Generalized Quadratic Forms

Abstract

In this chapter we consider some distributional problems associated with generalized quadratic forms in normal matrix variates and some of their applications in multivariate statistical analysis. The exact probability density function of a generalized qudratic form is given in Section 5.1. An alternate representation of the probability density function is obtained in terms of orthogonal polynomials of matrix arguments in Section 5.2. Section 5.3 gives some representations of the joint probability density function of the latent roots and of the largest latent root of a generalized quadratic form. Section 5.4 deals with the distributions of certain functions of a generalized quadratic form, including a matrix t-variate distribution. Hotelling’s generalized T ²₀ -statistic is a fundamental statistic in multivariate analysis of variance. Two representations of its exact probability density function are given in Section 5.5. The first one is obtained in terms of P-polynomials and Laguerre polynomials of matrix arguments and converges for a certain range; the second one is expressed in terms of invariant polynomials and converges everywhere. The asymptotic expansion of the distribution function of T ²₀ in the non-null case is also obtained and its range of convergence is determined. Anderson’s linear discriminant function plays a fundamental role in discriminant analysis. Section 5.6 gives its exact moments in terms of P-polynomials; the distribution of the normalized Anderson’s linear discriminant function is obtained as an Edgeworth expansion. The multivariate calibration problem is considered in Section 5.7 and the distributions of certain statistics are approximated by the central F distribution.