On extreme order statistics from heterogeneous beta distributions with applications

This paper treats the problem of stochastic comparisons for the extreme order statistics arising from heterogeneous beta distributions. Some sufficient conditions involved in majorization-type partial orders are provided for comparing the extreme order statistics in the sense of various magnitude orderings including the likelihood ratio order, the reversed hazard rate order, the usual stochastic order, and the usual multivariate stochastic order. The results established here strengthen and extend those including Kochar and Xu (2007 Kochar, S.C., Xu, M. (2007). Stochastic comparisons of parallel systems when components have proportional hazard rates. Probab. Eng. Inf. Sci. 21:597–609.[Crossref], [Web of Science ®] , [Google Scholar]), Mao and Hu (2010 Mao, T., Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Probab. Eng. Inf. Sci. 24:245–262.[Crossref], [Web of Science ®] , [Google Scholar]), Balakrishnan et al. (2014 Balakrishnan, N., Barmalzan, G., Haidari, A. (2014). On usual multivariate stochastic ordering of order statistics from heterogeneous beta variables. J. Multivariate Anal. 127:147–150.[Crossref], [Web of Science ®] , [Google Scholar]), and Torrado (2015 Torrado, N. (2015). On magnitude orderings between smallest order statistics from heterogeneous beta distributions. J. Math. Anal. Appl. 426:824–838.[Crossref], [Web of Science ®] , [Google Scholar]). A real application in system assembly and some numerical examples are also presented to illustrate the theoretical results.     

Examples of doubly stochastic measures supported on the graphs of two functions

A doubly stochastic measure (DSM) is a measure μ on the unit square so that μ([0, 1] × A) = μ(A × [0, 1]) = m(A) where m is Lebesgue measure. The set of DSMs forms a convex set in the space of measures. It is known that DSMs supported on the union of two graphs of invertible functions are extreme points of that convex set (Seethoff and Shiflett, 1977/78 Seethoff, T.L., Shiflett. (1977/78). Doubly stochastic measures with prescribed support. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41(4):283–288.[Crossref], [Web of Science ®] , [Google Scholar]). In general, there are few examples of extreme points in the literature. There are examples of so-called hairpins where the functions involved are inverses of each other, but there are also examples of the union of the graphs of a function and its inverse does not support a DSM (Sherwood and Taylor, 1988 Sherwood, H., Taylor, M.D. (1988). Doubly stochastic measures with hairpin support. Probab. Theory Related Fields 78(4):617–626.[Crossref], [Web of Science ®] , [Google Scholar]). In this paper, for a function f in a certain class, we find companion functions g so that the union of the graphs of f and g support a DSM even though the union of the graphs of f and f-inverse do not.     

Testing monotonic equilibrium residual entropy of N-State Random Evolution

In this study, we propose an information measure of uncertainty associated with the random equilibrium residual lifetime of a system driven by N-State Random Evolution. A U-statistic test driven by a moment inequality is proposed for testing the hypothesis that the uncertainty of equilibrium remaining life of a system remains unchanged (when system is in the steady state) against the alternative situation when system’s equilibrium residual life has increasing uncertainty over time (i.e., the life distribution has Increasing Equilibrium Residual Entropy property). Some numerical results such as tabulated critical values and empirical power of the proposed test statistic are presented as well.     

Improved likelihood inferences for Weibull regression model

A general procedure is developed for bias-correcting the maximum likelihood estimators (MLEs) of the parameters of Weibull regression model with either complete or right-censored data. Following the bias correction, variance corrections and hence improved t-ratios for model parameters are presented. Potentially improved t-ratios for other reliability-related quantities are also discussed. Simulation results show that the proposed method is effective in correcting the bias of the MLEs, and the resulted t-ratios generally improve over the regular t-ratios.     

Spatio‐temporal Ornstein–Uhlenbeck Processes: Theory,Simulation and Statistical Inference

Spatio‐temporal modelling is an increasingly popular topic in Statistics. Our paper contributes to this line of research by developing the theory, simulation and inference for a spatio‐temporal Ornstein–Uhlenbeck process. We conduct detailed simulation studies and demonstrate the practical relevance of these processes in an empirical study of radiation anomaly data. Finally, we describe how predictions can be carried out in the Gaussian setting.     

Threshold Estimation for Stochastic Processes with Small Noise

Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent but numerically unstable in the sense of large standard deviations under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivalent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However, in numerical study, it looked asymptotically normal in an example where filter was chosen suitably, and the noise was a Lévy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.     

Bivariate extension of (dynamic) cumulative past entropy

Recently, the concept of cumulative residual entropy (CRE) has been studied by many researchers in higher dimensions. In this article, we extend the definition of (dynamic) cumulative past entropy (DCPE), a dual measure of (dynamic) CRE, to bivariate setup and obtain some of its properties including bounds. We also look into the problem of extending DCPE for conditionally specified models. Several properties, including monotonicity, and bounds of DCPE are obtained for conditional distributions. It is shown that the proposed measure uniquely determines the distribution function. Moreover, we also propose a stochastic order based on this measure.     

Sparse Bayesian variable selection in multinomial probit regression model with application to high-dimensional data classification

Here we consider a multinomial probit regression model where the number of variables substantially exceeds the sample size and only a subset of the available variables is associated with the response. Thus selecting a small number of relevant variables for classification has received a great deal of attention. Generally when the number of variables is substantial, sparsity-enforcing priors for the regression coefficients are called for on grounds of predictive generalization and computational ease. In this paper, we propose a sparse Bayesian variable selection method in multinomial probit regression model for multi-class classification. The performance of our proposed method is demonstrated with one simulated data and three well-known gene expression profiling data: breast cancer data, leukemia data, and small round blue-cell tumors. The results show that compared with other methods, our method is able to select the relevant variables and can obtain competitive classification accuracy with a small subset of relevant genes.