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 Diskr. Mat.: Year: Volume: Issue: Page: Find

 Diskr. Mat., 1999, Volume 11, Issue 4, Pages 145–151 (Mi dm404)

Two remarks on the multidimensional $\chi^2$ statistic

B. I. Selivanov, V. P. Chistyakov

Abstract: We consider a sequence of polynomial trials with $N$ outcomes and construct the multivariate statistic $\chi^2$ with the use of samples of growing sizes $n_1,\ldots,n_r$, $1\le n_1<\ldots<n_r$, $r\ge 2$, such that each subsequent sample contains the previous one. We assume that $N$ is fixed, $n_1\to\infty$, and $n_i/n_{i+1}\to\rho_i^2$, $0<\rho_i<1$, $i=1,\ldots,r-1$.
For fixed (not close) alternatives to a simple hypothesis tested, we establish the weak convergence of the distribution of the vector statistic $\chi^2$, 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.

DOI: https://doi.org/10.4213/dm404

Full text: PDF file (514 kB)

English version:
Discrete Mathematics and Applications, 1999, 9:6, 645–651

Bibliographic databases:

Document Type: Article
UDC: 519.2

Citation: B. I. Selivanov, V. P. Chistyakov, “Two remarks on the multidimensional $\chi^2$ statistic”, Diskr. Mat., 11:4 (1999), 145–151; Discrete Math. Appl., 9:6 (1999), 645–651

Citation in format AMSBIB
\Bibitem{SelChi99} \by B.~I.~Selivanov, V.~P.~Chistyakov \paper Two remarks on the multidimensional $\chi^2$ statistic \jour Diskr. Mat. \yr 1999 \vol 11 \issue 4 \pages 145--151 \mathnet{http://mi.mathnet.ru/dm404} \crossref{https://doi.org/10.4213/dm404} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1761020} \zmath{https://zbmath.org/?q=an:0960.62022} \transl \jour Discrete Math. Appl. \yr 1999 \vol 9 \issue 6 \pages 645--651