Sequential $\chi^2$ criteria

Independent trials with $m$ outcomes are considered. Let the probability of the $j$th outcome be $p_j$ under the null hypothesis $H$ but be $\widetilde p_j$ under the alternative hypothesis $\widetilde H$, $j=1,2,\dots,m$. For testing the hypothesis $H$ samples with increasing size $n_1<n_2<\dots<n_r$ are formed. We denote the number of times that the $j$th outcome appears in the first $n_i$ trials by $\nu_{ij}$. The statistics $\chi_i^2$ are introduced by formula (1.2). The hypothesis $H$ is rejected if $\chi_i^2>x_i^*$ for all $i=1,2,\dots,r$, where $x_i^*$ is some critical value, and is accepted in the remaining cases. The limit, for $n_i\to\infty$, of the distribution of $\chi^2$ under the hypotheses $H$ and $\widetilde H$ is given in the paper. These are used for the computation of the errors of the first and second kind, $\alpha$ and $\beta$, according to formulas (1.4) and (1.5). These distributions are multivariate generalizations of the central and noncentral $\chi^2$-distributions.
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