Probabilities and Large Quantiles of Noncentral Generalized Chi-Square Distributions

Abstract

Exact values of probability integrals for noncentral generalized chi-square distributions are numerically evaluated based upon new geometric representation formulae for these distributions. Using iterative numerical methods exact quantiles can be calculated then. Explicit quantile approximation formulae are deduced from an asymptotic expansion for related probabilities of large deviations. Though this method is originally directed to the construction of starting values for determining exact large quantiles it is of benefit for simply approximating large quantiles and for obtaining quantiles from the central part of the distributions, too. The accuracy of the explicit asymptotic approximation method can be improved by combining it with the geometric measure representation formulae. Several numerical studies compare the present results with results of other authors available in the special case of the classical noncentral chi-square distribution. As an application, critical test points as well as power functions for expectation tests in elliptically contoured sample distributions are considered and certain problems of sensitivity and robustness type are discussed.