Separable Statistics in a Polynomial Scheme. II

Medvedev, Yu. I.

doi:10.1137/1122075

Cited by 10 publications

(8 citation statements)

References 8 publications

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“…2) Results on the asymptotic distribution of Pearson's chisquare statistic and the GLRT statistic have been obtained in [37] and [38]. To the best of our knowledge, the proper normalization for the large deviations analysis has not been identified before in the small sample case.…”

Section: A Related Workmentioning

confidence: 99%

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Generalized Error Exponents for Small Sample Universal Hypothesis Testing

Huang

Meyn

2013

IEEE Trans. Inform. Theory

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The small sample universal hypothesis testing problem is investigated in this paper, in which the number of samples is smaller than the number of possible outcomes . The goal of this paper is to find an appropriate criterion to analyze statistical tests in this setting. A suitable model for analysis is the high-dimensional model in which both and increase to infinity, and . A new performance criterion based on large deviations analysis is proposed and it generalizes the classical error exponent applicable for large sample problems (in which ). This generalized error exponent criterion provides insights that are not available from asymptotic consistency or central limit theorem analysis. The following results are established for the uniform null distribution: 1) The best achievable probability of error decays as for some . 2) A class of tests based on separable statistics, including the coincidence-based test, attains the optimal generalized error exponents. 3) Pearson's chi-square test has a zero generalized error exponent and thus its probability of error is asymptotically larger than the optimal test. Index Terms-Bahadur efficiency, Chernoff efficiency, error exponent, hypothesis testing, large alphabet, large deviations, separable statistic, small sample.

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Section: A Related Workmentioning

confidence: 99%

“…Proof of Proposition 5: Applying Lemma 9 with , we obtain (38) where The rest of the proof is an application of the saddle point method [55]. It consists of two steps: the first step is to pick a particular contour around to carry out the integration.…”

Section: A) Approximation To the Logarithmic Moment Generating Functimentioning

confidence: 99%

Generalized Error Exponents for Small Sample Universal Hypothesis Testing

Huang

Meyn

2013

IEEE Trans. Inform. Theory

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“…Λ lot of articles is devoted to investigations of the statistic X 2 N . In particular, for the scheme of series for the case n, TV -» oo, the statistic Χ 2 Ν is considered in [5,7,8,[18][19][20], where the conditions of asymptotic normality of XJ; N are found. Put Let us consider more explicitly the case where the main hypothesis HQ is equiprobable, i.e., below we assume that po™ = l/TV, m = 1,2,..., Ν.…”

Section: (Nd 2 (2p(n))-l/2 -T U )mentioning

confidence: 99%

“…where TV denotes the number of boxes, p = maxp m ; or the condition up < 63 [2][3][4][5][6]21]; meN or the union of these conditions [7][8][9] is fulfilled. Only in [10] a wider class of polynomial schemes was considered under the condition np 2 -» 0, or…”

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Randomized decomposable statistics in the scheme of independent allocating particles into boxes

Mirakhmedov¹

1992

Discrete Mathematics and Applications

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Decomposable Statistics Generated by Multidimensional Pólya Distribution

L’vovskiy

Tsylova

2022

J Math Sci

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Exact and asymptotic formulas for the mathematical expectation and variance for decomposable statistics generated by multidimensional Pólya random walks (with stopping border G: N m=1 k m = = n) are proposed, as well as factorization for the double generating function of these statistics. Scenarios are described when decomposable statistics in the scheme of multidimensional Pólya random walks are asymptotically normal (integral and local normal limit theorems are presented). Also, minimal (natural) constraints are stipulated on the parameter α of the Pólya random walk.

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Separable Statistics in a Polynomial Scheme. II

Cited by 10 publications

References 8 publications

Generalized Error Exponents for Small Sample Universal Hypothesis Testing

Generalized Error Exponents for Small Sample Universal Hypothesis Testing

Randomized decomposable statistics in the scheme of independent allocating particles into boxes

Decomposable Statistics Generated by Multidimensional Pólya Distribution

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