Local asymptotic types

The local theory of asymptotic types is elaborated. It appears as coordinate-free version of part of Gohberg-Sigals theory of the inversion of finitely meromorphic, operator-valued functions at a point.Mathematics Subject Classification(2000): Primary 47A56; Secondary 06A11, 47L15     

On optimal asymptotic tests of composite hypotheses with several constraints

Summary A locally asymptotically most powerful test for composite hypotheses with t independent linear constraints on the parameters has been developed for the case where the observed random variables {X _nk , k=1,2,...,n} are independently but not necessarily identically distributed. However, their distributions depend on two vector parameters, one = ( ₁, ₂, ..., _t) being under test, and the other = ( ₁, ₂, ..., _s) being the nuisance parameter.This investigation was supported (in part) by a research grant (No. GM-10525(2)) from the National Institutes of Health, Public Health Service.On leave from UniversitÄt Heidelberg, Germany, and supported by a NATO research scholarship.     

Local asymptotic normality of families of Gaussian distributions

In the paper, in terms of spectral language, one finds conditions guaranteeing the LAN property.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 136, pp. 13–26, 1984.     

Equicontinuous families of Markov operators in view of asymptotic stability

The relation between the equicontinuity – the so-called e-property – and the stability of Markov operators is studied. In particular, it is shown that any asymptotically stable Markov operator with an invariant measure such that the interior of its support is non-empty satisfies the e-property.     

Characterization of asymptotic distribution functions of ratio block sequences

In this paper we give necessary and sufficient conditions for the block sequence of the set X = {x ₁ < x ₂ < … < x _n < …} ⊂ ℕ to have an asymptotic distribution function in the form x ^λ.     

An asymptotic minimax testing of nonparametric hypotheses on the density of the distribution of an independent sample

One investigates problems of testing the hypothesis of the distribution density of an independent homogeneous sample and of the homogeneity of two such samples.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 136, pp. 74–96, 1984.     

Optimal convergence rates and asymptotic efficiency of point estimators under truncated distribution families

For regular and irregular truncated distribution families, the optimal convergence rates of consistent point estimators have been found and the corresponding asymptotic efficiencies established. Also, it has been justified that commonly used estimators are all efficient. The efficiencies here are compared to the efficiencies of asymptotically median unbiased estimators, providing a lot of counter estimator examples such that those estimators are efficient in the former sense, but not in the latter.     

达到渐近GV界的一种线性映射族构造方法——生成Shannon好码渐近序列的新进展

给定有限域F_q(q≥2)、任意正整数n和k(n>k),F_q上的线性映射序列(代数族){σ_i}(σ_i:F_q^k→F_qⁿ(i→∞))的构造方法已经成为信息科学中编码理论的一个中心问题,一般称为实现Shannon理想的代数族途径.迄今为止,发现这种代数族{σ_i}的更好结构,并由它导出Shannon好码渐近序列{[n_i,k_i,d_i]}仍是一个尚未彻底解决的挑战性难题和持续不断的努力目标.衡量这种代数族的好坏,除了看{σ_i}的构造是否有利于通信工程实现(构造简明,执行复杂度低)之外,最重要的一个基本标准是看{σ_i}导出的序列{[n_i,k_i,d_i]}诸参数的渐近极限结果是否不至于衰减到渐近GilbertVarshamov(GV)界之下,该问题吸引了许多数学工作者的关注.本文从矩阵映射的观点给出一种生成任意有限域F_q上代数族{σ_i}的新方法,并表明由{σ_i}导出的渐近码序列{[n_i,k_i,d_i]}可达渐近GV界之上.这种新的代数族生成途径对于信息编码理论及其工程应用都具有很重要的意义.     

Characterization of the simple cubic multivariate exponential families

The Letac-Mora class of real cubic natural exponential families has been characterized by a property of 2-orthogonality of an associated sequence of polynomials (see [G. Letac, M. Mora, Natural real exponential families with cubic variance functions, Ann. Statist. 18 (1990) 1-37; A. Hassairi, M. Zarai, Characterization of the cubic exponential families by orthogonality of polynomials, Ann. Probab. 32 (2004) 2463-2476]). The present paper introduces a notion of transorthogonality for a sequence of polynomial on Rd to extend the characterization to the multivariate version of the Letac-Mora class of real natural exponential families.     

Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types

The goal of this paper is to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space ${\mathbb{H}^3}$ . Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known B?r?czky?CFlorian density upper bound for ??congruent horoball?? packings of ${\mathbb{H}^3}$ remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.     

Characterization of distributions by the local asymptotic optimality property of tests statistics

Let X₁, X₂,... be a sequence of independent, identically distributed random variables with density f(x–), R¹. We consider the problem of testing the hypothesis H₀=0 against H₁ 0 on the basis of a sequence of test statistics {T_n(X₁,...,X_n). In accordance with the Bahadur theory, a measure of the asymptotic efficiency of {T_n} is its exact slope C_t(). One says that {T_n} is locally asymptotically optimal in the Bahadur sense if C_t() ip 2K(), 0, where.The purpose of the paper is to characterize densities f for which the property of local asymptotic optimality is shared by such commonly used statistics as the sample mean, the Kolmogorov-Smirnov statistic, the sign statistic, ², etc. Under certain restrictions on f one proves, for example, that the sequence of statistics u is locally asymptotically optimal only for the hyperbolic cosine distribution, while the Kolmogorov statistic only for the Laplace distribution. At the end of the paper one obtains similar results for the two-sample case, in particular, for a large class of linear rank statistics.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 108, pp. 119–133, 1981.     

Isomorphism types of Rogers semilattices for families from different levels of the arithmetical hierarchy

We investigate differences in isomorphism types for Rogers semilattices of computable numberings of families of sets lying in different levels of the arithmetical hierarchy. Supported by RFBR grant No. 05-01-00819 and by INTAS grant No. 00-499. Supported by NSFC grant No. 60310213. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 637–654, November–December, 2006.     

Characterization of three types of planar spaces with invisible planes

In this paper we characterize a class of finite planar spaces whose planes have two sizes.Dedicato alla memoria dell'amico e Maestro Giuseppe TalliniWork supported by G.N.S.A.G.A. of C.N.R. and the projects M.U.R.S.T. 40 %     

The asymptotic of the necessary sample size in testing the hypotheses on the shape parameter of a distribution close to the normal one

We study the problem of testing the hypothesis on the “approximate normality” formulated in terms of large values of the shape parameter of an asymptotically normal underlying distribution. Considering the examples of gamma-and generalized Birnbaum—Saunders distributions, we propose one way to obtain the asymptotic of the necessary sample size for testing the mentioned hypothesis. Our approach differs from those based on contiguous alternatives or on the use of the large deviations theory for distributions of sums of independent random variables. Our method yields remarkably precise approximate formulas, what is illustrated by numerical data.