A note on Pitman's measure of closeness with balanced loss function

In this note, we consider the problem of estimating an unknown parameter θ in the sense of the Pitman's measure of closeness (PMC) using the balanced loss function (BLF). We show that the PMC comparison of estimators under the BLF can be reduced to the PMC comparison under the usual absolute error loss. The Pitman-closest estimators of the location and scale parameters under BLF are also characterized. Illustrative examples are given to show the broad range applications of the obtained results.     

Comparison of estimators of the location parameter using pitman's closeness criterion

Necessary and sufficient conditions for a linear estimator to dominate another linear estimator of a location parameter under the Pitman's criterion of comparison are discussed. Consequently it is demonstrated that a linear biased estimator can not dominate a linear unbiased estimator under Pitman's criterion and that the sample mean is the Closest Linear Unbiased Estimator (CLUE). It is also shown that the ridge regression estimator with a known biasing constant can not dominate the ordinary least squares estimator. If an estimator δdominates an estimator δin the average loss sense then sufficient conditions are obtained under which δis also preferred over δunder Pitman's criterion. Further we obtain sufficient conditions under which preference under the Pitman's criterion will lead to preference under the mean squared error sense.     

Preference regions and their probabilities based on a rectangular grid

In this article we consider a town in which the thoroughfares are laid out in a rectangular grid. Using the l₁ metric, we determine the Dirichlet regions for competitive convenience stores. Under the assumption of normality, we exemplify a technique for calculating the probability of the Dirichlet region associated with each convenience store. This method is generalized to the case where intersecting thoroughfares are oblique. Finally the example is used to illustrate the calculation of posterior Pitman's measure of closeness for various Bayesian estimators.     

Preface

The appearance of this issue will mark almost two years since its inception by the late Editor, Professor Don B. Owen. Professor Malay Ghosh suggested a special issue on Pitman's Measure of Closeness (PMC) to Professor Owen in autumn of 1989. After a thorough review process the issue was finalized in June 1991. It is with remorse that we publish this special issue in memory of Professor Owen.

The completion of the issue coincided with a special conference, “Pitman's Measure of Cfoseness: Celebrating a Decade of Renaissance,” held on June 15, 1991 at the University of Texas at San Antonio (UTSA). The papers in this special issue, except those of Professor Kubokawa and Drs. Bertuzzi and Gandolfi, were presented and discussed. The conference participants included C.R. Rao (Pennsylvania State University), Colin Blyth (Queen's University), and H.T. David (Iowa State University). Further, P.K. Sen (University of North Carolina) and Malay Ghosh (University of Florida) gave keynote addresses that respectively set the themes for the morning and afternoon sessions.

The conference banquet held in the Regent's Room at the University of Texas at San Antonio, featured stimulating addresses by C.R. Rao and Colin Blyth on some of the major controversies of PMC such as intransitiveness and Berkson's conjecture. We are grateful to the University of Texas at San Antonio for making this conference a reality. In particular, we thank Professor Shair Ahmad, Director of the Division of Mathematics, Computer Science, and Statistics at UTSA for funding travel expenses not only for the invited speakers but also for many of the younger researchers. We also acknowledge University President Samuel Kirkpatrick who made the KIVA room available for the technical sessions and the Regent's Room for the Banquet. We also acknowledge the financial support of Bell Helicopter Textron, Inc. as a cosponsor of the conference.     

Estimation of an exponential quantile under pitman's measure of closeness

We consider the problem of estimating a quantile of an exponential distribution with unknown location and scale parameters under Pitman's measure of closeness (PMC). The loss function is required to satisfy some mild conditions but is otherwise arbitrary. An optimal estimator is obtained in the class of location-scale-equivariant estimators, and its admissibility in the sense of PMC is investigated.     

On the estimation of missing values in AR(1) model with exponential innovations

We consider estimation of a missing value for a stationary autoregressive process of order one with exponential innovations and compare two methods of estimation of the missing value, with respect to Pitman's measure of closeness (PMC).     

A New Test Procedure for Decreasing Mean Residual Life

A new test statistic for testing the strict DMRL property of life distribution is developed. The asymptotic normality is established and the comparison between the test proposed and some other related ones in literature is conducted through evaluating the Pitman's asymptotic relative efficiency. Edge-worth expansion is also employed to improve the accuracy of the convergence rate of the test statistic. Some numerical results are presented as well to demonstrate the performance and the asymptotic normality of the new testing procedure.     

Forecasting With Bayesian Vector Autoregressions—Five Years of Experience

The results obtained in five years of forecasting with Bayesian vector autoregressions (BVAR's) demonstrate that this inexpensive, reproducible statistical technique is as accurate, on average, as those used by the best known commercial forecasting services. This article considers the problem of economic forecasting, the justification for the Bayesian approach, its implementation, and the performance of one small BVAR model over the past five years.     

Bayesian Interpretations of Heteroskedastic Consistent Covariance Estimators Using the Informed Bayesian Bootstrap

This article provides Bayesian interpretations for White's heteroskedastic consistent (HC) covariance estimator, and various modifications of it, in linear regression models. An informed Bayesian bootstrap provides a useful framework.     

The small sample properties of the restricted principal component regression estimator in linear regression model

In regression analysis, to deal with the problem of multicollinearity, the restricted principal components regression estimator is proposed. In this paper, we compared the restricted principal components regression estimator, the principal components regression estimator, and the ordinary least-squares estimator with each other under the Pitman's closeness criterion. We showed that the restricted principal components regression estimator is always superior to the principal components regression estimator, under certain conditions the restricted principal components regression estimator is superior to the ordinary least-squares estimator under the Pitman's closeness criterion and under certain conditions the principal components regression estimator is superior to the ordinary least-squares estimator under the Pitman's closeness criterion.     

Nonparametric hierarchical Bayes analysis of binomial data via Bernstein polynomial priors

For binomial data analysis, many methods based on empirical Bayes interpretations have been developed, in which a variance‐stabilizing transformation and a normality assumption are usually required. To achieve the greatest model flexibility, we conduct nonparametric Bayesian inference for binomial data and employ a special nonparametric Bayesian prior—the Bernstein–Dirichlet process (BDP)—in the hierarchical Bayes model for the data. The BDP is a special Dirichlet process (DP) mixture based on beta distributions, and the posterior distribution resulting from it has a smooth density defined on [0, 1]. We examine two Markov chain Monte Carlo procedures for simulating from the resulting posterior distribution, and compare their convergence rates and computational efficiency. In contrast to existing results for posterior consistency based on direct observations, the posterior consistency of the BDP, given indirect binomial data, is established. We study shrinkage effects and the robustness of the BDP‐based posterior estimators in comparison with several other empirical and hierarchical Bayes estimators, and we illustrate through examples that the BDP‐based nonparametric Bayesian estimate is more robust to the sample variation and tends to have a smaller estimation error than those based on the DP prior. In certain settings, the new estimator can also beat Stein's estimator, Efron and Morris's limited‐translation estimator, and many other existing empirical Bayes estimators. The Canadian Journal of Statistics 40: 328–344; 2012 © 2012 Statistical Society of Canada     

A test for the variance residual life

In this paper we develop a test based on the empirical distribution function for the alternative representing 'decreasing variance residual life1 property. The test is consistent with asymptotically normal test statistic and is shown to perform well in the Pitman's asymptotic relative efficiency sense.     

Tests for the total time on test transform order

In this paper we consider two test statistics for testing the strict TTT transform order between two life distributions of interest. We give their asymptotic distributions and compare our tests with some other related tests in terms of Pitman's asymptotic efficiency. Also we present some results to show the performance and the asymptotic normality of our tests.     

Performance of extrapolation based on Pitman's measure of closeness in spatial regression models with extended skew t innovations

Pitman's measure of closeness and mean square error of prediction are two well-known criteria for comparison between estimators and also between predictors. In a stationary first order multiplicative spatial autoregressive model, interpolation and extrapolation are compared based on these two criteria. A wide class of different innovation types are also studied containing Gaussian, exponential, asymmetric Laplace and extended skew t distributions.     

Bayesian Estimation of the Number of Change Points in Simple Linear Regression Models

A Bayesian approach is considered to detect the number of change points in simple linear regression models. A normal-gamma empirical prior for the regression parameters based on maximum likelihood estimator (MLE) is employed in the analysis. Under mild conditions, consistency for the number of change points and boundedness between the estimated location and the true location of the change points are established. The Bayesian approach to the detection of the number of change points is suitable whether the switching simple regression is continuous or discontinuous. Some simulation results are given to confirm the accuracy of the proposed estimator.     

Bayesian partial likelihood approach for tied observations

The Bayesian analysis based on the partial likelihood for Cox's proportional hazards model is frequently used because of its simplicity. The Bayesian partial likelihood approach is often justified by showing that it approximates the full Bayesian posterior of the regression coefficients with a diffuse prior on the baseline hazard function. This, however, may not be appropriate when ties exist among uncensored observations. In that case, the full Bayesian and Bayesian partial likelihood posteriors can be much different. In this paper, we propose a new Bayesian partial likelihood approach for many tied observations and justify its use.     

Empirical study of some non-parametric tests for dispersion of correlated data

Results of a computer simulation study of power and robustness of three competitor tests for comparing scales, for use with correlated data: Rothstein, Richardson and Bell (RRB), Arvesen, and Pitman, are presented. It is found that unless one could ímprove the approximate null distributions for Arvesen's and Pitman's test, RRB's procedure is best, having simulated probabilities of Type I error closest to the test's nominal α and being reasonably robust and powerful, for all distributions considered.     

Tests for exponentiality against new better than old in expectation and new better than some used in expectation alternatives

We present statistical procedures for testing exponentiality againt New Better than Old in Expectation (NBOE) and New Better than Some Used in Expectation (NBSUE) alternatives. The test statistics devised for the purpose are U-Statistics and hence asymptotically normally distributed. Pitman's asymptotic relative efficiency results have been obtained and Monte Carlo study presented to compare power of the test proposed with the other tests.     

A class of closeness criteria

A generalized class of closeness criteria for the pairwise comparison of esti¬mators is defined. This class contains an infinite number of members including Pitman's measure of closeness and at least one transitive criterion. Several specific members of the class are examined, and their relationships to Rao concentration and stochastic domination are shown. Graphical and analyti¬cal characterizations are shown for these members of the class. Examples are given which illustrate the behavior of some of these criteria.     

Bayesian–frequentist hybrid model with application to the analysis of gene copy number changes

Gene copy number (GCN) changes are common characteristics of many genetic diseases. Comparative genomic hybridization (CGH) is a new technology widely used today to screen the GCN changes in mutant cells with high resolution genome-wide. Statistical methods for analyzing such CGH data have been evolving. Existing methods are either frequentist's or full Bayesian. The former often has computational advantage, while the latter can incorporate prior information into the model, but could be misleading when one does not have sound prior information. In an attempt to take full advantages of both approaches, we develop a Bayesian-frequentist hybrid approach, in which a subset of the model parameters is inferred by the Bayesian method, while the rest parameters by the frequentist's. This new hybrid approach provides advantages over those of the Bayesian or frequentist's method used alone. This is especially the case when sound prior information is available on part of the parameters, and the sample size is relatively small. Spatial dependence and false discovery rate are also discussed, and the parameter estimation is efficient. As an illustration, we used the proposed hybrid approach to analyze a real CGH data.