Patrice Bertail

On Subsampling Estimators with Unknown Rate of Convergence

Abstract Politis and Romano have put forth a general subsampling methodology for the construction of large-sample confidence regions for a general unknown parameter θ associated with the probability distribution generating the stationary sequence X 1,…,X n . The subsampling methodology hinges on approximating the large-sample distribution of a statistic T n = T n (X 1,…, X n ) that is consistent for θ at some known rate τ n . Although subsampling has been shown to yield confidence regions for θ of asymptotically correct coverage under very weak assumptions, the applicability of the methodology as it has been presented so far is limited if the rate of convergence τ n happens to be unknown or intractable in a particular setting. In this article we show how it is possible to circumvent this limitation by (a) using the subsampling methodology to derive a consistent estimator of the rate τ n , and (b) using the estimated rate to construct asymptotically correct confidence regions for θ based on subsampling.

Fruit and Vegetable Consumption Patterns: A Segmentation Approach

In this article, we study the heterogeneity of fruit and vegetable consumption patterns in France. A finite mixture of AIDS models is used to describe food demand patterns revealing different preferences over distinct classes. We obtained six different clusters, which reflect specific socio-demographic characteristics and different income and price elasticities. This approach is appropriate for targeting specific public nutritional policies. Our main results show that unlike the other clusters in which the usual price and income policy tools may be used, the lowest income cluster with the lowest consumption, remains insensitive to economic variables.

Sharp Bounds for the Tails of Functionals of Markov Chains

This paper is devoted to establishing sharp bounds for deviation probabilities of partial sums

Subsampling the distribution of diverging statistics with applications to finance

\sum_{i=1}^{n}f(X_{i})

Approximate regenerative-block bootstrap for Markov chains

, where

Regeneration-based Statistics for Harris Recurrent Markov Chains

X=(X_{n})_{n\in{\bf N}}

A Renewal Approach to Markovian U-statistics

is a positive recurrent Markov chain and f is a real valued function defined on its state space. Combining the regenerative method to the Esscher transformation, these estimates are shown in particular to generalize probability inequalities proved in the independent i.i.d. case to the Markovian setting for (not necessarily uniformly) geometrically ergodic chains.

Statistical analysis of a dynamic model for dietary contaminant exposure

In this paper we propose a subsampling estimator for the distribution of statistics diverging at either known or unknown rates when the underlying time series is strictly stationary and strong mixing. Based on our results we provide a detailed discussion of how to estimate extreme order statistics with dependent data and present two applications to assessing financial market risk. Our method performs well in estimating Value at Risk and provides a superior alternative to Hills estimator in operationalizing Safety First portfolio selection.

Scaling up M-estimation via sampling designs: The Horvitz-Thompson stochastic gradient descent

The (approximate) regenerative block-bootstrap for bootstrapping general Harris Markov chains has recently been developed. It is built on the renewal properties of the chain, or of a Nummelin extension of the latter. It has theoretical properties that surpass other existing methods within the Markovian framework. The practical issues related to the implementation of this specific resampling method are discussed. Various simulation studies for investigating its performance and comparing it to other bootstrap resampling schemes, standing as natural candidates in the Markov setting are presented.

Empirical φ ∗ -Divergence Minimizers for Hadamard Differentiable Functionals

In this paper an attempt is made to present how renewalproperties of Harris recurrent Markov chains or of specific extensions of thelatter may be practically used for statistical inference in various settings.In the regenerative case, procedures can be implemented from data blockscorresponding to consecutive observed regeneration times for the chain. Themain idea for extending the application of these statistical techniques to generalHarris chains X consists in generating first a sequence of approximaterenewal times for a regenerative extension of X from data X1; :::; Xn and theparameters of a minorization condition satisfied by its transition probabilitykernel. Numerous applications of this estimation principle may be consideredin both the stationary and nonstationary (including the null recurrentcase) frameworks. This article deals with some important procedures basedon (approximate) regeneration data blocks, from both practical and theoreticalviewpoints, for the following topics: mean and variance estimation,confidence intervals, U-statis