### 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.

Original language | English (US) |
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Pages (from-to) | 215-230 |

Number of pages | 16 |

Journal | Annals of the Institute of Statistical Mathematics |

Volume | 52 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2000 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability

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## Cite this

*Annals of the Institute of Statistical Mathematics*,

*52*(2), 215-230. https://doi.org/10.1023/A:1004105603806