THE EXACT DENSITY FUNCTION OF THE RATIO OF TWO DEPENDENT LINEAR COMBINATIONS OF CHI-SQUARE VARIABLES

SERGE B. PROVOST AND EDMUND M. RUDIUK

Department of Statistical and Actuarial Sciences, The University of Western Ontario, London, Ontario, Canada N6A 5B7

(Received January 11, 1993; revised September 8, 1993)

Abstract. A computable expression is derived for the raw moments of the random variable Z = N/D where N = ∑_{1}^{n} m_{i}X_{i} + ∑_{n+1}^{s} m_{i}X_{i}, D = ∑_{n+1}^{s} l_{i}X_{i} + ∑_{s+1}^{r} n_{i}X_{i}, and the X_{i}’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.

Key words and phrases: Exact density, approximate density, moments, ratios of quadratic forms, Mellin transform.