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Moment-based approximations of distributions using mixtures: Theory and applications. (English) Zbl 0959.62016
Summary: 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 $$\chi^2$$ 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.

##### MSC:
 6.2e+18 Approximations to statistical distributions (nonasymptotic) 6.2e+16 Exact distribution theory in statistics
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