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Большой семинар кафедры теории вероятностей МГУ
24 сентября 2008 г. 16:45, г. Москва, ГЗ МГУ, ауд. 16-24
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Convergence of multinomial goodness-of-fit statistics to chi–square distribution
Zh. Assylbekov Hiroshima University
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Количество просмотров: |
Эта страница: | 190 |
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Аннотация:
Let $\boldsymbol{Y}=(Y_1,Y_2,\dots,Y_k)'$ be a random vector with multinomial distribution. In the talk we investigate the convergence rate of so-called power divergence family of statistics $\{I^\lambda(\boldsymbol{Y}),\lambda\in\mathbb{R}\}$ introduced by Cressie and Read (1984) to chi-square distribution. It is proved that for every $k\ge4$
$$
\mathsf{P}(2nI^\lambda(\boldsymbol{Y})<c)=G_{k-1}(c)+O(n^{-1+\mu(k-1)}),
$$
where $G_r(c)$ is the distribution function of chi-square random variable with $r$ degrees of freedom, $\mu(r)={6}/{(7r+4)}$ for $3\le r\le 7$, $\mu(r)={5}/{(6r+2)}$ for $r\ge 8$. This refines Zubov and Ulyanov's result (2008). The proof uses Krätzel-Nowak's theorem (1991) on the number of integer points in a convex body with smooth boundary.
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