The ultimate aim of the research work presented in this thesis is to show that various semiparametric M-estimation methods provide a very useful means of estimating a regression model for count data, namely Poisson regression. It is also intended to show that semiparametric methods are also valuable in testing problems. The main contribution of this thesis to this respect is threefold.

First, we propose a robust procedure with respect to the numerical instability inherent to the application of M-estimation methods to real data. In our Poisson regression setting, the objective function to be optimized is the pseudo likelihood function and the resulting estimator of the direction vector is called Pseudo Maximum Likelihood (PML) estimator. We investigate, by simulation arguments, the practical validity of the PML estimator asymptotic behavior and of the associated regression function estimator. In particular, it appears that the asymptotic results should not be used unless a huge number of observations is available. We propose a bootstrap procedure for approximating the variance of the direction estimator and a variant of bagging method introduced by Breiman (1996), in order to numerically stabilize the PML estimation procedure. Our method gives reasonable results even for moderate sized samples and therefore it can be used for doing statistical inference in practical situations.

Second, we derive two alternative M-estimation methods, based on risk estimation. For estimating the risk associated with the weighted average squared error, we propose two data-driven selectors: weighted least-squares (WLS) and double smoothing (DS). The first criterion is a weighted least-squares criterion plus a term which prevents undersmoothing in small samples, whereas the second method makes use of a double smoothing idea, as in Wand and Gutierrez (1997). Simulations are used to investigate the behavior of the above data-driven estimation methods in the single index Poisson model. In small samples, our weighted least-squares and double smoothing methods out-perform both the pseudo maximum likelihood method and the weighted least-squares cross-validation method of Härdle, Hall and Ichimura (1993).

Finally, we provide a procedure for testing the validity of the Poisson assumption. We propose a test statistics for overdispersion and derive its asymptotic distribution under the null hypothesis of Poisson model. The distributional approximation is assessed by simulations in several regression scenarios. The results indicate that the asymptotical normal approximation is not satisfactory unless the sample size is very large. Therefore, we propose a bootstrap approach for conducting the overdispersion test in small samples.

The bootstrap procedures for the estimation of the direction vector variance and for the overdispersion test are illustrated on a real data sample.

In order to control the health care costs, the Belgian government introduced in 1995 a regulation of the hospital payment system taking into account the pathologies measured by the Diagnosis Related Groups (DRG). In this context, the mean is usually estimated by a trimmed mean, i.e. a mean computed and the interquartile range like Q₁₍₃₎-(+)k^*(Q3-Q1). This thesis proposes to measure the variability of these bounds, in particular the higher bound and to take into account the characteristics of the patients.

First, the distribution of the total expenses to a pathology is estimated by a survival curve and the confidence interval of this curve is defined using bootstrap technique. We look for the corresponding value of the bounds described above on the survival curve. When analysing myocardial infarction, the variability of the bounds measured by the length of the confidence interval may result in 3.8% of difference in the estimation of the mean.

We also propose the use of frontier models in order to rank hospital stays in function of their expenses taking into account the severity level of the patient. In this case, the hospital stays with the lowest total expenses for a given level of severity characterise the frontier of possibilities and are thus considered as being efficient. We work with deterministic models, parametric and nonparametric. In an univariate approach, we try to find the minimal achievable value of LOS or the minimal value of the total expenses and in a multivariate approach, we try to find simultaneously the minimal value of both LOS and medical fees, for a given level of the severity of the patients. The efficiencies obtained by the parametric estimator are lower the nonparametric model, more flexible than the parametric model with his restrictive hypothesis, envelops the data more closely and should be preferred for this type of analysis.

But, deterministic models are very sensitive to the extreme stays so that some efficient stays could be in fact "too" efficient and considered as outliers. We try to highlight these stays using a simplified version of a method proposed by Wilson (1993) (B-W method). The too efficient stays are characterised by lower resource variables and higher severity level. As expected, the exclusion of the too efficient stays, the inefficient stays can be hightlighted. The stays to be excluded from the hospital financing could be searched among these inefficient stays. The inefficient stays present higher value of the resources variables for a lower severity level.

An alternative method allowing to detect the too efficient stays has been proposed by Simar (2001). It is based on a robust estimator of the frontier. The method is tested on the nonparametric model in the univariate approach minimizing the total expenses and the results are compared to the previous method. The too efficient stays hightlighted by the order-m forntier have an higher compact on the efficiencies of the other stays. Moreover the order-m frontier proposes an estimation of a plausible frontier for a given level of severity. In some cases, when there are very few stays for a given level of severity, the order-m frontier fails to hightlight the "too efficient" stays. For these cases, the B-W method offers an interesting alternative. So, a good way of work would be using first the order-m frontier and second the B-W method when there is no stay with at least the same characteristics.

The recent development of wavelet transforms based on multiresolution analysis suggests new techniques for nonparametric function estimation. Indeed, wavelet procedures achieve denoising of a curve and adaptivity through thresholding of the empirical wavelet coefficients. The localization both in space and frequency allows for the wavelet estimators to perform well in regions of high regularity, without deteriorating in the neighborhood of discontinuities. However, standard wavelet estimators are optimal only in presence of equispaced samples.

This thesis proposes some ways to remove this restriction by constructing wavelets which automatically adapt to the design at hand. Within the framework of second-generation wavelets, we use some instances of the lifting scheme to achieve this objective. This automatic adaptation avoids the use of some pre-processing steps, whose aim is to come back to the evenly spaced design case, and which may deteriorate the quality of the final estimator.

In the first part of this thesis, we treat the stochastic and the autoregressive univariate models by using a particular weighted wavelet basis, that is, a wavelet basis of the space weighted by the empirical measure of the regressors. Wavelet bases of this space are called "design-adapted". The approximation properties of an estimator based on a simple, Haar-like, design-adapted basis are investigated, and thereafter this initial basis is improved with the lifting scheme. This leads to smooth design-adapted wavelets. The resulting estimator is tested on simulated and real data sets. Under some conditions on the distribution of the regressors, the rate of decay of the smooth design-adapted wavelet coefficients is the same as in the classical case, and the risk, in the wavelet domain, of a linear wavelet estimator is also the same as in the classical setting. We show which thresholding scheme to use in case of an autoregressive model in order to remove all the noise, and we establish an upper bound for the corresponding l₂-risk in the wavelet domain.

The use of design-adapted wavelets can easily be extended to half-regular grids. By "half-regular", we mean a grid which is the Cartesian product of two irregular one-dimensional grids. In this construction, tensor products of spaces are used. A denoising scheme is proposed, and the resulting estimator is compared with a locally weighted regression procedure. In the last part, several wavelet transforms adapted to an irregular bivariate design are proposed. There, another instance of the lifting scheme has to be utilized. Coupled with a Bayesian denoising scheme, these wavelet transforms provide different wavelet estimators. A simulation study shows their good behaviour even in presence of discontinuities in the regression function.

Our interest is to estimate a density from an i.i.d. sample that has been contaminated by a measurement error. This problem, usually referred to as a deconvolution problem, has applications in many different fields such as astronomy, public health, chemistry, microfluorimetry or electrophoresis. The contaminating density, or error density, is often assumed to be known. In this context, a so-called deconvolution kernel density estimator has been proposed in the literature. This estimator requires the choice of a smoothing parameter called the bandwidth, which plays a crucial role in the estimation process. Despite this fact, very few papers in the literature deal with the problem of how to choose the bandwidth of the deconvolution kernel density estimator in practice. Part one of this thesis aims to provide practical methods of bandwidth selection in this deconvolution problem.

After having presented the cross-validation method of Stefanski and Carroll (1990), we propose several other methods of bandwidth selection that can be used in practice: a method based on exact calculations for the sinus cardinal kernel, a normal reference method, a plug-in procedure, a solve-the-equation bandwidth selector and a bootstrap method. We illustrate each method on some simulated examples, and prove the consistency of the plug-in and bootstrap procedures. Finally, we compare the performances of the various methods via simulated examples and apply the methods on some real data. From our simulation results it appears that the plug-in and bootstrap procedures compete and both outperform the other data-driven bandwidth selection procedures. As a by-product, we study the estimation of integrated squared density derivatives.

Another situation which arises quite often in real life examples is that the domain of variation of the uncontaminated data is not the whole real line: the possible values are between a finite minimum and/or a finite maximum. In the error-free case, we know that the kernel density estimator is not consistent in a finite endpoint of its support. In order to have a consistent estimator of the density of interest in an endpoint, one has to make some modifications on the kernel estimator, that take the support into account. This property extends to the error case, i.e. the deconvolution kernel estimator is not a good estimator at a (finite) boundary point. Hence it is important to be able to estimate the endpoints of the support if these are unknown. This boundary estimation problem also has applications in stochastic frontier estimation.

In the error-free case, various methods have been proposed to estimate the endpoints of a density, but this problem has been studied little in the error case. Part two of this thesis studies the boundary estimation problem for contaminated data. We propose a deconvolving kernel estimator of an endpoint, inspired by methods that exist in the error-free case to estimate a discontinuity point. The idea of the method is to estimate the location of a boundary point by the maximiser of a certain diagnostic function. The methodology can also be used to estimate a discontinuity point. We prove the consistency of the method and give some ideas of how to apply it in practice. We illustrate the performance of the method on some real and simulated examples.

This PhD thesis is concerned with measuring and modelling stochastic dependence between random variables, and the main focus is on discrete random variables. In our study on dependence, concepts such as concordance and copula are widely used.

We first propose a copula­type representation for random couples with Bernoulli margins. Some association 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.

We propose a systematic study of the monotonicity property of Kendall's t and Spearman's r with respect to the concordance ordering of pairs of discrete as well as continuous random variables.
This extends and completes results of Yanagimoto and Okamoto (1969) and Tchen (1980). Analytic expressions are given for the most extreme values of Kendall's t and Spearman's r associated with discrete uniform variables. Some other measures of concordance are also studied. A certain number of examples are used to highlight the drawbacks of some concordance measures noticed for Bernoulli random variables and still present for more general discrete random variables. Corrections for some measures of monotone dependence for discrete random variables are proposed to obtain a margins free range.

We also prove that various relationships between Kendall's t and Spearman's r mentioned in Nelsen (1999) remain valid for discrete variables. In particular, results of Capéraá and Genest (1993) are extended to the case of discrete random pairs. We also establish that some useful stochastic dependence properties used in the actuarial literature are conserved for discrete random variables. In particular some results in Dhaene and Goovaerts (1996) continue to hold for discrete random variables. Furthermore, we propose a copula for discrete random variables using a principle given in Denuit and Lambert (2002) which transforms an integer­valued random variable to a continuous one via convolution with a uniform random variable on [0,1].

Finally, we consider a loglinear model to construct a joint distribution with uniform discrete margins. This new distribution will preserve a local association of the joint distribution with arbitrary margins. The construction is based on a copula­type representation of an n × m table of joint probabilities with fixed margins. The idea arises from the decomposition of the joint probability in the original table into margins and association effects leading to the uniform representation. The technique uses an iterative procedure which allows to preserve local association by means of a set of local odds­ratio. In the continuous case any joint distribution function H, with margins F and G, admits a unique copula representation C, which is the part representing the dependence structure of H. While Sklar's continuous copula C preserves any dependence structure in H, the discrete uniform representation is shown to conserve only local dependence structures and properties.

Decision trees are one of the most widely used tools for "supervised classification" problems: provided a sample of correctly classified data is available, they allow to establish rules for explaining and/or predicting a categorical response (membership to classes) on the basis of observed values of predictive variables. Besides efficiency, this hierarchical method is endowed with many additional advantages. In particular decision trees are very flexible and can handle heterogeneous data types, they are invariant under monotone transformations of the data, they perform internal feature selection as an integral part of the procedure, and they generate solutions allowing qualitative understanding of the generated prediction rules. All these properties, often essential in data mining applications, explain the major place occupied by tree methods in this field.

A plethora of tree algorithms have been developed, most of them being but variants of the original CART and C4.5 algorithms. A large part of academic research has focused on incremental modifications to current machine learning methods, and on the speed up of existing algorithms. The main contribution of this work has been to consider a theoretically sound decision tree classifier, initiated by Devroye, Györfi and Lugosi, and to evaluate its expected performance and efficacy as compared to traditional tree methods by evolving it into a practical tool. It is presented in a larger framework including an overview of related approaches issued from different fields, and potential developments.

Enhanced problems related to the splitting method of ordinary trees include the instability of the solutions, the unnecessarily complex expression of the identified structure or the failure of the classifier when complex interrelations between predictors and class cannot be detected. Hyperrectangular Space Partitioning trees on the other hand are based on a different cutting method, and despite their greedy construction, they have been proven to be consistent without any additional tricks. However their practical implementation is a challenge due the computational burden involved. We explore its specific features, strong and weak points, by means of a massive search working tool, while a two-stage approach has been developed for approximating HSP tree solutions, allowing to tackle moderate dimensional problems from commonly used data bases.

This thesis is devoted to the nonparametric estimation of the class of density functions with a known bounded support. We distinguish two cases of bounded support.

For the first case, the support is [0,1]. For such a class of density functions we consider two estimators: the beta histogram and the beta kernel estimator. Different types of convergence are proved for these estimators: uniform weak consistency and convergence to infinity at the endpoints if the density is unbounded at these points. The rate of convergence of the mean integrated absolute error is also established. The quality of the estimation of the beta histogram is very sensitive to the choice of the bandwidth parameter. In this work we adapt two methods already used in the selection of the optimal choice for the standard kernel, the likelhood cross validation and the least square cross validation. The two estimators are already used in regression estimation. It will be interesting to use them in other nonparametric estimations such as the quantile and the hazard function.

For the second case, the support is [0,¥). For such a class, we have studied the consistency of the gamma histogram, the gamma kernel, the inverse gaussian and the reciprocal inverse gaussian estimators. We give the rate of convergence of the mean integrated absolute error of gamma histogram and gamma kernel estimators. The uniform weak and the uniform strong convergence is also proved. When the density is unbounded at the origin we proved the weak convergence to infinite under some conditions on the density function f of gamma histogram at this points. All the results proved for these estimators are generalised to the generalised kernel estimators. Simulations showed that the gamma histogram is better than the first gamma kernel estimator. We applied the gamma histogram and gamma kernel estimators on income data. The estimation requires a choice of an optimal bandwidth. Here the bandwidth selection method used leads to an asymptotically optimal window in the sense of minimising L₁ distance. The practical choice of the bandwidth parameter for the gamma histogram and gamma kernel estimators is an important problem.