Local Limit Theorems for Distributions of Sums of Independent Random Vectors

Mukhin, A. B.

doi:10.1137/1129047

Cited by 17 publications

(9 citation statements)

References 3 publications

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“…In particular it would be interesting to solve this problem for likelihood ratio test, although a formal comparison of its correction term f = φ(ί) 5 -2t 2 £N ~ vi/V 3 (see (10) and Theorem 6) with (20) in this zone provides a reason to suppose that the test is not efficient of the second order.…”

Section: Theorem 6 If Conditionmentioning

confidence: 99%

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

Ivchenko¹,

Mrakhmedov²

1992

Discrete Mathematics and Applications

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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/ Ν(0

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Section: Theorem 6 If Conditionmentioning

confidence: 99%

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

Ivchenko¹,

Mrakhmedov²

1992

Discrete Mathematics and Applications

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“…where α is the distance from α to the nearest integer and X * denotes a symmetrization of X. In Mukhin [17] and [18] , the two-sided inequality…”

Section: Introduction and Main Resultmentioning

confidence: 99%

Approximate local limit theorems with effective rate and application to random walks in random scenery

Giuliano¹,

Weber²

2017

Bernoulli

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We show that the Bernoulli part extraction method can be used to obtain approximate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term, that is with explicit parameters and universal constants. We also show that our estimates allow to recover Gnedenko and Gamkrelidze local limit theorems. We further establish by this method a local limit theorem with effective remainder for random walks in random scenery.2010 Mathematics Subject Classification: Primary: 60F15, 60G50 ; Secondary: 60F05.

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“…Koenker and Bassett then argue that an extension of the result of Stone (1965) (based on the central limit theorem and subsequently provided by Mukhin (1984)) implies (1.12)…”

Section: Bo { R P" E Po(yi-xi)=min}mentioning

confidence: 92%

Asymptotic Behavior of the Number of Regression Quantile Breakpoints

Portnoy¹

1991

SIAM J. Sci. and Stat. Comput.

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In the general regression model yi=xfl+ei, for i= 1,..., n and tiER p, the "regression quantile"/(0) estimates the coefficients of thelinear regression function parallel to x/3 and roughly lying above a fraction 0 of the data. As introduced by Koenker and Bassett [Econometrica, 46 (1978), pp. 33-50], these regression quantiles are analogous to order statistics and provide a natural and appealing approach to the analysis of the general linear model. Computation of/(0) can be expressed as a parametric linear programming problem with J, distinct extremal solutions as 0 goes from zero to one. That is, there will be J, breakpoints {0j}, for j-1,... ,J, such that/(0j) is obtained from/(0_) by a single simplex pivot. Each/(0) is characterized by a specific subset of p observations. Although no previous result restricts J to be less than the upper bound (,)= O(n P), practical experience suggests that J grows roughly linearly with n. Here it is shown that, in fact, Jn O(n log n) in probability, where the distributional assumptions are those typical of multiple regression situations. The result is based on a probabilistic rather than combinatoric approach which should have general application to the probabilistic behavior of the number of pivots in (parametric) linear programming problems. The conditions are roughly that the constraint coefficients form random independent vectors, and that the number of variables is fixed while the number of constraints tends to infinity.

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Local Limit Theorems for Distributions of Sums of Independent Random Vectors

Cited by 17 publications

References 3 publications

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

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

Approximate local limit theorems with effective rate and application to random walks in random scenery

Asymptotic Behavior of the Number of Regression Quantile Breakpoints

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