MOMENT-BASED APPROXIMATIONS OF DISTRIBUTIONS USING MIXTURES: THEORY AND APPLICATIONS

BRUCE G. LINDSAY^{1}, RAMANI S. PILLA^{2} AND PRASANTA BASAK^{3}

1Department of Statistics, 326 Classroom Building, Pennsylvania State University, University Park, PA-16802, U.S.A. 2National Institutes of Health, 6100 Executive Blvd., Room 7B-13, MSC 7510, Bethesda, MD 20892-7510, U.S.A.

3120 Eiche Library, The Pennsylvania State University, Altoona Campus, Altoona, PA-16601, U.S.A.

(Received September 29, 1997; revised November 10, 1998)

Abstract. There are a number of cases where the moments of a distribution are easily obtained, but theoretical distributions are not available in closed form. This paper shows how to use moment methods to approximate a theoretical univariate distribution with mixtures of known distributions. The methods are illustrated with gamma mixtures. It is shown that for a certain class of mixture distributions, which include the normal and gamma mixture families, one can solve for a p-point mixing distribution such that the corresponding mixture has exactly the same first 2p moments as the targeted univariate distribution. The gamma mixture approximation to the distribution of a positive weighted sums of independent central χ^{2} variables is demonstrated and compared with a number of existing approximations. The numerical results show that the new approximation is generally superior to these alternatives.

Key words and phrases: Cumulants, cumulative distribution function, gamma mixtures, mixture distribution, moment matrix, p-point mixture, tail probability, weighted sums of chi-squares.

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