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 χ2 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.
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Lindsay, B.G., Pilla, R.S. & Basak, P. Moment-Based Approximations of Distributions Using Mixtures: Theory and Applications. Annals of the Institute of Statistical Mathematics 52, 215–230 (2000). https://doi.org/10.1023/A:1004105603806
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DOI: https://doi.org/10.1023/A:1004105603806