Philippe Barbe

Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications

We establish some asymptotic expansions for infinite weighted convolution of distributions having regular varying tails. Various applications to statistics and probability are developed.

Asymptotic expansions of convolutions of regularly varying distributions

In this paper we derive precise tail-area approximations for the sum of an arbitrary finite number of independent heavy-tailed random variables. In order to achieve second-order asymptotics, a mild regularity condition is imposed on the class of distribution functions with regularly varying tails. Higher-order asymptotics are also obtained when considering a semiparametric subclass of distribution functions with regularly varying tails. These semiparametric subclasses are shown to be closed under convolutions and a convolution algebra is constructed to evaluate the parameters of a convolution from the parameters of the constituent distributions in the convolution. A Maple code is presented which does this task.

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.

Statistical Analysis of Mixtures and the Empirical Probability Measure

We consider the problem of estimating a mixture of probability measures in an abstract setting. Twelve examples are worked out, in order to show the applicability of the theory.

Note on Functional Large Deviation Principle for Fractional ARIMA Processes

Let Zt,t=1,2,... be a discrete time fractional ARIMA process. We prove a large deviation result for the process r→Z[Tr], 0≤r≤1 when T→∞. This result is the basic tool to obtain Chernoff and Bahadur efficiencies for such models.

Limiting distribution of the maximal spacing when the density function admits a positive minimum

Let X1, X2,... be a sequence of random variables with common distribution F, and let f be the density function of F. Let X1,n [less-than-or-equals, slant] ... [less-than-or-equals, slant] Xn,n be the order statistics of X1,..., Xn and let Mn = max 2 [less-than-or-equals, slant] i [less-than-or-equals, slant] nXi,n - Xi-1,n be the maximal spacing. We assume that f has a positive minimum in x0 and that f(x0 + h) = f(x0) + hrd sgn(h)(1 + o(1)) when h --> 0. We prove that limn-->[infinity]P[nMn [less-than-or-equals, slant] x + an] = exp(-e-[phi]x) where [phi] = f(x0 and an = [phi]-1 log n - [phi]-1r-1 log log n + [phi]-1 log(r-1d-1/r[Gamma](1/r)[phi]1/r).

A conditional limit theorem for a bivariate representation of a univariate random variable and conditional extreme values

We first consider a real random variable X represented through a random pair (R,T) and a deterministic function u as X = R⋅u(T). Under quite weak assumptions we prove a limit theorem for (R,T) given X>x, as x tends to infinity. The novelty of our paper is to show that this theorem for the representation of the univariate random variable X permits us to obtain in an elegant manner conditional limit theorems for random pairs (X,Y) = R⋅(u(T),v(T)) given that X is large. Our approach allows to deduce new results as well as to recover under considerably weaker assumptions results obtained previously in the literature. Consequently, it provides a better understanding and systematization of limit statements for the conditional extreme value models.

Joint approximation of processes based on spacings and order statistics

Let [omega]1, [omega]2, ... be a sequence of i.i.d. r.v. with E[omega]1 [not equal to] 0 and Var [omega]i = 1. Under some weak conditions on the distribution of the [omega]is, we give a joint approximation of the empirical process corresponding to and the empirical process corresponding to the spacings empirical distribution function . We apply this result to show that a large class of statistics based on spacings and order statistics of a uniform sample are asymptotically independent, leading to some improved goodness of fit tests.

A Critical Reanalysis of Maryland State Police Searches

This article argues that previous analyses of the Maryland State Police search data may be unreliable, since nonstationarity of these data precludes the use of standard statistical inference techniques. In contrast, proper statistical graphics seem better suited to capture the complexities of the racial bias issue.

Bootstrapping the renewal spacings processes

Abstract We define the generalized bootstrapped version of the empirical and quantile renewal spacing processes. We show that the asymptotic theory of the renewal spacings processes holds for the bootstrap version.