Michel Broniatowski

Parametric estimation and tests through divergences and the duality technique

We introduce estimation and test procedures through divergence optimization for discrete or continuous parametric models. This approach is based on a new dual representation for divergences. We treat point estimation and tests for simple and composite hypotheses, extending the maximum likelihood technique. Another view of the maximum likelihood approach, for estimation and tests, is given. We prove existence and consistency of the proposed estimates. The limit laws of the estimates and test statistics (including the generalized likelihood ratio one) are given under both the null and the alternative hypotheses, and approximations of the power functions are deduced. A new procedure of construction of confidence regions, when the parameter may be a boundary value of the parameter space, is proposed. Also, a solution to the irregularity problem of the generalized likelihood ratio test pertaining to the number of components in a mixture is given, and a new test is proposed, based on @g^2-divergence on signed finite measures and the duality technique.

Dual divergence estimators and tests: Robustness results

The class of dual @f-divergence estimators (introduced in Broniatowski and Keziou (2009) [5]) is explored with respect to robustness through the influence function approach. For scale and location models, this class is investigated in terms of robustness and asymptotic relative efficiency. Some hypothesis tests based on dual divergence criteria are proposed and their robustness properties are studied. The empirical performances of these estimators and tests are illustrated by Monte Carlo simulation for both non-contaminated and contaminated data.

The mean residual life function at great age: Applications to tail estimation

Abstract The limit behaviour of the mean residual life function of a distribution gives important information on the tail of that distribution. In this paper this is shown through new Abelian- and Tauberian-type results on the transform linking both distribution function and mean residual life function. We use these analytic results to derive tail heaviness and extreme quantile estimators. Some basic asymptotic results for these estimators are given.

On the estimation of the Weibull tail coefficient

Abstract Let X1, …, Xn be a sample from the distribution F(x)=P(X⩽x), such that 1−F(x)=exp(-x 1 α l(x)) , where l(·) is slowly varying at infinity and αϵ(0,∞). We introduce an estimate of the Weibull tail coefficient α based on a weighted average of the kn upper order statistics of the sample where 1⩽kn⩽n is an integer sequence such that n-1kn→0 as n→∞. We prove the consistency of our estimate under general assumptions, as well as its asymptotic normality. We also discuss the optimal choice of kn and present simulation results.

Long runs under a conditional limit distribution.

This paper presents a sharp approximation of the density of long runs of a random walk conditioned on its end value or by an average of a functions of its summands as their number tends to infinity. In the large deviation range of the conditioning event it extends the Gibbs conditional principle in the sense that it provides a description of the distribution of the random walk on long subsequences. Approximation of the density of the runs is also obtained when the conditioning event states that the end value of the random walk belongs to a thin or a thick set with non void interior. The approximations hold either in probability under the conditional distribution of the random walk, or in total variation norm between measures. Application of the approximation scheme to the evaluation of rare event probabilities through Importance Sampling is provided. When the conditioning event is in the zone of the central limit theorem it provides a tool for statistical inference in the sense that it produces an effective way to implement the Rao-Blackwell theorem for the improvement of estimators; it also leads to conditional inference procedures in models with nuisance parameters. An algorithm for the simulation of such long runs is presented, together with an algorithm determining the maximal length for which the approximation is valid up to a prescribed accuracy.

Simulation in exponential families

An acceptance-rejection algorithm for the simulation of random variables in statistical exponential families is described. This algorithm does not require any prior knowledge of the family, except sufficient stati stics and the value of the parameter. It allows simulation from many members of the exponential family. We present some bounds on computing time, as well as the main properties of the empirical measures of samples simulated by our methods (functional Glivenko-Cantelli and central limit theorems). This algorithm is applied in order to evaluate the distribution of M-estimators under composite alternatives; we also propose its use in Bayesian statistics in order to simulate from posterior distributions.

On Sharp Large Deviations for Sums of Random Vectors and Multidimensional Laplace Approximation

Let

Extended large deviations

X, X_i,i\ge 1

Note on Functional Large Deviation Principle for Fractional ARIMA Processes

, be a sequence of independent and identically distributed random vectors in

A self-adaptive technique for the estimation of the multifractal spectrum

{\bf R}^d