Gauri Sankar Datta

Probability matching priors : higher order asymptotics

Introduction and the Shrinkage Argument.- Matching Priors for Posterior Quantiles.- Matching Priors for Distribution Functions.- Matching Priors for Highest Posterior Density Regions.- Matching Priors for Other Credible Regions.- Matching Priors for Prediction.

Some Remarks on Noninformative Priors

Abstract This article focuses primarily on a comparison between the reference priors of Berger and Bernardo and the reverse reference priors suggested by J. K. Ghosh. Sufficient conditions are given that provide agreement between the two classes of priors. Several examples are given showing the agreement or disagreement between the two. In addition, these priors are compared under a criterion that requires the frequentist coverage probability of the posterior region of a real-valued parametric function to match a nominal level with a remainder of O(n −1), where n denotes the sample size. The latter priors, first introduced by Welch and Peers, are obtained by solving a differential equation due to Peers. Finally, in the presence of several parameters of interest, a general class of priors that satisfies the matching criterion separately for each parameter is constructed, and examples are given to illustrate how reference or reverse reference priors fit within this class of priors.

Hierarchical Bayes Estimation of Unemployment Rates for the States of the U.S.

Abstract Under a federal-state cooperative program, the U.S. Bureau of Labor Statistics (BLS) publishes monthly unemployment rate estimates for its 50 states and the District of Columbia. The primary source of data for this estimation problem is the Current Population Survey (CPS). However, the CPS state unemployment rate estimates are unreliable, because the survey provides relatively few observations per state. Various federal agencies use state-level unemployment rate estimates for policy making and fund allocation. Thus it is important to improve on the CPS estimates. For this, we propose a hierarchical Bayes (HB) method using a time series generalization of a widely used cross-sectional model in small-area estimation. The proposed method is compared in detail with the corresponding HB method, which uses the HB analog of the well-known Fay-Herriot cross-sectional model. A third model based on a time series approach to repetitive surveys is found to be very hard to implement for these data; the resulti...

Empirical bayes estimation of median income of four-person families by state using time series and cross-sectional data

Abstract The Department of Health and Human Services uses estimates of the median income of four-person families for all the fifty states and the District of Columbia to formulate its energy assistance program for low income families. Such estimates are provided by the US Census Bureau on an annual basis. A hierarchical time series model is considered to combine information from three relevant sources: (a) Current Population Survey (CPS), (b) Decennial Censuses and (c) Bureau of Economic Analysis. An empirical Bayes (EB) method is used to smooth the CPS estimates of the median income of four-person families for the states. The proposed method is an advancement over the EB method currently used by the US Bureau of the Census in the sense that it uses a more realistic model, provides maximum likelihood and residual maximum likelihood method of variance components estimation and provides a valid measure of uncertainty of the proposed estimates which captures all different sources of variations. Compared to the corresponding hierarchical Bayes estimation, the method is very easy to implement and saves a tremendous amount of computer time. The proposed EB method is compared with rival estimators using the 1989 four-person median income figures obtained from the 1990 Census.

Model-Based Approach to Small Area Estimation

Publisher Summary This chapter reviews both frequentist and Bayesian approaches to model based small area estimation. Although the frequentist approach is still more popular among practitioners, the Bayesian approach is also gaining popularity and acceptability. The difficulty in the Bayesian approach is prior specification and computation. Although the former is still a difficult issue, enormous progress in recent years has been achieved on computational issues. It is worthwhile to point out that frequentist solutions based on jackknife or bootstrap are also computer intensive. One advantage with the Bayesian approach is that it automatically incorporates all sources of uncertainty associated with an inference problem. There are many more important applications of small area estimation encountered by various government agencies. Indirect estimates of small area means that borrow strength from other areas are referred to as cross-sectional estimates. On the other hand for a survey which is repeated regularly, one can obtain indirect estimates of small area means by borrowing strength both from other areas and the time series.

Some new Results on Probability Matching Priors

The paper has three components. First, for a realvalued parameter of interest orthogonal (Cox and Reid, 1987) to the nuisance parameter vector, we find a necessary and sufficient condition for the equivalence of second order quantile matching priors and highest posterior density regions matching priors within the class of first order quantile matching priors. Examples are presented to illustrate the result. Second, we develop a quantile matching prior in a normal hierarchical Bayesian model. This prior turns out to be different from the one proposed earlier by Morris (1983). Third, we obtain an exact matching result when the objective is prediction of a real-valued random variable from a location family of distributions. AMS (2000) Subject Classification: 62F15, 62F25, 62E20

Noninformative priors for maximal invariant parameter in group models

SummaryFor an Euclidean groupG acting freely on the parameter space, we derive, among several noninformative priors, the reference priors of Berger-Bernardo and Chang-Eaves for our parameter of interest θ1, a scalar maximal invariant parametric function. Identifying the nuisance parameter vector with the group element, we derive a simple structure of the information matrix which is used to obtain different noninformative priors. We compare these priors using the marginalization paradox and the probability-matching criteria. The Chang-Eaves and the Berger-Bernardo reference priors appear to be the most attractive choice. Several illustrative examples are considered.

Model Selection by Testing for the Presence of Small-Area Effects, and Application to Area-Level Data

The models used in small-area inference often involve unobservable random effects. While this can significantly improve the adaptivity and flexibility of a model, it also increases the variability of both point and interval estimators. If we could test for the existence of the random effects, and if the test were to show that they were unlikely to be present, then we would arguably not need to incorporate them into the model, and thus could significantly improve the precision of the methodology. In this article we suggest an approach of this type. We develop simple bootstrap methods for testing for the presence of random effects, applicable well beyond the conventional context of the natural exponential family. If the null hypothesis that the effects are not present is not rejected then our general methodology immediately gives us access to estimators of unknown model parameters and estimators of small-area means. Such estimators can be substantially more effective, for example, because they enjoy much faster convergence rates than their counterparts when the model includes random effects. If the null hypothesis is rejected then the next step is either to make the model more elaborate (our methodology is available quite generally) or to turn to existing random effects models. This article has supplementary material online.

Empirical best linear unbiased and empirical Bayes prediction in multivariate small area estimation

Abstract Small area estimation plays a prominent role in survey sampling due to a growing demand for reliable small area estimates from both public and private sectors. Popularity of model-based inference is increasing in survey sampling, particularly, in small area estimation. The estimates of the small area parameters can profitably ‘borrow strength’ from data on related multiple characteristics and/or auxiliary variables from other neighboring areas through appropriate models. Fay (1987, Small Area Statistics, Wiley, New York, pp. 91–102) proposed multivariate regression for small area estimation of multiple characteristics. The success of this modeling rests essentially on the strength of correlation of these dependent variables. To estimate small area mean vectors of multiple characteristics, multivariate modeling has been proposed in the literature via a multivariate variance components model. We use this approach to empirical best linear unbiased and empirical Bayes prediction of small area mean vectors. We use data from Battese et al. (1988, J. Amer. Statist. Assoc. 83, 28 –36) to conduct a simulation which shows that the multivariate approach may achieve substantial improvement over the usual univariate approach.

Bayesian CAR models for syndromic surveillance on multiple data streams: Theory and practice

Syndromic surveillance has, so far, considered only simple models for Bayesian inference. This paper details the methodology for a serious, scalable solution to the problem of combining symptom data from a network of US hospitals for early detection of disease outbreaks. The approach requires high-end Bayesian modeling and significant computation, but the strategy described in this paper appears to be feasible and offers attractive advantages over the methods that are currently used in this area. The method is illustrated by application to ten quarters worth of data on opioid drug abuse surveillance from 636 reporting centers, and then compared to two other syndromic surveillance methods using simulation to create known signal in the drug abuse database.