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The exact density function of the ratio of two dependent linear combinations of chi-square variables. (English) Zbl 0817.62005

Summary: A computable expression is derived for the raw moments of the random variable \(Z = N/D\) where \[ N = \sum^ n_ 1m_ i X_ i + \sum^ s_{n+1} m_ iX_ i, \quad D = \sum^ s_{n + 1} l_ iX_ i + \sum^ r_{s+1} n_ iX_ i, \] 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.

MSC:

62E15 Exact distribution theory in statistics
62H10 Multivariate distribution of statistics
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
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