Numerical computation for the exact distribution of Roy's largest root statistic under linear alternative

TL;DR

This paper develops a numerical method to compute the exact distribution of Roy's largest root statistic in MANOVA, enabling precise power analysis under linear alternatives.

Contribution

It derives an exact expression for the largest eigenvalue distribution using zonal polynomials and provides an efficient algorithm for numerical computation.

Findings

Exact distribution formula for Roy's largest root under linear alternative

Algorithm for expanding zonal polynomial products for numerical calculation

Facilitates precise power analysis in multivariate tests

Abstract

This paper discusses the computation of exact powers for Roy's test in multivariate analysis of variance~(MANOVA). We derive an exact expression for the largest eigenvalue of a singular noncentral Beta matrix in terms of the product of zonal polynomials. The numerical computation for that distribution is conducted by an algorithm that expands the product of zonal polynomials as a linear combination of zonal polynomials. Furthermore, we provide an exact distribution of the largest eigenvalue in a form that is convenient for numerical calculations under the linear alternative.