This study introduces a method of selecting a subset of k populations containing the best when the populations are ranked in terms of the population means. It is assumed that the populations have an unknown location family of distribution functions. The proposed method involves estimating the constant in Gupta’s subset selection procedure by bootstrap. It is shown that estimating this constant amounts to estimating the distribution function of a certain function of random variables. The proposed bootstrap method is shown to be consistent and second-order correct in the sense that the accuracy of bootstrap approximation is better than that of the approximation based on limiting distribution. Results of a simulation study are given.

Osatohanmwen, Oyegue and Ogbonmwan (2017) introduced the Gumbel-Burr XII distribution, studied its properties including estimation issues and provided a data application. However, the likelihood function given appears incorrect, and this might have affected subsequent inference results and the real data application. Here, we provide the correct likelihood function.

This paper deals with uncertainty quantification (UQ) for a class of robust estimators of population parameters of a stationary, multivariate random field that is observed at a finite number of locations s1,…, sn, generated by a stochastic design. The class of robust estimators considered here is given by the so-called M-estimators that in particular include robust estimators of location, scale, linear regression parameters, as well as the maximum likelihood and pseudo maximum likelihood estimators, among others. Finding practically useful UQ measures, both in terms of standard errors of the point estimators as well as interval estimation for the parameters is a difficult problem due to presence of inhomogeneous dependence among irregularly spaced spatial observations. Exact and asymptotic variances of such estimators have a complicated form that depends on the autocovariance function of the random field, the spatial sampling density, and also on the relative rate of growth of the sample size versus the volume of the sampling region. Similar complex interactions of these factors are also present in the sampling distributions of these estimators which makes exact calibration of confidence intervals impractical. Here it is shown that a version of the spatial block bootstrap can be used to produce valid UQ measures, both in terms of estimation of the standard error as well as interval estimation. A key advantage of the proposed method is that it provides valid approximations in very general settings without requiring any explicit adjustments for spatial sampling structures and without requiring explicit estimation of the covariance function and of the spatial sampling density.

In a Bayesian context, prior specification for inference on monotone densities is conceptually straightforward, but proving posterior convergence theorems is complicated by the fact that desirable prior concentration properties often are not satisfied. In this paper, I first develop a new prior designed specifically to satisfy an empirical version of the prior concentration property, and then I give sufficient conditions on the prior inputs such that the corresponding empirical Bayes posterior concentrates around the true monotone density at nearly the optimal minimax rate. Numerical illustrations also reveal the practical benefits of the proposed empirical Bayes approach compared to Dirichlet process mixtures.

In this paper, a new three-parameter lifetime model, called the new odd log-logistic Lindley (NOLL-L) distribution, is introduced. Some structural properties of the new distribution including ordinary and incomplete moments, quantile and generating functions and order statistics are obtained. The new density function can be expressed as a linear mixture of exponentiated Lindley densities. The different methods are discussed to estimate the model parameters and a simulation study is done to show the performance of the new distribution. The importance and flexibility of the new model are also illustrated empirically by means of two real data sets.

We consider the problem of Bayesian discriminant analysis using a high dimensional predictor. In this setting, the underlying precision matrices can be estimated with reasonable accuracy only if some appropriate additional structure like sparsity is assumed. We induce a prior on the precision matrix through a sparse prior on its Cholesky decomposition. For computational ease, we use shrinkage priors to induce sparsity on the off-diagonal entries of the Cholesky decomposition matrix and exploit certain conditional conjugacy structure. We obtain the contraction rate of the posterior distribution for the mean and the precision matrix respectively using the Euclidean and the Frobenius distance, and show that under some milder restriction on the growth of the dimension, the misclassification probability of the Bayesian classification procedure converges to that of the oracle classifier for both linear and quadratic discriminant analysis. Extensive simulations show that the proposed Bayesian methods perform very well. An application to identify cancerous breast tumorbased on image data obtained using find needle aspirate is considered.

Stochastic random phenomena studied in probability theory constitute only a part of all random phenomena, as was pointed out by Borel (1956) and Kolmogorov (1986). The need to study nonstochastic randomness led to new models. In particular, Ivanenko and Labkovsky (Sankhya A 77, 2, 237–248. 2015) defined a set of finitely additive probability measures as a set of accumulation points of a sequence or a net of frequency distributions. Here we prove the existence theorem for a nonstochastic random process described by a system of weak* closed sets of finite-dimensional distributions. Concretely, we show that a corresponding system of random variables can be defined on a probability space with a probability measure determined up to some set of measures, provided that the sets of finite-dimensional distributions are consistent.

In this paper we revisit the so-called non-stationary regression models for repeated categorical/multinomial data collected from a large number of independent individuals. The main objective of the study is to obtain consistent and efficient regression estimates after taking the correlations of the repeated multinomial data into account. The existing (1) ‘working’ odds ratios based GEE (generalized estimating equations) approach has both consistency and efficiency drawbacks. Specifically, the GEE-based regression estimates can be inconsistent which is a serious limitation. Some other existing studies use a MDL (multinomial dynamic logits) model among the repeated responses. As far as the estimation of the regression effects and dynamic dependence (i.e., correlation) parameters is concerned, they use either (2) a marginal or (3) a joint likelihood approach. In the marginal approach, the regression parameters are estimated for known correlation parameters by solving their respective marginal likelihood estimating equations, and similarly the correlation parameters are estimated by solving their likelihood equations for known regression estimates. Thus, this marginal approach is an iterative approach which may not provide quick convergence. In the joint likelihood approach, the regression and correlation parameters are estimated simultaneously by searching the maximum value of the likelihood function with regard to these parameters together. This approach may encounter computational drawback, specially when the number of correlation parameters gets large. In this paper, we propose a new estimation approach where the likelihood function for the regression parameters is developed from the joint likelihood function by replacing the correlation parameter with a consistent estimator involving unknown regression parameters. Thus the new approach relaxes the dimension issue, that is, the dimension of the correlation parameters does not affect the estimation of the main regression parameters. The asymptotic properties of the estimates of the main regression parameters (obtained based on consistent estimating functions for correlation parameters) are studied in detail.

Dimension reduction procedures have received increasing consideration over the past decades. Despite this attention, the effect of data contamination or outlying data points in dimension reduction is, however, not well understood, and is compounded by the issue that outliers can be difficult to classify in the presence of many variables. This paper formally investigates the influence of data contamination for sliced inverse regression (SIR), which is a prototypical dimension reduction procedure that targets a lower-dimensional subspace of a set of regressors needed to explain a response variable. We establish a general theory for how estimated reduction subspaces can be distorted through both the number and direction of outlying data points. The results depend critically on the regressor covariance structure and the most harmful types of data contamination are shown to differ in cases where this covariance structure is known or unknown. For example, if the covariance structure is estimated, data contamination is proven to produce an estimated subspace that is automatically orthogonal to the directions of outlying data points, constituting a potentially serious loss of information. Our main results demonstrate the degree to which data contamination indeed causes incorrect dimension reduction, depending on the amount, magnitude, and direction of contamination. Further, by metricizing distances between dimension reduction subspaces, worst case results for data contamination can be formulated to define a finite sample breakdown point for SIR as a measure of global robustness. Our theoretical findings are illustrated through simulation.

This note concerns a family of distributions on the unit sphere obtained by inverse stereographic projection of elliptical distributions. We give some properties of these distributions with emphasis on the study of unimodality. This construction encompasses many known families of distributions on the unit sphere. Finally we show that it is possible to define new families of unimodal distributions that allow anisotropy.