Sudip Bose

Pitman's measure of closeness for symmetric stable distributions

Abstract: This paper considers symmetric stable distributions with different exponents y (0 < y Q 2) and studies Pitman’s measure of closeness of sample averages (both weighted and unweighted) based on different sample sizes. The behavior of measures of concentration of such averages around the point of symmetry is also studied. It is found that while Pitman’s closeness criterion is compatible with the measure of concentration for 1 d y Q 2, this need not always be so for 0 < y < 1. The relationship of our results with the ones given in Blyth and Pathak (1985) is also pointed out. AMS 1980 Subject Classification: 62FlO. Keywords: Pitman-closeness; measure of concentration; stable distributions; symmetric; sample averages; weighted; normal; Cauchy. 1. Introduction Pitman’s measure of closeness, originally introduced in Pitman (19371, is, in effect, a comparison between two estimators of a parameter of interest based on the joint distribution of resulting losses. This is in sharp contrast to the usual decision-theory analysis based on the marginal distributions of the losses. It is shown in Hwang (1985) that one estimator dominates the other uniformly under any symmetric bowl-shaped loss if and only if the former has greater concentration around the parameter of interest than the latter. On the other hand, even though an estimator has greater concentration around the parameter than another, the former can be vastly inferior to the latter according to Pitman’s closeness criterion.

Maximin efficiency-robust tests and some extensions

Abstract The Maximin Efficiency-Robust Test idea of Gastwirth (1966) was to maximize the minimum asymptotic power (for fixed size) versus special local families of alternatives over some specially chosen families of score statistics. This approach is reviewed from a general decision-theoretical perspective, including some Bayesian variants. For two-sample censored-data rank tests and stochastically ordered but not proportional-hazard alternatives, the MERT approach leads to customized weighted-logrank tests for which the weights depend on estimated random-censoring distributions. Examples include statistics which perform well against both Lehmann and logistic alternatives or against families of alternatives which include increasing, decreasing, and ‘bathtub-shaped’ hazards.

Bayesian robustness with more than one class of contaminations

Abstract This paper generalizes the usual e-contamination class Γ(π 0 , e, Q )={π: π=(1−e)π 0 +eq, for some q∈ Q } . The class considered is Γ(e 0 , e 1 ,…,e m , Q 0 , Q 1 ,…, Q m ) which is defined to be {π: π=e 0 q 0 +e 1 q 1 +⋯+e m q m , for some q 0 ∈ Q 0 , q 1 ∈ Q 1 ,…, q m ∈ Q m } . This class encompasses many of the classes of priors used in Bayesian robustness and sensitivity analyses. Also, by taking Q 0 ⊂ Q 1 ⊂⋯⊂ Q m one can model partial prior information of ‘graded’ precision, i.e.where, with decreasing confidence, one can specify the prior more precisely. One can specify quantiles of the prior, either exactly or within bounds, moments of the prior, shape or smoothness constraints, or that the prior is within neighborhoods (of varying sizes), metric or otherwise, of a base prior. One need not assume an inclusion ordering among the Q i s, and thus could model conflicting partial prior information with various degrees of prior belief in the information. Extremes of posterior expectations can be determined for this class by using the linearization techniques of Lavine (J. Amer. Statist. Assoc.86 (1991) 143–156) and Lavine et al. (J. Statist. Plann. Inference. to appear).

Are Nonhomogeneous Poisson Process Models Preferable to General-Order Statistics Models for Software Reliability Estimation?

As alternatives to general order statistics (GOS) models, several nonhomogeneous Poisson process (NHPP) models have been proposed in the literature for software reliability estimation. It has been known that an NHPP model in which the expected number of events (μ) over (0, ∞) is finite (called an NHPP-I process) is a Poisson mixture of GOS processes. We find that the underlying GOS model better fits the data in the sense that it has a larger likelihood than the NHPP-I model. Also, among unbiased estimators for an NHPP-I model, if an estimator is optimal for estimating a related feature of the underlying GOS model, then it is also optimal for the NHPP-I estimation problem. We conducted a simulation study to compare maximum likelihood estimators for one NHPP-I model and for its corresponding GOS model. The results show for small μ and small debugging time the estimators for NHPP-I model are unreliable. For longer debugging time the estimators for the two models behave similarly. These results and certain logical issues suggest that compared to an NHPP-I model, its underlying GOS model may better serve for analyzing software failure data.

Non-dependence of the predictive distribution on the population size

We show that, given the number of defectives in a sample, the predictive distribution of the number of defectives in a subsequent sample does not depend on the population size provided a simple relationship holds between the prior distributions on the number of defectives in the population, for varying population sizes.

On Smoothness Constraints with Shape Constraints in a Robust Bayesian Analysis

We examine the role of the likelihood in Bayesian robustness with the density ratio class (DeRobertis and Hartigan, 1981. Ann. Stat.) and show how to impose smoothness on the density ratio class after imposing shape constraints. We discuss how to impose shape constraints on the density bounded class (Lavine, 1991. JASA)

On the Imposition of Shape Constraints in a Robust Bayesian Analysis

We consider the problem of imposing shape constraints on a neighborhood class — the density ratio class (DeRobertis and Hartigan, 1981). Bose (1994) used mixture distributions to impose shape and smoothness constraints simultaneously. We discuss how one may impose either or both unimodality and symmetry without requiring simultaneous imposition of a smoothness constraint.

Robustness in Bayesian nonparametrics

We consider Bayesian robustness in the context of Bayesian Nonparametrics, and specifically for the Dirichlet Process prior. We show how to find an optimal procedure, based on C-minimax posterior regret (CMPR) for a class of priors C. We consider regret based on squared error loss. The neighborhood classes considered are the density ratio (DR) class and the epsilon-contamination class. We find optimal robust estimators in a Nonparametric Bayesian setting.Minimax (with respect to a class of priors) posterior regret is the optimality criterion.We consider density ratio classes and epsilon-contamination classes.

When Population Size Does Not Matter: Robust Bayesian Testing for Categorical Populations

We present a strongly robust Bayesian test of the hypothesis of equality of distributions for populations consisting of data in finitely many categories. The Bayes factors are the same for infinite and finite populations. Under the hypothesis of inequality, one distribution can be viewed as an exponentially tilted or exponentially distorted version of another. We illustrate the method by calculating Bayes factors for a range of data values.