Exact Distribution of the Noncentral Complex Roy's Largest Root Statistic via Pieri's Formula

TL;DR

This paper derives the exact distribution and moments of the noncentral complex Roy's largest root statistic using complex zonal polynomials and Pieri's formula, enabling precise power calculations in complex MANOVA.

Contribution

It provides explicit formulas for the distribution and moments of Roy's largest root in the complex case, utilizing combinatorial techniques for the first time.

Findings

Exact distribution derived using zonal polynomials

Explicit computation of linearization coefficients via Pieri's formula

Application to power analysis in complex MANOVA

Abstract

In this study, we derive the exact distribution and moment of the noncentral complex Roy's largest root statistic, expressed as a product of complex zonal polynomials. We show that the linearization coefficients arising from the product of complex zonal polynomials in the distribution of Roy's test under a specific alternative hypothesis can be explicitly computed using Pieri's formula, a well-known result in combinatorics. These results were then applied to compute the power of tests in the complex multivariate analysis of variance (MANOVA).