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 1

Estimation in the interval censoring model is considered. A class of smooth functionals is introduced, of which the mean is an example. The asymptotic information lower bound for such functionals can be represented as an inner product of two functions. In case 1, i.e. one observation time per unobservable event time, both functions can be given explicitly. We mainly consider case 2, with two observation times for each unobservable event time, in the situation that the observation times can not become arbitrarily close to each other. For case 2, one of the functions in the inner product can only be given implicitly as solution to a Fredholm integral equation. We study properties of this solution and, in a sequel to this paper, prove that the nonparametric maximum likelihood estimator of the functional asymptotically reaches the information lower bound.     

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.     

Diagnostic analysis for a vector autoregressive model under Student′s t‐distributions

In this paper, we use the local influence method to study a vector autoregressive model under Student^′s t‐distributions. We present the maximum likelihood estimators and the information matrix. We establish the normal curvature diagnostics for the vector autoregressive model under three usual perturbation schemes for identifying possible influential observations. The effectiveness of the proposed diagnostics is examined by a simulation study, followed by our data analysis using the model to fit the weekly log returns of Chevron stock and the Standard & Poor's 500 Index as an application.     

Minimax‐Optimal Strength of Statistical Evidence for a Composite Alternative Hypothesis

While the likelihood ratio measures statistical support for an alternative hypothesis about a single parameter value, it is undefined for an alternative hypothesis that is composite in the sense that it corresponds to multiple parameter values. Regarding the parameter of interest as a random variable enables measuring support for a composite alternative hypothesis without requiring the elicitation or estimation of a prior distribution, as described below. In this setting, in which parameter randomness represents variability rather than uncertainty, the ideal measure of the support for one hypothesis over another is the difference in the posterior and prior log‐odds. That ideal support may be replaced by any measure of support that, on a per‐observation basis, is asymptotically unbiased as a predictor of the ideal support. Such measures of support are easily interpreted and, if desired, can be combined with any specified or estimated prior probability of the null hypothesis. Two qualifying measures of support are minimax‐optimal. An application to proteomics data indicates that a modification of optimal support computed from data for a single protein can closely approximate the estimated difference in posterior and prior odds that would be available with the data for 20 proteins.     

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).     

Small‐scale Inference: Empirical Bayes and Confidence Methods for as Few as a Single Comparison

Empirical Bayes methods of estimating the local false discovery rate (LFDR) by maximum likelihood estimation (MLE), originally developed for large numbers of comparisons, are applied to a single comparison. Specifically, when assuming a lower bound on the mixing proportion of true null hypotheses, the LFDR MLE can yield reliable hypothesis tests and confidence intervals given as few as one comparison. Simulations indicate that constrained LFDR MLEs perform markedly better than conventional methods, both in testing and in confidence intervals, for high values of the mixing proportion, but not for low values. (A decision‐theoretic interpretation of the confidence distribution made those comparisons possible.) In conclusion, the constrained LFDR estimators and the resulting effect‐size interval estimates are not only effective multiple comparison procedures but also they might replace p‐values and confidence intervals more generally. The new methodology is illustrated with the analysis of proteomics data.