Premium
The effect of mixing‐distribution misspecification in conjugate mixture models
Author(s) -
Gustafson Paul
Publication year - 1996
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315741
Subject(s) - mixing (physics) , mathematics , conjugate prior , posterior predictive distribution , distribution (mathematics) , parametric statistics , conditional probability distribution , mixture distribution , compound probability distribution , inverse chi squared distribution , marginal distribution , statistical physics , categorical distribution , estimator , statistics , bayesian probability , bayesian linear regression , asymptotic distribution , mathematical analysis , probability distribution , prior probability , probability density function , distribution fitting , bayesian inference , random variable , physics , quantum mechanics
Abstract Parametric mixture models are commonly used in the analysis of clustered data. Parametric families are specified for the conditional distribution of the response variable given a cluster‐specific effect, and for the marginal distribution of the cluster‐specific effects. This latter distribution is referred to as the mixing distribution. If the form of the mixing distribution is misspecified, then Bayesian and maximum‐likelihood estimators of parameters associated with either distribution may be inconsistent. The magnitude of the asymptotic bias is investigated, using an approximation based on infinitesimal contamination of the mixing distribution. The approximation is useful when there is a closed‐form expression for the marginal distribution of the response under the assumed mixing distribution, but not under the true mixing distribution. Typically this occurs when the assumed mixing distribution is conjugate, meaning that the conditional distribution of the cluster‐specific parameter given the response variable belongs to the same parametric family as the mixing distribution.