Extensions of Wilks' integral equations and distributions of test statistics

Abstract

SummaryWilks [26] introduced two integral equations in connection with distribution problems in statistics. He called them Type A and Type B equations. Tretter and Walster ([22], [24]) solved the Type B equation and obtained the null and non-null distributions of the likelihood ratio criterion for testing linear hypotheses in the multinormal case. In this article we present several types of solutions of these equations along with new equations called Types C, D, E and F with their solutions. These include the integral equations satisfied by the density of a random variable which is (a) product of independent real gamma variates; (b) products of independent real beta variates; (c) ratio of products of independent beta and gamma variates ; (d) arbitrary powers of products of gamma and beta variates; (e) arbitrary powers of products and ratios of beta and gamma variates, and more general cases.

Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring

“…See [13] and references therein. Mellin transform technique was then utilized in a number of papers by Mathai who later rediscovered Nørlund's coefficients in a form similar to (2.9) in [29]. Mathai's and other contributions until 1973 are described in his survey paper [28].…”

Wilks' Integral Equations in Multivariate Distribution Theory

“…In comparison, the representations related to the g(w) above of Mathai ( (1984Mathai ( ( ), (1971] are reported to converge more quickly than mixtures, however, the computational forms appear to have the potential for major numerical instability, Mathai and Katiyar (1979). Again, the G and H-function approach to obtaining distributions should also lead to alternative computational forms and these efforts should benefit from the availability of supercomputers.…”

“…Mathai (1984) also points out that previous work with a G-function, Mathai and Saxena (1973) , Mathai (1973), In 1977, Gupta used the Type B integral equation and the approach suggested by Wilks, implemented by Tretter and Walster (1975), to obtain the exact distri- integral equations . What is needed now is a complete numerical study of these representations.…”

“…Concurrent to those developments was an approach that used the powerful and encompassing theory of G and H-functions, Saxena [ (1973), (1978) ]. An excellent survey of this approach as related to Wilks' integral equations is given in Mathai (1984). In that paper Mathai demonstrates the power of the approach by obtaining extensions of Wilks' integral equations in the form of new types.…”