On singularities of sampling distributions, in particular for ratios of quadratic forms

Abstract

SUMMARY A general theory of singularities in sampling distributions, given previously for statistical functions on a Euclidean space, is here extended to functions on a hypersphere; in particular, to distributions of ratios of quadratic forms. The parent distributions chiefly considered are normal and nonsingular. The theory expresses the density of a sampling distribution as the sum of a singular function and a smoother remainder. Cases are adduced in which the remainder vanishes: in more general cases, approximations to the remainder may be based on moments or other supplementary information. Numerical illustrations of approximations based mainly on singularities are given for distributions of noncircular serial correlation coefficients with known mean in samples of sizes 5 and 10. An appendix extends Hotelling's expansion for a sampling distribution near an end of its range, and provides an integral remainder and bounds for this remainder.