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\@writefile{toc}{\contentsline {section}{\numberline {2}Priors for $D$}{4}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Conjugate prior}{4}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Non-conjugate reference priors}{5}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Hierarchical priors}{5}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.1}Wishart hierarchical prior}{5}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.2}Hierarchical prior on the correlations}{6}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.3}Hierarchical prior on the Givens angles}{6}}
\@writefile{toc}{\contentsline {section}{\numberline {3}A simulation study of Risk}{7}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Objective}{7}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Details of Simulation}{8}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.2.1}Priors}{8}}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.2.2}True Covariance Matrices}{8}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Results and Conclusions}{9}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Computation in Hierarchical Models}{10}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Non-conjugate priors}{10}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Issues in computing}{12}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Examples}{13}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Example I: Guinea pig data}{13}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Example II: Hospital Data}{14}}
\@writefile{toc}{\contentsline {section}{\numberline {6}Discussion}{15}}
\@writefile{toc}{\contentsline {section}{\numberline {Appendix\nobreakspace  {}A}Correlation matrices for simulation}{17}}
\@writefile{toc}{\contentsline {section}{\numberline {Appendix\nobreakspace  {}B}Derivation of Importance Weights}{18}}
\newlabel{app}{{B.1}{18}}
\@writefile{toc}{\contentsline {section}{References}{19}}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Risk for Several Estimators. Simulation Standard Errors are in parentheses. The true values of $D$ are described in Section 3.2.2.}}{22}}
\newlabel{risk}{{1}{22}}
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Computational comparison of Gibbs sampler with Importance Sampling. The numbers in each column are the ratios of Monte Carlo variances for 2000 iterations. MCMC$^{\star }$ denotes posterior simulation without using fact that the full conditional of $D^{-1}$ is Wishart. }}{23}}
\newlabel{nonlinear}{{2}{23}}
\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces Posterior means, standard deviations, and $95\%$ credible intervals for the third (Delaware) and the fifth state (Florida) in Example II. Here, ``Standard Wishart'' is a Wishart distribution with estimated scale matrix and df$=4$, and ``Wishart HP'' and ``Givens HP'' are respectively, the Wishart and Givens-angle hierarchical priors.}}{24}}
\newlabel{hospital}{{3}{24}}