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Extensions of Wilks' integral equations and distributions of test statistics

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Summary

Wilks [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.

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  1. MoGill University, Montreal, Canada

    A. M. Mathal

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  1. A. M. Mathal
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Mathal, A.M. Extensions of Wilks' integral equations and distributions of test statistics. Ann Inst Stat Math 36, 271–288 (1984). https://doi.org/10.1007/BF02481970

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  • DOI: https://doi.org/10.1007/BF02481970

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