Optimally robust credible sets with a class of priors

We determine a credible set A   that is the “best” with respect to the variation of the prior distribution in a neighborhood Γ$\Gamma$ of the starting prior _π0(θ)${\pi }_0(\theta )$. Among the class of sets with credibility γ$\gamma$ under _π0${\pi }_0$, the “optimally robust” set will be the one which maximizes the minimum probability of including θ$\theta$ as the prior varies over Γ$\Gamma$. This procedure is also Γ-minimax$\Gamma \text{-minimax}$ with respect to the risk function, probability of non-inclusion. We find the optimally robust credible set for three neighborhood classes Γ$\Gamma$, the ε-contamination$\epsilon \text{-contamination}$ class, the density ratio class and the density bounded class. A consequence of this investigation is that the maximum likelihood set is seen to be an optimal credible set from a robustness perspective.     

On batch queueing systems: A combinatorial approach

We consider batch queueing systems M/M^H/1$M/M^H/1$ and M^H/M/1$M^H/M/1$ with catastrophes. The transient probability functions of these queueing systems are obtained by a Lattice Path Combinatorics approach that utilizes randomization and dual processes. Steady state distributions are also determined. Generalization to systems having batches of different sizes are discussed.     

Bayesian variable selection using an adaptive powered correlation prior

The problem of selecting the correct subset of predictors within a linear model has received much attention in recent literature. Within the Bayesian framework, a popular choice of prior has been Zellner's g$g$-prior which is based on the inverse of empirical covariance matrix of the predictors. An extension of the Zellner's prior is proposed in this article which allow for a power parameter on the empirical covariance of the predictors. The power parameter helps control the degree to which correlated predictors are smoothed towards or away from one another. In addition, the empirical covariance of the predictors is used to obtain suitable priors over model space. In this manner, the power parameter also helps to determine whether models containing highly collinear predictors are preferred or avoided. The proposed power parameter can be chosen via an empirical Bayes method which leads to a data adaptive choice of prior. Simulation studies and a real data example are presented to show how the power parameter is well determined from the degree of cross-correlation within predictors. The proposed modification compares favorably to the standard use of Zellner's prior and an intrinsic prior in these examples.     

Addition of runs to an s-level supersaturated design

The purpose of the present work is to extend the work of Gupta et al. (2010) to s  -level column balanced supersaturated designs. Addition of runs to an existing E(χ²)-optimal$E({\chi }^2)\mathrm{-}\mathrm{optimal}$ supersaturated design and to study the optimality of the resulting design is an important issue. This paper considers the study of the optimality of the resulting design. A lower bound to E(χ²)$E({\chi }^2)$ has been obtained for the extended supersaturated designs. Some examples and a small catalogue of E(χ²)-optimal$E({\chi }^2)\mathrm{-}\mathrm{optimal}$ supersaturated designs are also presented.     

Empirical Bayes regression analysis with many regressors but fewer observations

In this paper, we consider the prediction problem in multiple linear regression model in which the number of predictor variables, p, is extremely large compared to the number of available observations, n  . The least-squares predictor based on a generalized inverse is not efficient. We propose six empirical Bayes estimators of the regression parameters. Three of them are shown to have uniformly lower prediction error than the least-squares predictors when the vector of regressor variables are assumed to be random with mean vector zero and the covariance matrix (1/n)X^tX$(1/n){\mathit{X}}^{\mathrm{t}}\mathit{X}$ where X^t=(_x1,…,_xn)${\mathit{X}}^{\mathrm{t}}=({\mathit{x}}_1,\dots ,{\mathit{x}}_n)$ is the p×n$p\times n$ matrix of observations on the regressor vector centered from their sample means. For other estimators, we use simulation to show its superiority over the least-squares predictor.     

Statistical analysis for rounded data

When random variables do not take discrete values, observed data are often the rounded values of continuous random variables. Errors caused by rounding of data are often neglected by classical statistical theories. While some pioneers have identified and made suggestions to rectify the problem, few suitable approaches were proposed. In this paper, we propose an approximate MLE (AMLE) procedure to estimate the parameters and discuss the consistency and asymptotic normality of the estimates. For our illustration, we shall consider the estimates of the parameters in AR(p)$\mathit{AR}(p)$ and MA(q)$\mathit{MA}(q)$ models for rounded data.     

Analysis of one-way layout of count data in the presence of over or under dispersion

This article develops test statistics for the homogeneity of the means of several treatment groups of count data in the presence of over-dispersion or under-dispersion when there is no likelihood available. The C(α)$C(\alpha )$ or score type tests based on the models that are specified by only the first two moments of the counts are obtained using quasi-likelihood, extended quasi-likelihood, and double extended quasi-likelihood. Monte Carlo simulations are then used to study the comparative behavior of these C(α)$C(\alpha )$ statistics compared to the C(α)$C(\alpha )$ statistic based on a parametric model, namely, the negative binomial model, in terms of the following: size; power; robustness for departures from the data distribution as well as dispersion homogeneity. These simulations demonstrate that the C(α)$C(\alpha )$ statistic based on the double extended quasi-likelihood holds the nominal size at the 5% level well in all data situations, and it shows some edge in power over the other statistics, and, in particular, it performs much better than the commonly used statistic based on the quasi-likelihood. This C(α)$C(\alpha )$ statistic also shows robustness for moderate heterogeneity due to dispersion. Finally, applications to ecological, toxicological and biological data are given.     

Constrained Bayes and empirical Bayes estimation under random effects normal ANOVA model with balanced loss function

The paper develops constrained Bayes and empirical Bayes estimators in the random effects ANOVA model under balanced loss functions. In the balanced normal–normal model, estimators of the Bayes risks of the constrained Bayes and constrained empirical Bayes estimators are provided which are correct asymptotically up to O(m^-1)$\mathrm{O}(m^{-1})$, that is the remainder term is o(m^-1)$\mathrm{o}(m^{-1})$, m$m$ denoting the number of strata.     

Optimal block designs for three treatments when observations are correlated

E$E$-optimal designs for comparing three treatments in blocks of size three are identified, where intrablock observations are correlated according to a first order autoregressive error process with parameter ρ∈(0,1)$\rho \in (0,1)$. For number of blocks b   of the form b=3n+1$b=3n+1$, there are two distinct optimal designs depending on the value of ρ$\rho$, with the best design being unequally replicated for large ρ$\rho$. For other values of b$b$, binary, equireplicate designs with specified within-block assignment patterns are best. In many cases, the stronger majorization optimality is established.