The exact density function of the ratio of two dependent linear combinations of chi-square variables |
| |
Authors: | Serge B Provost Edmund M Rudiuk |
| |
Institution: | (1) Department of Statistical and Actuarial Sciences, The University of Western Ontario, N6A 5B7 London, Ontario, Canada |
| |
Abstract: | A computable expression is derived for the raw moments of the random variableZ=N/D whereN=
1
n
m
iXi+
n
+1s
m
iXi,D=
n
+1s
l
iXi+
s
+1r
n
iXi, and theX
i's are independently distributed central chi-square variables. The first four moments are required for approximating the distribution ofZ by means of Pearson curves. The exact density function ofZ is obtained in terms of sums of generalized hypergeometric functions by taking the inverse Mellin transform of theh-th moment of the ratioN/D whereh is a complex number. The casen=1,s=2 andr=3 is discussed in detail and a general technique which applies to any ratio having the structure ofZ 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 quardratic forms is evaluated at various points and then compared with simulated values. |
| |
Keywords: | Exact density approximate density moments ratios of quadratic forms Mellin transform |
本文献已被 SpringerLink 等数据库收录! |