George Casella Point and Interval Estimation in Mixed Models with Dirichlet Process Random Effects
University of Florida
Estimation, and prediction, in mixed models has a long history, going back to the work of
Henderson and colleagues in the 1950s. Recently, the Dirichlet process has emerged as an
alternative to the usual assumption of normality of the random effect distribution,
providing a richer class of distributions to model this unobservable random variable.
Here we describe a variety of problems, and some solutions, about the estimation of
parameters of this model. Although much work has focused on Bayesian estimation
(see, for example, Kyung et. al 2009 Annals of Statistics), we also consider
estimation from the classical perspective. First, we describe a useful
representation of the model, and look at the properties of Bayes estimates
and credible intervals. Then we look at the model from a classical
perspective, and apply the Gauss-Markov theorem to get the Best Linear
Unbiased Estimator (BLUE). Further examination of the resulting covariance
structure reveals conditions under which the least squares estimator is BLUE,
and how the Dirichlet process precision parameter affects the covariance structure of the model.