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- W1994464771 abstract "Let $(N_1, cdots, N_k)$ be a multinomial vector with $n = sum N_i$ and with parameter $(p_1, cdots, p_k), sum p_i = 1$. Let $f_1, cdots, f_k$ be real-valued functions defined on the integers ${0, 1, cdots, n}$, and let $S_k = sum^k_{i=1} f_i(N_i)$. Suppose $k rightarrow infty$, and as $k rightarrow infty$, that $n rightarrow infty$ and $max_{1leqq i leqq k}p_i rightarrow 0$. Conditions on the ${f_i}$ are given which guarantee that $S_k$, suitably centered and scaled, has a normal limit in law. An application shows that if $min_{1leqq i leqq k} (np_i)$ is bounded away from zero and the ${f_i}$ are polynomials of bounded degree as $k rightarrow infty$, that $S_k$ is asymptotically normal provided only that a uniformly asymptotically negligible (uan) condition on the ${f_i}$ holds. For testing the specified simple hypothesis $p_i = p_i^0$ for all $1 leqq i leqq k$, Pearson's chi-square statistic and the likelihood ratio statistic can be written in the form of $S_k$. It is shown that these two statistics are asymptotically normal as $k rightarrow infty$ provided they satisfy simple conditions which are equivalent to their respective uan conditions." @default.
- W1994464771 created "2016-06-24" @default.
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- W1994464771 date "1975-01-01" @default.
- W1994464771 modified "2023-09-24" @default.
- W1994464771 title "Central Limit Theorems for Multinomial Sums" @default.
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- W1994464771 doi "https://doi.org/10.1214/aos/1176343006" @default.
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