AISM 52, 215-230

## Moment-based approximations of distributions using mixtures: Theory and applications

### Bruce G. Lindsay^{1}, Ramani S. Pilla^{2} and Prasanta Basak^{3}

^{1}Department of Statistics, 326 Classroom
Building, Pennsylvania State University, University Park, PA-16802, U.S.A.

^{2}National Institutes of Health, 6100 Executive Blvd.,
Room 7B-13, MSC 7510, Bethesda, MD 20892-7510, U.S.A.

^{3}120, 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 2*p* moments as
the targeted univariate distribution. The gamma mixture approximation
to the distribution of a positive weighted sums of independent central
*chi*^{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.

**Source**
( TeX , DVI )