Lucien Le Cam

Convergence of stochastic empirical measures

Let Pn be a random probability measure on a metric space S. Let P^n be the empirical measure of kn iid random variables, each distributed according to Pn. Our main theorem asserts that if {Pn} converges in distribution, as random probability measures on S, then so does {P^n}. Applications of the result to the study of bootstrap and other stochastic procedures are given.

On the standard asymptotic confidence ellipsoids of Wald

En 1943 Wald avait demontre que certaines formes quadratiques du maximum de vraisemblance ont des proprietes asymptotiques optimales. Toutefois plusieurs auteurs ont donne des donne des exemples ou la technique de Wald conduit a des resultats evidemment mauvais. On proprose une methode basee sur la distance de Hellinger qui evite certaines des difficultes et qui est asymptotiquement equivalente a celle de Wald sous les hypotheses faites par cet auteur. La difference entre les deux methodes est illustree par un exemple de Vaeth et par un exemple gaussien heteroschedastique

An Infinite Dimensional Convolution Theorem

The classical Hajek-LeCam convolution theorem assumes that the underlying parameter space is a locally compact group. Extensions to Hilbert spaces with Gaussian measures were given by Moussatat, Millar and von der Vaart. We propose an extension covering cylinder measures subject to a domination restriction on their finite dimensional projections. The proof is complex and leaves open a number of problems.

ON THE RISK OF BAYES ESTIMATES

Publisher Summary This chapter discusses the behavior of Bayes estimates for the case of independent observation that are not necessarily identically distributed. The parameter space Θ is metrized by a certain Hilbertian distance H obtainable from Hellinger distances on component spaces. It is assumed that (Θ, H) satisfies suitable dimensionality restrictions. It is shown that if a prior measure μ is sufficiently spread out, the corresponding Bayes estimates satisfy inequalities of the type E [H2 ( , where C (D, μ) is a number depending only on the dimension of the space Θ and on certain features of the measure μ. The chapter presents some recent results of L. Birge on tests between Hellinger balls and products of Hellinger balls. These tests and the inequalities are used to obtain bounds on the risk of Bayes procedures. It also contains a construction of prior measures with properties adapted to the present problem.

Locally Asymptotically Normal Families

The classical theory of asymptotics in Statistics relies heavily on certain local quadratic approximations to the logarithms of likelihood ratios. Such approximations will be studied here but in a restricted framework.

On the asymptotic behavior of mixtures of poisson distributions

for a certain random variable T, whose distribution is allowed to vary with n in a fairly arbitrary fashion. Let P, be the product measure joint distribution of(X 1, 9 9 9 X,) for the Poisson case and let Q, be the corresponding measure for the mixture case. We give necessary and sufficient conditions for the contiguity of the sequences {P,} and {Q,}. In this contiguous case we characterize the possible limiting distributions , dQ. for the logarithm of likelihood ratio A, = Jog ~-~-p. More precisely we show that the

Stochastic models of lesion induction and repair in yeast

Abstract In the past decade several stochastic models for the effects of radiation on cell survival have been proposed. We survey them briefly and consider their possible application to some experimental results of Frankenberg-Schwager and coauthors on irradiated yeast. One possible model is a slight modification of the model proposed by Yang and Swenberg. It is shown that the modified model does not actually fit well and that the repair mechanism requires additional complications for adequate description.

Gaussian Shift and Poisson Experiments

This chapter is an introduction to some experiments that occur in their own right but are also very often encountered as limits of other experiments. The gaussian ones have been used and overused because of their mathematical tractability, not just because of the Central Limit Theorem. The Poisson experiments cannot be avoided when one models natural phenomena. We start with the common gaussian experiments. By “gaussian” we shall understand throughout they are the “gaussian shift” experiments, also called homoschedastic. There is a historical reason for that appellation: In 1809 Gauss introduced them (in the one-dimensional case) as those experiments where the maximum likelihood estimate coincides with the mean of the observations. This fact, however, will not concern us.

Local Asymptotic Normality

The classical theory of asymptotics in statistics relies heavily on certain local quadratic approximations to the logarithms of likelihood ratios. Such approximations will be studied here but in a restricted framework.

Limit Laws for Likelihood Ratios

In this chapter we shall consider a double sequence { e n,j ; j = 1, 2, …; n = 1, 2, … } of experiments e n,j = { p t,n,j ; t ∈ Θ }. Let be e n the direct product in j of the hat is, n consists of performing the independently of each other. The measures that constitute e n are the product measures Pt,n = IIj; Pt,n,j • It will usually be assumed that j runs through a finite set.