Limit Distributions of the $\chi ^2 $ Statistic

Selivanov, B. I.

doi:10.1137/1129015

Cited by 5 publications

(8 citation statements)

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“…It is clear that σ 2 > 0 for all alternatives to the hypothesis HQ under consideration and, as observed in [3], the equality σ 2 = 0 holds true for the above alternatives HA only.…”

mentioning

confidence: 58%

“…In this paper, which complements the results obtained in [1], we give limit distributions of the vector statistic χ 2 for not close (fixed) alternatives. In Section 2, results of [2,3] related to the usual univariate Pearson statistic χ 2 given in (3) are extended to the multivariate case. We establish the weak convergence of the vector statistic χ 2 , whose components are appropriately centered and normalized, to multivariate normal and chi-square laws.…”

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confidence: 99%

“…Brought to you by | Stockholms Universitet Authenticated Download Date | 7/10/15 11:44 PM If Η τ^ HA, then the well-known expressions of the mathematical expectation and variance of Pearson's statistic χ 2 cited in [2,3] yield W! where ι = 1, .…”

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confidence: 99%

“…The first part of the theorem is thus proved. Now let Η = HA-In this case, as observed in [3], the random variables η* defined by (9) are reduced to v?.…”

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confidence: 99%

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Two remarks on the multivariate statistic χ 2

Selivanov¹,

Chistyakov²

1999

Discrete Mathematics and Applications

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We consider a sequence of polynomial trials with Ν outcomes and construct the multivariate statistic χ 2 with the use of samples of growing sizes n\,... ,Λ Γ , 1 < n\ < ... < n r , r > 2, such that each subsequent sample contains the previous one. We assume that N is fixed, n\ -> <», and rt//n f +i -* p?, 0 < p,· < 1, / = 1,... ,r-1.For fixed (not close) alternatives to a simple hypothesis tested, we establish the weak convergence of the distribution of the vector statistic χ 2 , whose components are appropriately centered and normalized, to multivariate normal and chi-square laws. In the case of convergence to the normal law, the components of the limiting normal random vector form a non-homogeneous Markov chain; the densities of transition probabilities of this chain are found.This research was supported by the Russian Fiundation for Basic Research, grants 96-01-00531, 96-15-96092.

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“…It is clear that σ 2 > 0 for all alternatives to the hypothesis HQ under consideration and, as observed in [3], the equality σ 2 = 0 holds true for the above alternatives HA only.…”

mentioning

confidence: 58%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…The first part of the theorem is thus proved. Now let Η = HA-In this case, as observed in [3], the random variables η* defined by (9) are reduced to v?.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Two remarks on the multivariate statistic χ 2

Selivanov¹,

Chistyakov²

1999

Discrete Mathematics and Applications

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“…Here d The calculation of the distributions of the statistic .m/ n can be easily organised with the use of relations (11) and (13), but not with the use of explicit formulas similar to those given for the case m D 2 in Corollary 1. It is seen from (11) that in order to find the polynomial Q .m/ nC1 .y/, n 1, first it is needed to calculate and store in the memory of a computer the polynomials d In conclusion, we give some results on the limit distributions of the statistics .m/ n .…”

Section: Theorem 1 Polynomials (8) Satisfy the Recurrence Relationmentioning

confidence: 99%

On calculation of exact distributions of decomposable statistics in the multinomial scheme

Selivanov¹

2006

discrete math appl

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We present several recursive procedures for calculating exact (non-limit) distributions of decomposable statistics in the multinomial scheme. In order to obtain recurrence relations, we make use of the generating functions of the statistics under consideration. In the general case, the method of calculating reduces to the algorithm suggested by A. Zubkov in 1996 which is presented in the language of Markov chains.

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