The distribution of the characteristic roots ofS 1 S 2 −1 under violations in the complex case and power comparisons of four tests

Abstract

SummaryThe joint density function of the latent roots of S1S? ~ under violations is obtained where S~ has a complex non-central Wishart distribution We (p, n~, 2,~, [J) and $2, an independent complex central Wishart, Wo(p, n2, Z2, 0). The density and moments of Hotelling's trace are also derived under violations. Further, the non-null distributions of the following four criteria in the two-roots case are studied for tests of three hypotheses: Hotelling's trace, Pillai's trace, Wilks' criterion and Roy's largest root. In addition, tabulations of powers are carried out and power comparisons for tests of each of three hypotheses based on the four criteria are made in the complex case extending such work of Pillai and Jayachandran in the classical Gaussian case. The findings in the complex Gaussian are generally similar to those in the classical.

The Moment Generating Function of Pillai’s Criterion Concerning Three Hypotheses Under Violations in the Complex case

“…:;onstants and v is a partition on n. The relation in (2.9) will be of use while deriving the mgf in this paper. Further, upon using (2.9) integrating term by term by the aid of (2.6), hence we obtain from (2.8) the following The joint pdf of rh ... , rp bas been obtained by Pillai and Hsu (1979) under violation, it is in the form : It is known that Pillai's V(PI criterion is given by,…”

“…The lower order moments of VOl) were studied by Khatri and Pillai (1965), (1967), (1968) in the linear and planar cases. Now we shall consider the following hypotheses: In connection with (a) and (b), Pillai and Hsu (1979) have obtained the joint pdf of the latent roots rio r2, .'" rp or R=Si'2 S-1 SI/2 under violation in the complex case, it is given by eqn (3.1).…”

“…1) is carried out under the assump lion that A=2:1 2:;1 is "random". About the assumption of randomness and motivation behind robustness, one may refer to Pillai (I975), also Appendix C ofPillai and Hsu (1979).…”