Decomposable Statistics and Hypothesis Testing for Grouped Data

Ivchenko, G. I.; Medvedev, Yu. I.

doi:10.1137/1125065

“…However, first, such tests (for equiprobable HQ) ought to be generated by non-symmetric DS necessarily, and secondly, they will not be goodness-offit tests, since they cannot distinguish between HQ and all such alternatives (a detailed investigation of this problem see in [14]). Κ we restrict ourselves to the class of symmetric DS, they allow us to generate a goodness-of-fit test for HQ, but their power does not degenerate asymptotically (if the significance level is fixed) namely under the alternatives of the form (11).…”

Section: Theorem 6 If Conditionmentioning

confidence: 99%

“…Thus the chi-square test is still AMP as a -> 0 and α -> oo among the symmetric tests with condition (14). Besides there are other AMP tests in these zones, namely Brought to you by | University of Arizona Authenticated Download Date | 5/28/15 6:00 PM those for which t N = Δτν/\/2 -η ω + o (l) = t + o(l), since there is a possibility that p(f> a) -> 1 as α -> 0 or α -» oo.…”

Section: Theorem 6 If Conditionmentioning

confidence: 99%

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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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“…However, it is also of great interest, from both the practical and theoretical points of view, to investigate the limit behaviour of these divisible statistics when both n and N tend to infinity. Research in this direction includes [10], [9], [27] etc.…”

Section: The Role Of λ-"Rate Per Cell"mentioning

confidence: 99%

Large Number of Rare Events: Diversity Analysis in Multiple Choice Questionnaires and Related Topics

Kvizhinadze¹

0

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<p>The statistical analysis of a large number of rare events, (LNRE), which can also be called statistical theory of diversity, is the subject of acute interest both in statistical theory and in numerous applications. A careful eye will quickly see the presence of a large number of very rare objects almost everywhere: large numbers of rare species in ecosystems, large numbers of rare opinions in any opinion pool, large numbers of small admixtures in any solution and large numbers of rare words in any text are only few examples. In studying such objects, the interest for mathematical statisticians lies in the fact that most of the frequencies are small and, therefore, difficult to deal with. It is not immediately clear how one should be able to derive consistent and reliable inference from a large number of such frequencies. In this thesis we study the diversity of questionnaires with multiple answers. It has been demonstrated that this is a particular model of LNRE theory. In our analysis, the theories of large deviation, contiguity and Edgeworth expansion were employed, and limit theorems have been established.</p>

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“…1: Spectral statistics for James Joyce's Ulysses Total number of words n = 264217Vocabulary µ n = 30030 µ n (1) µ n (2) µ n (3) µ n (4) µ n (5) µ n (6) µ n (7) µ n (8) µ n (9) µ n(10) …”

mentioning

confidence: 99%

Large Number of Rare Events: Diversity Analysis in Multiple Choice Questionnaires and Related Topics

Kvizhinadze¹

1

0

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<p>The statistical analysis of a large number of rare events, (LNRE), which can also be called statistical theory of diversity, is the subject of acute interest both in statistical theory and in numerous applications. A careful eye will quickly see the presence of a large number of very rare objects almost everywhere: large numbers of rare species in ecosystems, large numbers of rare opinions in any opinion pool, large numbers of small admixtures in any solution and large numbers of rare words in any text are only few examples. In studying such objects, the interest for mathematical statisticians lies in the fact that most of the frequencies are small and, therefore, difficult to deal with. It is not immediately clear how one should be able to derive consistent and reliable inference from a large number of such frequencies. In this thesis we study the diversity of questionnaires with multiple answers. It has been demonstrated that this is a particular model of LNRE theory. In our analysis, the theories of large deviation, contiguity and Edgeworth expansion were employed, and limit theorems have been established.</p>

show abstract

Decomposable Statistics and Hypothesis Testing for Grouped Data

Cited by 24 publications

References 4 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

Large Number of Rare Events: Diversity Analysis in Multiple Choice Questionnaires and Related Topics

Large Number of Rare Events: Diversity Analysis in Multiple Choice Questionnaires and Related Topics

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