Abstract
This article aims to make use of moment-generating functions (mgfs) to derive the density of mixture distributions from hierarchical models. When the mgf of a mixture distribution doesn’t exist, one can extend the approach to characteristic functions to derive the mixture density. This article uses a result given by E.R. Villa, L.A. Escobar, Am. Stat. 60 (2006), 75–80. The present work complements E.R. Villa, L.A. Escobar, Am. Stat. 60 (2006), 75–80 article with many new examples.
Highlights
A random variable X is said to have mixture distribution if it depends on a quantity that has a distribution
The mixture distribution negative-binomial can arise in the distribution of the sum of N independent random variables, each having the same logarithmic distribution and N having a Poisson distribution; this mixture distribution was used in modeling biological spatial data; see Gurland [16], Bagui and Mehra [14]
Because of the high importance of mixture distributions, students should be exposed to mixture distributions as soon as they have familiarity with conditional expectations
Summary
A random variable X is said to have mixture distribution if it depends on a quantity that has a distribution. The mixture distribution arises from a hierarchical model Karlis and Xekalaki [8] derived results related to Poisson mixtures models with applications in various other fields. To fit plant quadrat data on the blue-green sedge, Skellam [12] used a mixture of binomial with varying sample sizes modeled with Poisson distributions. The Gamma mixture of Poisson r.v.’s yield negative-binomial, while Green and Yule [13] used this mixture distribution to model “accident proneness”; see Bagui and Mehra [14]. The mixture distribution negative-binomial can arise in the distribution of the sum of N independent random variables, each having the same logarithmic distribution and N having a Poisson distribution; this mixture distribution was used in modeling biological spatial data; see Gurland [16], Bagui and Mehra [14]
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