Exact nonnull distributions of sphericity tests for trivariate normal population with power comparison

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. Khatri and Srivastava’s formula for the nonnull density functions are shown. Power comparisons are made and Grieve’s conjecture on power of two tests is confirmed. Harris and Peers’s condition on asymptotic powers is also confirmed. Asymptotic formulas by normal and non-central chisquare distributiosn are compared.