(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 = \sumn1miXi + \sumsn+1miXi, D = \sumsn+1liXi + \sumrs+1niXi, 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.
Key words and phrases: Exact density, approximate density, moments, ratios of quadratic forms, Mellin transform.
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