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.

翻译：本文件讨论罗伊在多变量分析差异~(MANOVA)中测试的精确功率的计算。 我们从区域多元性产品中得出一个单非中位贝塔矩阵最大半数值的精确表达法。 该分布的数值计算由一种算法进行,该算法将区域多元性产品作为区域多元性线性组合的线性组合扩大。 此外,我们提供了最大半数值的精确分布法,其形式便于在线性替代品下进行数字计算。