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\noindent Statistica Sinica {\bf 11}(2001), 77-95
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\centerline{\large\bf PROPRIETY OF POSTERIORS WITH IMPROPER PRIORS}
\vspace{2pt}
\centerline{\large\bf IN HIERARCHICAL LINEAR MIXED MODELS}
\vspace{.25cm}
\centerline{Dongchu Sun, Robert K. Tsutakawa and Zhuoqiong He}
\vspace{.4cm}
\centerline{\it University of Missouri-Columbia}
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\begin{quotation}
\noindent {\it Abstract:}
This paper examines necessary and sufficient conditions for the propriety of
the posterior distribution in hierarchical linear mixed effects models for
a collection of improper prior distributions. In addition to the flat prior
for the fixed effects,  the collection includes various limiting forms of the
invariant gamma distribution for the variance components, including cases
considered previously by Datta and Ghosh (1991), and Hobert and Casella (1996).
Previous work is extended by considering a family of correlated random effects
which include as special cases the intrinsic autoregressive models of Besag,
York and  Molli\'{e} (1991), the Autoregressive (AR) Model of Ord (1975),
and the Conditional Autoregressive (CAR) Models of Clayton and Kaldor (1987),
which have been found useful in the analysis of spatial effects.
Conditions are then presented for the propriety of the posterior distribution
for a generalized linear mixed model, where the first stage distribution
belongs to an exponential family.\par
\vspace{9pt} \noindent {\it Key words and phrases:}
Generalized linear mixed models, Gibbs sampling, linear mixed models,
Markov chain Monte Carlo, multivariate normal, variance components.
\end{quotation}
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