On limit theorems for decomposable statistics and efficiency of the corresponding statistical tests

Abstract

New results on asymptotic expansions in the central limit theorem for decomposable statistics in a generalized scheme of random allocations are obtained, and applications of these results to the analysis of asymptotic efficiency of statistical tests in the multinomial scheme are discussed.1. Let η = T/i, . . . , η Ν be a random vector (r. vec.) with the distribution of the formwhere ζ = 6>···>6ν is a r.vec. with independent non-negative integer-valued components, (N = Em=iU and Ρ{ζ Ν = η} > 0. The r.vec. η defines a generalized scheme of a random allocation of n particles into N cells [1], In applications some of the most important statistics based on the r.vec. η can be represented in the form A*0i) = Σ /»(»?»). m = l where f m (y\ m = 1 } ...,TV, are functions of the integer-valued non-negative argument. Such statistics are called decomposable statistics (DS) if the functions / m (y), m = 1, . . . , Ν, are not random, and they are called randomized DS (RDS) if the functions f m (y), m = 1, . . . , TV, are random variables (r. v.) for each fixed y > 0.In the investigations of RDS we usually suppose that f\(y\\ . . . , /TVU/W) f°r ^Y given 2/1,... ,y;v are independent r.v. which in turn are independent of £ and η. Further the symbols Em> n m denote respectively the sum and the product over m from 1 to TV, and d, C are positive absolute constants which in general are different in different places.We will use the following notations: A m = E£ m , Bff = Em D£ m , χ = z(n, TV) = (nim),f m ), A N = 9m = <7m(£m) = /m(6n) ~ E/ <?N = Em D0m = Em D/m(fm) -9m = 9m/ W, |m = (fm ~ \m)/BN, ΦτηΛΓ^, T) = ΦΝ(*, r) = Hm «m*(t, τ), ν>Ν(0

Higher-order expansions and efficiencies of tests based on spacings

“…These results show that the well-known phenomenon of first-order efficiency generally implying second-order efficiency (see, e.g., Bickel et al 1981) does not hold true here. This effect arises mainly because we are dealing with the case when the step of the spacings s depends on the sample size n. This is similar to the phenomenon that has been observed for tests based on grouped data when the number of groups increases along with the sample size (see, e.g., Kallenberg 1985;Ivchenko and Mirakhmedov 1992). Downloaded by [Professor S. Rao Jammalamadaka] at 09:40 13 February 2013…”

On Edgeworth Expansions in Generalized Urn Models

“…There is extensive literature on DSs, much of it related to sampling with and without replacement from a finite population. We specifically mention a few: Mirakhmedov (1996) obtains a bound for the remainder term in CLT and Cramer's type large deviation result for a special class of GUM; Mirakhmedov (1992) and Ivchenko and Mirakhmedov (1992) consider a two-term expansion with applications to some special cases of DS in a multinomial scheme under somewhat restrictive conditions; Babu and Bai (1996) obtain Edgeworth expansion for mixtures of global and local distributions --results that can be used when the DS is a linear function of frequencies and a GUM is defined by identically distributed r.v.s m . Such results are clearly very restrictive on the parameters of the urn model and on the kernel functions , mN f .…”

Decomposable statistics in inverse urn problems

“…[5], where the corresponding references are also cited). Let us emphasize that the number of trials in model (1) is fixed, and such problems are known as the direct ones.…”