assessment of concentration levels of chemical substances in the environment has become a major issue for defining actual exposure in risk evaluation processes. Currently, different approaches are proposed to achieve this assessment. The first one, widely used, relies on multi-media fate models. On the other hand, monitoring data begin also to be used for this purpose but, up to now, few statistical methods are proposed to validate and summarize them adequately at a local or regional level. A very important characteristic of monitoring data is that many observations stay below the detection limits of the measurement devices and therefore the data highly censored.
This paper presents a practical methodology to estimate a regional distribution for the concentration of a chemical substance in the surface water of a given region. The estimate of the distribution is obtained from complete or summarized monitoring data collected over time in the region's sampling stations. From this estimate, it is possible to easily derive different statistical summaries as, for examples, the concentration mean, the standard deviation, the percentiles, etc.

Two approaches have been developed and are compared on UK mercury data. A first non-parametric approach derives the regional distribution by aggregating the observed data without taking any assumption on their statistical distribution. A second approach, which may be applied to complete or summarized local data, proceeds in two steps. First, a statistical distribution (e.g. lognormal, gamma,...) of concentration is fitted to the, possibly censored, data of each sampling station. These local distributions are then aggregated at a regional level and statistics of interest derived.

The paper is closed by a discussion on how to build confidence intervals for the estimates and how to fix the weights to combine the distributions from the local to the regional level.
 

0128.

COSSETTE, H., DENUIT, M. and E. MARCEAU, Distributional bounds for functions of dependent risks.
 

This paper aims to derive bounds for the cumulative distribution function of a function of dependent risks. The results presented here complement a recent work by Denuit, Genest and Marceau (1999) where sums of correlated random variables were considered.
 

0129.

LI, G. and I. VAN KEILEGOM, Likelihood ratio confidence bands in nonparametric regression with censored data.
 

Let (X,Y) be a random vector, where Y denotes the variable of interest possibly subject to random right censoring, and X is a covariate. We construct confidence intervals and bands for the conditional survival and quantile function of Y given X using a nonparametric likelihood ratio approach. This approach was introduced by Thomas and Grunkemeier (1975), who estimated confidence intervals of survival probabilities based on right censored data. The method is appealing for several reasons: it always produces intervals inside [0,1] , it does not involve variance estimation, and can produce asymmetric intervals. Asymptotic results for the confidence intervals and bands are obtained, as well as simulation results, in which the performance of the likelihood ratio intervals and bands is compared with that of the normal approximation method. We also propose a bandwidth selection procedure based on the bootstrap and apply the technique on a real data set.
 

0130.

GIJBELS, I. and A-C. GODERNIAUX, Data-driven discontinuity detection in derivatives of a regression function.
 

This paper provides a fully data­driven procedure for estimating the locations of jump discontinuities occuring in the k th derivative of an unknown regression function. The basic ingredients for the procedure are a two­steps method for estimating the locations of the jump discontinuities, a bootstrap procedure for selecting the smoothing parameters involved in this estimation, and a cross­validation method for estimating the number of discontinuities in a derivative function. The paper extends ideas developed for change point detection in the regression function itself by Gijbels and Goderniaux (2000). A simulation study illustrates the performance of the procedure, and applications to some real data demonstrate its use in practice.
 

0131.

PATILEA, V. and J-M. ROLIN, Product limit estimators of the survival function for doubly censored data.
 

Starting from left and right censored data, two product­limit estimators of the distribution function of a lifetime variable are introduced. They correspond to two latent models with independent variables L (left censoring), T (lifetime of interest) and R (right censoring). Basically, each of our estimators is obtained by combining two Kaplan­Meier type estimators. When the observations are generated according to the classical double censoring model introduced by Turnbull (1974), the product­limit estimators represent close upper and lower bounds for Turnbull's estimator. We deduce the strong convergence of our estimators on the whole real half­line without any additional assumption. Moreover, their asymptotic normality is obtained by the delta­method under conditions concerning only the observed distribution.
 

0132.

BEGUIN, C. and L. SIMAR, Confidence interval for the expenses linked to hospital stays : usefulness of bootstrapping survival curves.
 

The financing of hospitals in function of the pathologies uses the mean of the total expenses of length of stays by DRG after exclusion of extreme stays with trimming rules like Q₁ - k ^*(Q₃ - Q₁) or Q₃ + k ^*(Q₃ - Q₁) where Q₁ is the fist quartile, Q₃ is the third quartile and k usually takes the value 1.5. In order to validate these techniques, we analyse the behaviour fo the estimator of the survival curve of the total expenses obtained in Kaplan-Meier of Cox models, by using bootstrap techniques. The bootstrap method used by Efron in a similar context seems well adapted to this problem. The objective is to find confidence intervals for upper percentiles of the time-dependent variable of the survival function, in our case the total amount spent for a hospital stay, and to verify if the upper bound Q₃ + k ^*(Q₃ - Q₁) lies in this interval. These confidence intervals will be particularly useful for analysing the sampling variability of these bounds. Due to this variability, it appears that these bounds have to be considered with great care.
 

0133.

BEGUIN, C. and L. SIMAR, Detecting too long hospital stays : comparison of nonparametric and parametric frontier approaches.
 

Since 1995, length of stay by Diagnosis Related Groups is taken into account in Belgian hospital's financing. Usually, trimming rules are used before the estimation of the mean length of stay by pathology. In this study we propose to take into account the characteristics of the patients in order to highlight hospital stays with discrepancy between the severity and the resources the patient consumes. A deterministic parametric frontier model is used so that the efficiencies estimated, regarding the length of stay, allow to rank hospital stays, after exclusion of the too efficient stays. The results coming from parametric and nonparamtric fontier models are compared.
 

0134.

PURCARU, O. and M. DENUIT, On the dependence induced by frequency credibility models. I. Time-invariant random effects.
 

In casualty insurance, actuaries usually resort to random effects to take unexplained heterogeneity into account in the spirit of the Bühlmann­Straub model. This paper aims to study the kind of dependence induced by the introduction of common latent variables in the annual numbers of claims reported by policyholders. It will be seen that this classical construction entails strong positive dependence between the underlying random variables.
 

0135.

DHAENE, J., WOLTHUIS, H., DENUIT, M. and M. GOOVAERTS, Risk and savings contracts.
 

Following the "time­capital" approach of De Vylder (1997) it is shown that a fair life insurance contract can uniquely be separated into a fair savings and a fair pure risk contract. It is also shown that each fair life insurance contract can be separated into a fair associated stochastic savings contract and a corresponding fair associated risk contract with the same premium structure as the original contract.
 

0136.

DHAENE, J., DENUIT, M., GOOVAERTS, M.J., KAAS, R. and D. VYNCKE, The concept of comonotonicity in actuarial science and finance : theory.
 

In an insurance context, one is often interested in the distributionfunction of a sum of random variables. Such a sum appears when considering the aggregate claims of an insurance portfolio over a certain reference period. It also appears when considering discounted payments related to a single policy or a portfolio at different future points in time. The assumption of mutual independence between the components of the sum is very convenient from a computational point of view, but sometimes not realistic. We will determine approximations for sums of random variables, when the distributions of the terms are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. In this paper, the theoretical aspects are considered. Applications of this theory are considered in a subsequent paper. Both papers are to a large extent an overview of recent research results obtained by the authors, but also new theoretical and practical results are presented.
 

0137.

BROUHNS, N. and M. DENUIT, Risque de longévité et rentes viagères : I. Evolution de la mortalité en Belgique de 1880 à nos jours.
 

After having described the longevity risk inherent to any life annuity portfolio, we examine mortality trends in Belgium, on the basis of statistics provided by the NIS, relating to the period 1880­1999. It will be clear that the pricing of life annuities requires projected life tables, that will be established in the subsequent parts of this work.
 

0138.

BROUHNS, N. and M. DENUIT, Risque de longévité et rentes viagères : II. Tables de mortalité prospectives pour la population belge.
 

On the basis of data relating to Belgian population during 1960­1998, the mortality of those people aged 60 and over is forecast. This allows us to determine the net present value of pension benefits,in function of the retirement year. We quantify the impact of longevity improvements on the price of life annuities offered by social security systems.
 

0139.

BROUHNS, N. and M. DENUIT, Risque de longévité et rentes viagères : III. Elaboration de tables de mortalité prospectives pour la population assurée belge, et évaluation du coût de l'antisélection.
 

Adverse selection is an important phenomenon in the context of life annuities. We adapt here the projected life tables obtained in Brouhns & Denuit (2001b) to the data gathered be the Belgian regulatory authorities about annuitants. This allows us to build a pure price list taking into account future improvements in annuitants' mortality.
 

0140.

WALHIN, J-F. and M. DENUIT, On the pricing of Top & Drop excess of loss covers.
 

A Top & Drop cover is a reinsurance cover that may be found on the retrocession world. It offers some capacity that may be used either for a top layer or for a working layer. In the latter case one speaks of a drop. Within the collective risk model we show in this paper how to use the multivariate version of the Panjer's algorithm in order to price this cover. We also compare the exact price with the prices obtained with the Fréchet bounds, and with the price obtained with the incorrectly assumed hypothesis of independence. Generalization of the results in dimension higher than 2 is also provided.
 

0141.

GUO, W., DAI, M., OMBAO, H.C. and R. von SACHS, Smoothing spline ANOVA for time-dependent spectral analysis.
 

In this paper, a locally stationary process is proposed using a Smooth Localized Complex Exponential (SLEX) basis, whose spectrum is assumed to be smooth in both time and frequency. A smoothing Spline ANOVA (SS-ANOVA) is used to estimate and make inference on the time-varying log-spectrum. This approach allows the time and frequency domains to be modeled in an unified approach and jointly estimated. Because the SLEX basis is orthogonal and localized in both time and frequency, our method has good nite sample performance. It also allows for deriving desirable asymptotic properties. Inference procedures such as confidence intervals and hypothesis tests proposed for the SS-ANOVA can be adopted for the time-varying spectrum. Because of the smoothness assumption of the underlying spectrum, once we have the estimates on a time-frequency grid, we can calculate the estimate at any given time and frequency. This leads to a high computational efficiency as for large data sets we only need to estimate the initial raw periodograms at a much coarser grid. We present simulation results and apply our method to an EEG data recorded during an epileptic seizure.
 

0142.

TAJAR, A., MESFIOUI, M. and M. DENUIT, On the monotonicity of some nonparametric dependence measures with respect to concordance ordering.
 

This paper explores the monotonicity with respect to concordance ordering of Kendall's $\tau$ and Spearman's $\rho$ for counting random variables with an unbounded support. This generalizes some results of Schweizer and Wolff (1981) and Tchen (1980).
 

0143.

DENUIT, M., PITREBOIS, S. and J-F. WALHIN, Personnalisation des primes-fréquence en assurance automobile par régression poissonienne en présence de données longitudinales.
 

Souvent, les actuaires utilisent plusieurs années d'observation de leur portefeuille automobile afin de personnaliser les primes­fréquence. Ce faisant, ils négligent la dépendance sérielle existant entre les données relatives à un même individu. Cet article examine précisément à l'aide des "Generalized Estimating Equations'' (GEE) l'impact de cette méthodologie sur les estimations des fréquences des sinistres. Une illustration est proposée à l'aide du logiciel SAS sur base d'un portefeuille d'assurance automobile belge observé au cours de trois années.
 

0144.

GIJBELS, I. and A-C. GODERNIAUX, Bootstrap test for change points in nonparametric regression.
 

The objective of this paper is to test whether or not there is an abrupt change in the regression function itself or in its first derivative at certain (prespecified or not) locations. The test does not rely on asymptotics but approximates the sample distribution of the test statistic using a bootstrap procedure. The proposed testing method involves a data­driven choice of the smoothing parameters. The performance of the testing procedures is evaluated via a simulation study. Some comparison with two asymptotic tests, a test by Hamrouni (1999) and Grégoire and Hamrouni (2001b) and a test by Müller and Stadtmüller (1999), is provided. We also demonstrate the use of the testing procedures on some real data.
 

0145.

VANDENHENDE, F. and Ph. LAMBERT, Some dependence criteria for bivariate ordinal variables with given marginal distributions.
 

This paper presents a family of dependence measures for bivariate ordinal data. The joint distribution is expressed as a function of the margins. That function characterizes their dependence in between complete positive and complete negative association. When the margins are given or known, all measures are presented as functions of the expectation of the coupling function. They are all rank-based and verify selected properties for dependence (Renyi, 1959) and concordance (Scarsini, 1984) measures. Our criteria extend Schweizer's $\sigma$ (Schweizer & Wolff, 1981), Spearman's rho and Kendall's tau to the ordinal (non-continuous) case. Unlike these standard measures even after adjustment for ties, perfect dependence is always associated to the same values of $\pm$ 1 whatever the distribution of the margins. Estimates of these dependence measures are proposed together with their asymptotic distributions. A goodness of fit procedure is also given to evaluate the quality of parametric models for the dependence. An illustration is proposed using data from a migraine trial.
 

0146.

SIMAR, L., Detecting outliers in frontier models : a simple approach.
 

In frontier analysis, most of the nonparametric approaches (DEA,FDH) are based on envelopment ideas which suppose that with probability one, all the observed units belong to the attainable set. In these "deterministic'' frontier models, statistical theory is now mostly available (Simar and Wilson, 2000). In the presence of super­efficient outliers, envelopment estimators could behave dramatically since they are very sensitive to extreme observations. Some recent results from Cazals, Florens and Simar (2000) on robust nonparametric frontier estimators may be used in order to detect outliers by defining a new DEA/FDH "deterministic'' type estimator which does not envelop all the data points and so is more robust to extreme data points. In this paper we summarizes the main results of Cazals, Florens and Simar (2000) and we show how this tool can be used for detecting outliers when using the classical DEA/FDH estimators or any parametric techniques. We propose a methodology implementing the tool and we illustrate through some numerical examples with simulated and real data. The method should be used in a first step, as an exploratory data analysis, before using any frontier estimation.

3.2 Published Papers

124.

BOUCKAERT, A. and M. MOUCHART. Sure outcomes of random events: a model for clinical trials. Statistics in medicine. 20, 521-543, 2001.
 

125.

LU, Z. and Z. JIANG. L₁ geometric ergodicity of a multivariate nonlinear AR model with an ARCH term. Statistics & probability letters. 51, 121-130, 2001.
 

126.

LU, Z. and I. GIJBELS. Asymptotics for partly linear regression with dependent samples and ARCH errors : consistency with rates. Science in China. 44, 2, 168-183, 2001.
 

127.

GIJBELS, I. and V. ROUSSON. A nonparametric least-squares test for checking a polynomial relationship. Statistics & probability letters. 51, 253-261, 2001
 

128.

VAN KEILEGOM, I., AKRITAS, M.G. and N. VERAVERBEKE. Estimation of the conditional distribution in regression with censored data : a comparative study. Computational Statistics & Data Analysis. 35, 487-500, 2001.
 

129.

DENUIT, M. and C. GENEST. An extension of Osuna's model for stress caused by waiting. Journal of Mathematical Psychology. 45, 115-130, 2001.
 

130.

DENUIT, M., LEFEVRE, C. and M. SHAKED. On s-convex approximations. Adv. Appl. Prob. 32, 994-1010, 2000.
 

131.

DENUIT, M. and S. VAN BELLEGEM. On the stop-loss and total variation distances between random sums. Statistics & probability letters. 53, 153-165, 2001.
 

132.

ANTONIADIS, A., FAN, J. and I. GIJBELS. A wavelet method for unfolding sphere size distributions. The Canadian Journal of Statistics. 29, 251-268, 2001.
 

133.

DENUIT, M., DHAENE, J. and C. RIBAS. Does positive dependence between individual risks increase stop-loss premiums. Insurance: Mathematics & Economics. 28, 305-308, 2001.
 

134.

DENUIT, M. and C. LEFEVRE. Stochastic s-(increasing) convexity. Generalized convexity and generalized monotonicity. 168-182, 2001.
 

135.

DENUIT, M., PITREBOIS, S. and J.F. WALHIN. Méthodes de construction de systèmes bonus-malus en RC auto. actu-L. 1, 2-32, 2001.
 

136.

GIJBELS, I. and L. PENG. Estimation of a support curve via order statistics. Extremes. 3, 251-277, 2000.
 

137.

AKRITAS, M.G. and I. VAN KEILEGOM. ANCOVA methods for heteroscedastic nonparametric regression models. Journal of the American Statistical Association. 96, 453, 220-231, 2001.
 

138.

DENUIT, M., DHAENE, J., LE BAILLY de TILLEGHEM, C. and S. TEGHEM. Measuring the impact of a dependence among insured lifelengths. Belgian Actuarial Bulletin. 1, 18-39, 2001.
 

139.

NGAI, H-M. and J. ZHANG. Multivariate cumulative sum control charts based on projection pursuit. Statistica Sinica. 11, 3, 747-766, 2001.
 

140.

SIMAR, L. and P.W. WILSON. Testing restrictions in nonparametric efficiency models. Communications in statistics. 30, 1, 159-184, 2001.
 

141.

DENUIT, M., LEFEVRE, C. and M. SCARSINI. On s-convexity and risk aversion. Theory and Decision. 50, 239-248, 2001.
 

142.

COSSETTE, H., DENUIT, M., DHAENE, H. and E. MARCEAU. Stochastic approximations of present value functions. Mitteilungen der Schweiz. Aktuarvereinigung. 1, 15-28, 2001.
 

143.

OMBAO, H.C., RAZ, J.A., von SACHS, R. and B.A. MALOW. Automatic statistical analysis of bivariate nonstationary time series. Journal of the american statistical association. 96, 454, 543-560, 2001.
 

144.

M. DENUIT. Laplace transform ordering of actuarial quantities. Insurance: Mathematics and Economics. 29, 83-102, 2001.
 

145.

V. PATILEA. Convex models, MLE and misspecification. The Annals of Statistics. 29, 1, 94-123, 2001.
 

146.

HALL P., HUANG, L.S., GIFFORD, J.A. and I. GIJBELS. Nonparametric estimation of hazard rate under the constraint of monotonicity. Journal of Computational and Graphical Statistics. 10, 3, 592-614, 2001.
 

147.

CAZALS, C., FLORENS, J-P. and L. SIMAR. Nonparametric frontier estimation : a robust approach. Journal of Econometrics. 106, 1-25, 2002.
 

148.

AKRITAS, M. and I. VAN KEILEGOM, Non-parametric estimation of the residual distribution. Scandinavian Journal of Statistics. 28, 549-567, 2001.

3.3 Books published by members of the Institute

J.P. FLORENS, M. MOUCHART and J.M. ROLIN. Elements of Bayesian Statistics, 544 pp, New York: Marcel Dekker, 1990. W. HÄRDLE and L. SIMAR (editors). Computer Intensive Methods in Statistics, 175 pp, (Statistics and Computing, I). Berlin: Physica-Verlag, 1993.

W. HÄRDLE, S. KLINKE and B.A. TURLACH. XploRe: An Interactive Statistical Computing Environment, 387 pp, Statistics and Computing, Springer-Verlag: New York, 1995. J. FAN and I. GIJBELS. Local Polynomial Modelling and its Applications, 341 pp, Chapman and Hall: London, 1996.

3.4 Editing activities

M. DENUIT

Proceedings Editor for "Insurance: Mathematics and Economics"
Editor of the "Belgian Actuarial Bulletin"

I. GIJBELS

Associate Editor of "Journal of Computational and Graphical Statistics"
Editor of "Journal of Multivariate Analysis"