Exact Nonnull Distributions of Sphericity Tests for Trivariate Normal Population with Power Comparison

SYNOPTIC ABSTRACT

Exact formulas for the nonnull density function of the locally best invariant test and that of likelihood ratio test for testing sphericity in trivariate normal distribution are derived, extending the exact null density functions obtained by John (1972). Khatri and Srivastava(1971)'s formula for the nonnull density of the likelihood ratio test is shown to be useful if adjusted. The graphs of the density functions are shown. Power comparisons are made and Grieve(1984)'s conjecture on power of two tests is confirmed. Harris and Peers(1980)'s condition on asymptotic powers is also confirmed. Asymptotic formulas by normal and noncentral chisquare distributions are compared.

Key Words and Phrases:

• locally best invariant test
• likelihood ratio test
• asymptotic approximation
• zonal polynomials
• exact power
• Mellin transform