A Bayesian Regularization Procedure for a Better Extremal Fit

In Structural Reliability, special attention is devoted to model distribution tails. This is important when one wants to estimate the occurrence probability of rare events as critical failures, extreme charges, resistance measures, frequency of stressing events, etc. People try to find distribution models having a good overall fit to the data. Particularly, the distributions are strongly required to fit the upper observations and provide a good picture of the tail above the maximal observation. Specific goodness-of-fit tests such as the ET test can be constructed to check this tail fit. Then what can we do with distributions having a good central fit and a bad extremal fit ? We propose a regularization procedure, that is to say a procedure which preserves the general form of the initial distribution and allows a better fit in the distribution tail. It is based on Bayesian tools and takes the opinion of experts into account. Predictive distributions are proposed as model distributions. They are obtained as a mixture of the model family density functions according to the posterior distribution. Therefore, they are rather smooth and can easily be simulated. We numerically investigate this method on normal, lognormal, exponential, gamma and Weibull distributions. Our method is illustrated on both simulated and real data sets

The Half-sample Method for Testing Parametric Regressive and Autoregressive Models of Order 1

The half-sample method has been introduced by Stephens (1978) for testing parametric models of distribution functions. In this paper, we present a similar method for testing the goodness-of-fit of linear or nonlinear regression or autoregression functions for parametric models of order $1$, under minimal stationarity and ergodicity assumptions. Our procedure is based on a measure of the cumulated deviation process $\hat{A}_{n}$ between a weighted marked process of residuals and a parametric estimator of the cumulated conditional mean function (i.e.\ cumulated regression or autoregression function), under the null hypothesis $H_{0}$. We establish a functional limit theorem under $H_{0}$ for a variant $\hat{A}^{(\kappa)}_{n}$, $0 < \kappa \le 1$, of the process $\hat{A}_{n}$. The half-sample method corresponds to $\kappa = 1/2$. We show that the limiting distribution of $\hat{A}^{(1/2)}_{n}$ under $H_{0}$ takes a very simple form. Several easily implemented goodness-of-fit tests can be based on this result. We provide simple conditions under which their power converges to 1 as the sample size goes to $\infty$. Finally, we investigate the asymptotic behavior of $\hat{A}^{(\kappa)}_{n}$ as $n \to \infty$ under sequences of $O(n^{-1/2})$ local alternatives. This allows us to compare the corresponding local powers of tests based on $\hat{A}^{(1/2)}_{n}$ and on $\hat{A}^(1)_{n}$

A stochastic approximation type EM algorithm for the mixture problem

Projet CLORECRésumé disponible dans le fichier PD

On the Convergence of the ET Method for Extreme Upper Quantile Estimation

We examine the consistency of the Exponential Tail (ET) nonparametric method forestimating extreme quantiles of an unknown distribution. We show that, in general, the consistency of ET imposes strong limitations on the rate of convergence to $0$ of the estimated quantile order

Reconnaissance de mélange de densité et classification.Un algorithme d'apprentissage probabiliste : l'algorithme SEM

Projet SYSTO

Une version de type recuit simule de l'algorithme EM

L'algorithme EM est tres repandu pour l'estimation par le maximum de vraisemblance de parametres de modeles ou les donnees sont incompletes. Nous presentons une version de type recuit simule de l'algorithme EM. Cet algorithme, designe algorithme SAEM est une adaptation de l'algorithme stochastique SEM que nous avons precedemment developpe. Comme ce dernier, l'algorithme SAEM repond aux limitations bien connues de l'algorithme EM ; mais de plus il se comporte mieux pour traiter de petits echantillons. Par ailleurs, il est plus simple a apprehender que l'algorithme SEM dans la mesure ou il converge presque surement tandis que l'algorithme SEM converge en loi. Ici, on limite la presentation detaillee de l'algorithme SAEM au probleme des melanges de lois de probabilite. On etablit un theoreme qui assure que toute suite d'estimes par SAEM converge p.s. vers un maximum local de la fonction de vraisemblance. On conclut cet article par une etude comparative du comportement pratique des 3 algorithmes EM, SEM et SAEM