On Asymptotic Expansion in the Random Allocation of Particles by Sets

We consider a scheme of equiprobable allocation of particles into cells by sets. The Edgeworth type asymptotic expansion in the local central limit theorem for a number of empty cells left after allocation of all sets of particles is derived.Comment: 15 page

The Probabilities of Large Deviations Associated with Multinomial Distributions

We consider a multinomial distribution, where the number of cells increases and the cell-probabilities decreases as the number of observations grows. The probabilities of large deviations of statistics, which has form of a sum of Borel functions of cell-frequencies, are studied. Large deviation results for the power-divergence statistics and its most popular special variants, as well as for some count statistics are derived as consequences of general theorems

The probabilities of large deviations for a certain class of statistics associated with multinomial distribution

Let η = (η1, …, ηN) be a multinomial random vector with parameters n = η1 + ⋯ + ηN and pm > 0, m = 1, …, N, p1 + ⋯ + pN = 1. We assume that N →∞ and maxpm → 0 as n →∞. The probabilities of large deviations for statistics of the form h1(η1) + ⋯ + hN(ηN) are studied, where hm(x) is a real-valued function of a non-negative integer-valued argument. The new large deviation results for the power-divergence statistics and its most popular special variants, as well as for several count statistics are derived as consequences of the general theorems

On the intermediate asymptotic efficiency of goodness-of-fit tests in multinomial distributions

We consider goodness-of-fit tests for uniformity of a multinomial distribution by means of tests based on a class of symmetric statistics, defined as the sum of some function of cell-frequencies. We are dealing with an asymptotic regime, where the number of cells grows with the sample size. Most attention is focused on the class of power divergence statistics. The aim of this article is to study the intermediate asymptotic relative efficiency of two tests, where the powers of the tests are asymptotically non-degenerate and the sequences of alternatives converge to the hypothesis, but not too fast. The intermediate asymptotic relative efficiency of the χ2 test wrt an arbitrary symmetric test is considered in details

Edgeworth expansions for two-stage sampling with applications to stratified and cluster sampling

A two-term Edgeworth expansion for the standardized version of the sample total in a two-stagesampling design is derived. In particular, for the commonly used stratified and cluster sampling schemes,formal two-term asymptotic expansions are obtained for the Studentized versions of the sample total. Theseresults are applied in conjunction with the bootstrap to construct more accurate confidence intervals for theunknown population total in such sampling schemes.Les auteurs pr´esentent un d´eveloppement en deux termes d’une s´erie d’Edgeworth pour l’estimateurdu total bas´e sur un plan ´echantillonnal `a deux phases. Ils obtiennent en particulier des d´eveloppementsasymptotiques formels pour la version studentis´ee du total ´echantillonnal bas´e sur un ´echantillonnage stratifi´eet sur un ´echantillonnage en grappes. Les auteurs utilisent ces r´esultats et des m´ethodes de r´e´echantillonnageafin de construire des intervalles de confiance plus pr´ecis pour le total de la population dans le contexte deces plans d’´echantillonnage.Statistical methods for ecological research on data from national monitoring programs. Funded by the Swedish Research Council. Grant Number 340-2013-5076

A Cramér-type large deviation theorem for sums of functions of higher order non-overlapping spacings

Cramér condition, Exponential distribution, Uniform spacings, Uniform distribution, Large deviation,