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

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 form W I CN = n), (1) where ζ = 6>···>6ν is a r.vec. with independent non-negative integer-valued components, (N = Em=iU and Ρ{ζΝ = 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 fm(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 fm(y), m = 1, . , TV, are random variables (r. v.) for each fixed y > 0. In the investigations of RDS we usually suppose that f\(y\\ . . . , /TVU/W) f° ^Y given 2/1,. ,y;v are independent r.v. which in turn are independent of £ and η. Further the symbols Em> nm denote respectively the sum and the product over m from 1 to TV, and d, C are positive absolute constants which in general are different in different places. We will use the following notations: Am = E£m, Bff = Em D£m, χ = z(n, TV) = (n im),fm), AN = 9m = <7m(£m) = /m(6n) ~ E/ <? N = Em D0m = Em D/m(fm) 9m = 9m/ W, |m = (fm ~ \m)/BN, ΦτηΛΓ^, T) = ΦΝ(*, r) = Hm «m*(t, τ), ν>Ν(0 *UDC 519.2. Originally published in Diskretnaya Matematika (1991) 3, No. 3, 73-88 (in Russian). Translated by G. I. Ivchenko. 548 G. I. Ivchenko and Sh. A. Mirakhmedov From (1) we obtain the following representation of the characteristic function (ch. f.) φκ(ί) of normalized statistic RN^)N „} \τ\<ττΒΝ The asymptotic distributions of RDS were investigated in [2-7] with the use of this formula. In particular in [4,6] the central limit theorem (c.l.th.) was obtained for Ηκ(η) with the estimation of the remainder term in it; in [6] the asymptotic expansion (a.e.) in the local limit theorem (1.1. th.) for the r.v. Ηχ(η) was also found in the case of a lattice distribution (see Theorem 1 without the multiplier (1 + \ub\) below), probabilities of large deviations for RN&) were investigated in [5], [7]. New theorems on a. e. of the distributions of RDS Κκ(η) are obtained in the present paper, and applications of these results and the limit theorems from [4-7] to the analysis of efficiency of statistical tests in a multinomial scheme are discussed. 2. Formula (2) determines the construction of a.e. for ch. f. <pn(t). Namely the integrand function Φ;ν(^ )> which is the ch. f. of a sum of independent two-dimensional r. vec. , is replaced formally by its known a.e. (see, for example [8], Sections 7-8]) which is then integrated over r. Further we obtain the a.e. for \/2πΒΝ? {ζΝ = η}. Finally the a.e. for y? w(i) is obtained as the result of division of the previous expansions. In the concluding a.e. there appear specific terms that distinguish it from the analogous a.e. for the sums of independent r. v. Let us introduce necessary additional notations: = Em E|£m|>, jN = Em E|0m|', <*ijN = Em E & , 117 / X / Ν Λ * ~ 3U ( ~ *) (1 ~ U)* \̂ WN(u) = φ(η) l l -f — — «30^ -GLUM -2α21Ν ] ' f ( 1 — y 1 — x xy \ = J WN(u)au = $(y) + (p(y) ( — — ay>N + —^—aUN γ2ΐΛτ 1 , -00 ^ ' be the standard normal distribution function, (p(y) = Φ'(ί/), (α) be the distance between a e R and the nearest integer, X* be the symmetrization of the r.v. X,