Shrinkage estimation of <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">X</Emphasis><<Emphasis Type="Italic">Y</Emphasis>) in the exponential case with common location parameter

We consider the problem of estimating R=P(X<Y) where X and Y have independent exponential distributions with parameters and respectively and a common location parameter . Assuming that there is a prior guess or estimate R₀, we develop various shrinkage estimators of R that incorporate this prior information. The performance of the new estimators is investigated and compared with the maximum likelihood estimator using Monte Carlo methods. It is found that some of these estimators are very successful in taking advantage of the prior estimate available.Acknowledgments. The authors are grateful to the editor and to the referees for their constructive comments that resulted in a substantial improvement of the paper.     

Some model-restricted shrinkage estimators for contingency tables with missing data

For contingency tables with extensive missing data, the unrestricted MLE under the saturated model, computed by the EM algorithm, is generally unsatisfactory. In this case, it may be better to fit a simpler model by imposing some restrictions on the parameter space. Perlman and Wu (1999) propose lattice conditional independence (LCI) models for contingency tables with arbitrary missing data patterns. When this LCI model fits well, the restricted MLE under the LCI model is more accurate than the unrestricted MLE under the saturated model, but not in general. Here we propose certain empirical Bayes (EB) estimators that adaptively combine the best features of the restricted and unrestricted MLEs. These EB estimators appear to be especially useful when the observed data is sparse, even in cases where the suitability of the LCI model is uncertain. We also study a restricted EM algorithm (called the ER algorithm) with similar desirable features. Received: July 1999     

Asymptotically optimal estimation of smooth functionals for interval censoring, part 2

Estimation in the interval censoring model is considered. A class of smooth functionals is introduced, of which the mean is an example. We consider case 2, with two observation times for each unobservable event time, in the situation that the observation times cannot become arbitrarily close to each other. It is proved that the nonparametric maximum likelihood estimator of the functional asymptotically reaches the information lower bound.     

Performance of empirical Bayes estimators of random coefficients in multilevel analysis: Some results for the random intercept-only model

For a multilevel model with two levels and only a random intercept, the quality of different estimators of the random intercept is examined. Analytical results are given for the marginal model interpretation where negative estimates of the variance components are allowed for. Except for four or five level-2 units, the Empirical Bayes Estimator (EBE) has a lower average Bayes risk than the Ordinary Least Squares Estimator (OLSE). The EBEs based on restricted maximum likelihood (REML) estimators of the variance components have a lower Bayes risk than the EBEs based on maximum likelihood (ML) estimators. For the hierarchical model interpretation, where estimates of the variance components are restricted being positive, Monte Carlo simulations were done. In this case the EBE has a lower average Bayes risk than the OLSE, also for four or five level-2 units. For large numbers of level-1 (30) or level-2 units (100), the performances of REML-based and ML-based EBEs are comparable. For small numbers of level-1 (10) and level-2 units (25), the REML-based EBEs have a lower Bayes risk than ML-based EBEs only for high intraclass correlations (0.5).