The exact density function of the ratio of two dependent linear combinations of chi-square variables

Abstract

A computable expression is derived for the raw moments of the n X s random variable Z = N/D where N = }-~.1 rn~ i + }-~.n+l rniX~, D = 7" ~-~.~+1 liX~ + ~-~-s+l niXi, and the Xi's are independently distributed central chi-square variables. The first four moments are required for approximating the distribution of Z by means of Pearson curves. The exact density function of Z is obtained in terms of sums of generalized hypergeometric functions by taking the inverse Mellin transform of the h-th moment of the ratio N/D where h is a complex number. The case n = 1, s = 2 and r = 3 is discussed in detail and a general technique which applies to any ratio having the structure of Z is also described. A theoretical example shows that the inverse Mellin transform technique yields the exact density function of a ratio whose density can be obtained by means of the transformation of variables technique. In the second example, the exact density function of a ratio of dependent quadratic forms is evaluated at various points and then compared with simulated values.