ASYMPTOTIC NORMALITY FOR CHI-BAR-SQUARE DISTRIBUTIONS

Chi-bar-square distributions, which are mixtures of chi-square distributions, mixed over their degrees of freedom, often occur when testing hypotheses that involve inequality constraints. Here, necessary and sufficient conditions on the mixing or weighting distribution are found to ensure asymptotic normality of the corresponding chi-bar-square distribution. Essentially, asymptotic normality occurs for the chi-bar-square distribution if either the ratio of the mean to the variance of the mixing distribution goes to infinity, or the weighting distribution itself is asymptotically normal. Other than a combination of these two phenomena, this is also the only way for asymptotic normality to hold. Several examples of pertinent chi-bar-square distributions are shown to be asymptotically normal by the results in this paper.