Various Distributional Results For Ratios Of Quadratic Forms With Applications To Serial Correlation
Date of Award
1993
Degree Type
Dissertation
Degree Name
Doctor of Philosophy
Abstract
Quadratic forms in normal vectors are central building blocks in statistics, and ratios of quadratic forms arise in a variety of contexts. They are connected, for example, to regression and analysis of variance problems associated with linear models and to correlation analysis.;The matrices of the quadratic forms may be for example positive definite, positive semidefinite or indefinite; the normal vectors may be central or noncentral, nonsingular or singular. All these cases are considered for a single quadratic form as well as for the ratio of two quadratic forms. The distributional results also apply to bilinear forms, sums of quadratic and bilinear forms and the ratios thereof.;The inverse Mellin transform technique is used to obtain a representation of the density function of a nonnegative definite quadratic form. The density function of an indefinite quadratic form is then obtained in terms of Whittaker's function. Approximate density functions based on Patnaik and Pearson's approximations are also proposed. Computable expressions are then derived for the corresponding distribution functions. Three techniques are proposed for determining the distribution of ratios of quadratic forms: (1) the problem is transformed so that the results obtained for indefinite quadratic forms may be used; (2) the inverse Mellin transform technique is applied directly to the ratio in order to obtain the exact density function in terms of generalized hypergeometric functions; these techniques apply to ratios of quadratic forms which are not necessarily independently distributed; the moments which are used to approximate the distribution are also given; (3) the density function is obtained by differentiation; this approach requires the independence of the quadratic forms. These theoretical results are illustrated via several examples, and then corroborated by simulations. For the lag-k serial correlation coefficient, the exact distribution function and an approximation are obtained. The cumulants of the serial covariance are derived in two ways: by introducing a special operator and by solving a system of second order difference equations.
Recommended Citation
Rudiuk, Edmund M., "Various Distributional Results For Ratios Of Quadratic Forms With Applications To Serial Correlation" (1993). Digitized Theses. 2204.
https://ir.lib.uwo.ca/digitizedtheses/2204