We consider in this paper goodness of fit tests of the null hypothesis that the underlying distribution function of a sample $F(x)$ belongs to a given family of distribution functions $\scr F$. We propose a method for deriving approximate values of the power of a weighted Cramér-von Mises type test of goodness of fit. Our method relies on Karhunen-Loève expansions on $(0,1)$ for the weighted Brownian bridges.

In this work we present some results on the optimality of the empirical distribution function as an estimator of the invariant distribution function of an ergodic diffusion process. The results presented were obtained in different previous works under conditions that are rewritten in a unified form that make those results comparable. It is well known that the empirical distribution function is an unbiased and uniformly consistent estimator for the invariant distribution function of an ergodic diffusion process. It is also an efficient estimator in the sense that the risk of this estimator attains an asymptotic minimax lower bound. In this paper we review some results on the problem of the efficiency of the empirical distribution function considering three types of risk function. The first one is in a semi-parametric set-up. The second one is the integrated mean square error and the third is based on the sup norm.

Ce travail est consacré aux problèmes d’estimation pour différents modèles de processus de Poisson non homogènes. Nous supposons que la fonction d’intensité du processus de Poisson est discontinue par rapport aux paramètres inconnus. On montre que l’estimateur de la distance minimale est consistant et asymptotiquement normal. Des simulations sont faites pour chaque modèle.

We consider several problems of parameter estimation by observations of different models of inhomogeneous Poisson processes of discontinuous intensity functions. It is shown that the minimum distance estimators of these parameters are consistent and asymptotically normal. The numerical simulation results are presented as well.

Our study is focused on the research of a coding that would keep as best as can be the real configuration of a questionnaire containing ordered categorical variables which are graded according to approbation. The piecewise barycentric codings with picewisely acting pivots and an active zero describes both an improvement and and a generalisation of the personal equation coding keeping the order of given rating. As we have given to the middle pivot a distinct action on the different belonging functions for the four manners of response, we have established a more refined coding which distinguishes the ways to seperate the the initial data x(i,j) on the intervals [-a,0] and [0,b]. Hence, the re-creating formula of the initial data are more precise. The acting zero leads to an infinite numbers of coding characterized by homotopical and homologous ways.

Notre étude se focalise sur la recherche d’un codage conservant au mieux les configurations réelles d’un questionnaire clos à variables qualitatives ordonnées sur une échelle d’approbation. Les codages barycentriques linéaires par morceaux à pivots agissant par morceaux avec zéro actif traduisent une amélioration et une généralisation du codage suivant l’équation personnelle conservateur de l’ordre des modalités. Ayant conféré au pivot central zéro une action distincte sur les différentes fonctions d’appartenance aux quatre modalités, nous avons établi un codage plus fin distinguant les manières d’éclater les notes recadrées x(i,j) sur les intervalles [-a,0] et [0,b]. Ainsi les formules de reconstitution des données initiales se sont révélées plus précises. Le zéro actif fait apparaître une infinité de codages qui sont déterminés par des chemins homotopes et homologues.

Standard Monte Carlo simulation needs prohibitive time to achieve reasonable estimations. for untractable integrals (i.e. multidimensional integrals and/or intergals with complex integrand forms). Several statistical technique, called variance reduction methods, are used to reduce the simulation time. In this note, we propose a generalization of the well known antithetic variate method. Principally we propose a $K$−antithetic variate estimator (KAVE) based on the generation of $K$ correlated uniform variates. Some numerical examples are presented to show the improvenment of our proposition.