The log-likelihood ratio for sparse multinomial mixtures

The log likelihood ratio is expanded for testing a sequence of multinomial null hypotheses against a sequence of multinomial mixture close alternative hypothesis. As the number of categories grows without limit, the sample size increases and the variances of the mixing distributions tend to zero. The limiting form of the log likelihood ratio is functionally different from previously studied goodness of fit statistics. The statistic derived here exhibits moderate asymptotic power when Pearson's chi-square is biased.