On the Moment Generating Function of Pillai's $V^{(s)}$ Criterion

Abstract

The moment generating function (mgf) of Pillai's $V^{(s)}$ criterion in the central case has been considered by Pillai [8], [9] and James [3]. In the present paper, the mgf's of $V^{(s)}$ are considered in the following non-central cases: (i) MANOVA and (ii) canonical correlation. The lower order moments of $V^{(s)}$ for (i) and (ii) in the non-central case were obtained earlier by Pillai [10] for $s = 2$ in the linear case i.e. when there is only one non-zero population root. These results for (i) were extended by Khatri and Pillai [5], [6], [7] to general $s$ in the linear case and in the planar case i.e. when there are two non-zero population roots. Only the first two moments were considered in the planar case while the first four moments were obtained in the linear case. The results of this paper further facilitates the derivation of the general moments.