Estimation of multiple poisson means: A compromise between maximum likelihood and empirical bayes estimators

For estimatingp(⩾ 2) independent Poisson means, the paper considers a compromise between maximum likelihood and empirical Bayes estimators. Such compromise estimators enjoy both good componentwise as well as ensemble properties. Research supported by the NSF Grant Number MCS-8218091.     

Bayes Convolution

A general convolution theorem within a Bayesian framework is presented. Consider estimation of the Euclidean parameter θ by an estimator T within a parametric model. Let W be a prior distribution for θ and define G as the W -average of the distribution of T - θ under θ . In some cases, for any estimator T the distribution G can be written as a convolution G = K * L with K a distribution depending only on the model, i.e. on W and the distributions under θ of the observations. In such a Bayes convolution result optimal estimators exist, satisfying G = K . For location models we show that finite sample Bayes convolution results hold in the normal, loggamma and exponential case. Under regularity conditions we prove that normal and loggamma are the only smooth location cases. We also discuss relations with classical convolution theorems.     

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     

On the Risk Performance of Bays Empirical Bayes Procedures for Classification Between N(-1,l) and N(1,1)

In empirical Bayes decision making, the Bayes empirical Bayes approach is diccussed by Gilliland and Boyer (1979). In the finite state component case, the Bayes empirical Bayes procedures are shown to have optimal properties in a fairly general setting and believed to have small sample advantage over the classical rules. The flexibility of making desirable adjustments for these decision procedures by choice of prior enables one to set a proper strategy when dealing with actual problems.
The applications of Bayes empirical Bayes procedures, however, create some interesting theoretical and computational problems as they are fairly complicated in structure. This paper gives a brief introduction into the Bayes empirical Bayes approach, and, to illustrate it, explicit results are given for testing H₀: N(-1,1) against H₁: N(1,1).     

Smooth estimators for estimating order restricted scale parameters of two gamma distributions

We consider the problem of component-wise estimation of ordered scale parameters of two gamma populations, when it is known apriori which population corresponds to each ordered parameter. Under the scale equivariant squared error loss function, smooth estimators that improve upon the best scale equivariant estimators are derived. These smooth estimators are shown to be generalized Bayes with respect to a non-informative prior. Finally, using Monte Carlo simulations, these improved smooth estimators are compared with the best scale equivariant estimators, their non-smooth improvements obtained in Vijayasree, Misra & Singh (1995), and the restricted maximum likelihood estimators. Acknowledgments. Authors are thankful to a referee for suggestions leading to improved presentation.     

Robust linear static panel data models using ε-contamination

The paper develops a general Bayesian framework for robust linear static panel data models using$\epsilon$-contamination. A two-step approach is employed to derive the conditional type-II maximum likelihood (ML-II) posterior distribution of the coefficients and individual effects. The ML-II posterior means are weighted averages of the Bayes estimator under a base prior and the data-dependent empirical Bayes estimator. Two-stage and three stage hierarchy estimators are developed and their finite sample performance is investigated through a series of Monte Carlo experiments. These include standard random effects as well as Mundlak-type, Chamberlain-type and Hausman–Taylor-type models. The simulation results underscore the relatively good performance of the three-stage hierarchy estimator. Within a single theoretical framework, our Bayesian approach encompasses a variety of specifications while conventional methods require separate estimators for each case.     

Double Empirical Bayes Testing

Analysing data from large-scale, multiexperiment studies requires scientists to both analyse each experiment and to assess the results as a whole. In this article, we develop double empirical Bayes testing (DEBT), an empirical Bayes method for analysing multiexperiment studies when many covariates are gathered per experiment. DEBT is a two-stage method: in the first stage, it reports which experiments yielded significant outcomes and in the second stage, it hypothesises which covariates drive the experimental significance. In both of its stages, DEBT builds on the work of Efron, who laid out an elegant empirical Bayes approach to testing. DEBT enhances this framework by learning a series of black box predictive models to boost power and control the false discovery rate. In Stage 1, it uses a deep neural network prior to report which experiments yielded significant outcomes. In Stage 2, it uses an empirical Bayes version of the knockoff filter to select covariates that have significant predictive power of Stage 1 significance. In both simulated and real data, DEBT increases the proportion of discovered significant outcomes and selects more features when signals are weak. In a real study of cancer cell lines, DEBT selects a robust set of biologically plausible genomic drivers of drug sensitivity and resistance in cancer.     

Monotone empirical Bayes estimators for the continuous one-parameter exponential family

Abstract  A class of empirical Bayes estimators (EBE's) is proposed for estimating the natural parameter of a one-parameter exponential family. In contrast to related EBE's proposed and investigated until now, the EBE's presented in this paper possess the nice property of being monotone by construction. Based on an arbitrary reasonable estimator of the underlying marginal density, a simple algorithm is given to construct a monotone EBE. Two representations of these EBE's are given, one of which serves as a tool in establishing asymptotic results, while the other one, related with isotonic regression, proves useful in the actual computation.     

Bayesian estimation and prediction based on Rayleigh sample quantiles

Ordered data arise naturally in many fields of statistical practice. Often some sample values are unknown or disregarded due to various reasons. On the basis of some sample quantiles from the Rayleigh distribution, the problems of estimating the Rayleigh parameter, hazard rate and reliability function, and predicting future observations are addressed using a Bayesian perspective. The construction of β-content and β-expectation Bayes tolerance limits is also tackled. Under squared-error loss, Bayes estimators and predictors are deduced analytically. Exact tolerance limits are derived by solving simple nonlinear equations. Highest posterior density estimators and credibility intervals, as well as Bayes estimators and predictors under linear loss, can easily be computed iteratively.     

Methods for Default and Robust Bayesian Model Comparison: the Fractional Bayes Factor Approach

In the Bayesian approach to model selection and hypothesis testing, the Bayes factor plays a central role. However, the Bayes factor is very sensitive to prior distributions of parameters. This is a problem especially in the presence of weak prior information on the parameters of the models. The most radical consequence of this fact is that the Bayes factor is undetermined when improper priors are used. Nonetheless, extending the non-informative approach of Bayesian analysis to model selection/testing procedures is important both from a theoretical and an applied viewpoint. The need to develop automatic and robust methods for model comparison has led to the introduction of several alternative Bayes factors. In this paper we review one of these methods: the fractional Bayes factor (O'Hagan, 1995). We discuss general properties of the method, such as consistency and coherence. Furthermore, in addition to the original, essentially asymptotic justifications of the fractional Bayes factor, we provide further finite-sample motivations for its use. Connections and comparisons to other automatic methods are discussed and several issues of robustness with respect to priors and data are considered. Finally, we focus on some open problems in the fractional Bayes factor approach, and outline some possible answers and directions for future research.     

Relative efficiencies of some simple Bayes estimators of coefficients in a dynamic equation with serially correlated errors - II

Generalized least squares estimators, with estimated variance-covariance matrices, and maximum likelihood estimators have been proposed in the literature to deal with the problem of estimating autoregressive models with autocorrelated disturbances. In this paper we compare the small sample efficiencies of these estimators with those of some approximate Bayes estimators. The comparison is done with the help of a sampling experiment applied to a model specification. Though these Bayes estimators utilize very weak prior information, they out-perform the sampling theory estimators in every case we consider.     

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).     

Bayesian Calibration of p‐Values from Fisher's Exact Test

p‐Values are commonly transformed to lower bounds on Bayes factors, so‐called minimum Bayes factors. For the linear model, a sample‐size adjusted minimum Bayes factor over the class of g‐priors on the regression coefficients has recently been proposed (Held & Ott, The American Statistician 70(4), 335–341, 2016). Here, we extend this methodology to a logistic regression to obtain a sample‐size adjusted minimum Bayes factor for 2 × 2 contingency tables. We then study the relationship between this minimum Bayes factor and two‐sided p‐values from Fisher's exact test, as well as less conservative alternatives, with a novel parametric regression approach. It turns out that for all p‐values considered, the maximal evidence against the point null hypothesis is inversely related to the sample size. The same qualitative relationship is observed for minimum Bayes factors over the more general class of symmetric prior distributions. For the p‐values from Fisher's exact test, the minimum Bayes factors do on average not tend to the large‐sample bound as the sample size becomes large, but for the less conservative alternatives, the large‐sample behaviour is as expected.     

Limited deviation estimators of the Poisson parameter

Estimators for the Poisson parameter are proposed which perform well with respect to both a weighted, i.e. Bayes, and unweighted risk criterion. The estimators follow the Bayes rule (with respect to a conjugate gamma prior) as closely as possible subject to a restraint imposed on the allowable deviation from the minimax estimate. The resulting class of rules maintains good performance with respect to the Bayes criterion while at the same time possessing bounded risk functions. The excess Bayes risk incurred is compared to a lower bound on the optimal restricted Bayes risk.     

Bayes sequential estimation for an exponential family of processes: A discrete time approach

The problem of Bayes sequential estimation of the mean value parameter of continuous time processes with stationary independent increments having exponential-type likelihood functions is considered. Using a weighted square error loss and observing cost involving both a time cost and a state cost, the explicit solutions to the problem are derived. A discrete time approach is taken in which decisions are made at the end of time intervals having lengtht. The examples of optimal procedures in the case when the cost of observation includes a squared state cost are also given.     

Objective Bayes Factors for Inequality Constrained Hypotheses

This paper will present a Bayes factor for the comparison of an inequality constrained hypothesis with its complement or an unconstrained hypothesis. Equivalent sets of hypotheses form the basis for the quantification of the complexity of an inequality constrained hypothesis. It will be shown that the prior distribution can be chosen such that one of the terms in the Bayes factor is the quantification of the complexity of the hypothesis of interest. The other term in the Bayes factor represents a measure of the fit of the hypothesis. Using a vague prior distribution this fit value is essentially determined by the data. The result is an objective Bayes factor.     

Hybrid estimators for small diffusion processes based on reduced data

We deal with the Bayes type estimators and the maximum likelihood type estimators of both drift and volatility parameters for small diffusion processes defined by stochastic differential equations with small perturbations from high frequency data. From the viewpoint of numerical analysis, initial Bayes type estimators for both drift and volatility parameters based on reduced data are required, and adaptive maximum likelihood type estimators with the initial Bayes type estimators, which are called hybrid estimators, are proposed. The asymptotic properties of the initial Bayes type estimators based on reduced data are derived and it is shown that the hybrid estimators have asymptotic normality and convergence of moments. Furthermore, a concrete example and simulation results are given.     

An Example of Data Technology Product: A Control Chart for Weibull Processes

This work proposes a new Shewhart-type control chart of the Weibull percentile (i.e. the reliable life) as a practical example of a product attained following the Data Technology (DT) approach. DT is briefly introduced as a new discipline defined apart from Information Technology (IT). Following this approach, some specific Bayes estimators are selected from literature and then used to build the above new chart. These estimators allow to improve the control making use of any available kind of data (statistical and non-statistical). The operative steps of DT approach are fully explained. The results are illustrated by means of a real applicative example.     

Monotonizing empirical Bayes estimators for a class of of discrete distributions with monotone likelihood ratio

A procedure is given for the construction of a monotone estimator that dominates a given estimator for a class of discrete distributions with monotone likelihood ratio. This procedure is applied to some empirical Bayes estimators. Monte Carlo results are given that demonstrate the usefulness of monotonizing.     

Modelling of repairable systems with various degrees of repair

The availability of a stochastic repairable system depends on the failure behaviour and on repair strategies. In this paper, we deal with a general repair model for a system using auxiliary counting processes and corresponding intensities which include various degrees of repair (between minimal repair and perfect repair). For determining the model parameters we need estimators depending on failure times and repair times: maximum likelihood (ML) estimator and Bayes estimators are considered. Special results are obtained by the use of Weibull-type intensities and random observation times.