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THE EXACT DENSITY FUNCTION OF THE RATIO

OF TWO DEPENDENT LINEAR COMBINATIONS

OF CHI-SQUARE VARIABLES

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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* = \sum^{n}_{1}*m*_{i}*X*_{i} + \sum^{s}_{n+1}*m*_{i}*X*_{i},
*D* =
\sum^{s}_{n+1}*l*_{i}*X*_{i} + \sum^{r}_{s+1}*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.

**Source**
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