Approximations for the distributions of the extreme latent roots of three matrices

In this paper we present simple approximations for the distributions of the extreme latent roots of three matrices occurring in multivariate analysis. The matrices considered are (i)S1S2−1 whereS1 andS2 are independent Wishart matrices estimating different covarance matrices, (ii)S1S2−1 whereS1 andS2 are independent and estimate the same covariance matrix, withS2 having the Wishart distribution andS1 having the noncentral Wishart distribution, and (iii) the noncentral Wishart matrix. The approximations take the form of upper and lower bounds for the distribution functions of the largest and smallest latent roots respectively. For the three matrices considered above these bounds are expressed very simply in terms of products of (i)F, (ii) noncentralF and (iii) noncentralX2 probabilities.