Estimating Random Effects via Adjustment for Density Maximization

Abstract

We develop and evaluate point and interval estimates for the random effects $\theta_i$, having made observations $y_i|\theta_i\stackrel{\m athit{ind}}{\sim}N[\theta_i,V_i],i=1,...,k$ that follow a two-level Normal hierarchical model. Fitting this model requires assessing the Level-2 variance $A\equiv\operatorname {Var}(\theta_i)$ to estimate shrinkages $B_i\equiv V_i/(V_i+A)$ toward a (possibly estimated) subspace, with $B_i$ as the target because the conditional means and variances of $\theta_i$ depend linearly on $B_i$, not on $A$. Adjustment for density maximization, ADM, can do the fitting for any smooth prior on $A$. Like the MLE, ADM bases inferences on two derivatives, but ADM can approximate with any Pearson family, with Beta distributions being appropriate because shrinkage factors satisfy $0\le B_i\le1$. Our emphasis is on frequency properties, which leads to adopting a uniform prior on $A\ge0$, which then puts Stein's harmonic prior (SHP) on the $k$ random effects. It is known for the "equal variances case" $V_1=...=V_k$ that formal Bayes procedures for this prior produce admissible minimax estimates of the random effects, and that the posterior variances are large enough to provide confidence intervals that meet their nominal coverages. Similar results are seen to hold for our approximating "ADM-SHP" procedure for equal variances and also for the unequal variances situations checked here. For shrinkage coefficient estimation, the ADM-SHP procedure allows an alternative frequency interpretation. Writing $L(A)$ as the likelihood of $B_i$ with $i$ fixed, ADM-SHP estimates $B_i$ as $\hat{B_i}=V_i/(V_i+\hat{A})$ with $\hat{A}\equiv \operatorname {argmax}(A*L(A))$. This justifies the term "adjustment for likelihood maximization," ALM.