Madhuri S. Mulekar Pengarang

Bayesian and Classical Estimation of the I nverse Pareto Distribution and Its Application to Strength-Stress Models Lei G uo and Wenhao G ui D epar t ment of Mathemat ics, Beijing J iaotong Universit y, B eijing, China SYNOPTIC ABSTR AC T Th is ar ticle d eals with the estimation o f R = P(Y < X) when X and Y are two independent random variables following inverse Pareto distributions with diffrent parameters. The maximum-likelihood estimator of R and its asymptotic sampling distribution are proposed. The asymptotic sampling distribution is used to construct an asymptotic confience interval for R. The exact confience interval and bootstrap confience interval for R are also presented. Bayes estimate and credible interval are studied using Gibbs sampling technique. Monte Carlo simulations are performed to compare the performance of diffrent proposed estimation methods. Analysis of a real data set is presented for illustrative purposes. KEY WO RDS AND PHR ASES I nverse Pa reto distribution; maximum likelihood estimator; a symptotic d i s t r i b u t i o n ; re l i a b i l i t y; s t re s s - s t ren g t h m o d e l ; G i b b s s a m p l i n g Baye sian Estimation of Dynamic Cumulative R esidual Entropy for Classical Pareto Distribution K . R . R e n j i n i ,a E. I. Abdul Sathar,a and G. Rajeshb aDepartment of Statistics, University of Kerala, Trivandrum, India; bDepartment of Statistics, CUSAT, Kochi, India SYNOPTIC ABSTR AC T Dynamic cumulative residual entropy plays a signifiant role in reliability and survival analysis to model and analyze the data. This article presents Bayesian estimation of the dynamic cumulative residual entropy of the classical Pareto distribution using informative and non-informative priors. The Bayes estimators and their associated posterior risks are calculated under diffrent symmetric and asymmetric loss functions. A numerical example is given to illustrate the results derived. Based on a Monte Carlo simulation study, comparisons are made between the proposed estimators. The objective of this article is to identify the combination of a loss function and a prior having the minimum Bayes risk in order to estimate effiently the dynamic cumulative residual entropy of Pareto distribution. KEY WO RDS AND PHR ASES Bayesian estimation; Pareto distribution; dynamic cumulative residual entropy; prior/posterior distribution; loss functions Baye sian Predic tive I nfe re nce for Ze ro -I nflated Poisson (ZIP) Distribution with Applications Suntaree Unhapipat,a , b , c Montip Tiensuwan,b , c and Nabendu Pal a aDepartment of Mathematics, University of Louisiana at Lafayette, Louisiana, USA; bDepartment of Mathematics, Faculty of Science, Mahidol University, Bangkok, Thailand; cCenter of Excellence in Mathematics, Commission on Higher Education, Bangkok, Thailand SYNOPTIC ABSTR AC T Count data (over time or space) that has an unusually large number of zeros cannot be modelled by the usual Poisson distribution. Typically, a better fi to such data is provided by a two parameter zero-inflted Poisson (ZIP) distribution. This work deals with Bayesian predictive inference under the ZIP model where various types of prior distributions have been considered. The applicability and usefulness of our proposed Bayesian techniques under the ZIP model have been demonstrated by four examples with real-life datasets, ranging from public health to natural disasters to vehicle accidents. KEY WORDS AND PHR ASES Cross validation; empirical Bayes method; Jeffery’s prior; noninformative prior; predictive distribution Estimation-Equivalent and Dispersion-Equivalent Error Covariance M atrices for the G e neral Linear M odel Phil D. Younga and Dean M. Youngb aDepartment of Information Systems, Baylor University, Waco, Texas, USA; bDepartment of Statistical Science, Baylor University, Waco, Texas, USA SYNOPTIC ABSTR AC T We give a new, very concise derivation of an explicit characterization representation of the general nonnegative-defiite error covariance matrix for a Gauss-Markov model, such that the best linear unbiased estimator is identical to the least-squares estimator. Our characterization derivation is very concise, and we use only elementary matrix properties in the proof. We also characterize the general symmetric nonnegativedefiite error covariance matrix of a Gauss-Markov model, such that the covariance matrices of the best linear unbiased estimator, the least squares estimator, and the independently and identically-distributed least-squares estimator have identical covariance structures. KEY WO RDS AND PHR ASES Matrix equations; general common symmetric nonnegative-definite solutions; estimation of covariance structures; Moore-Penrose inverse Means, Medians, and Multivariate Mixed Design Dat a Liber tie B. M antilla and J ef T. Terpstra D epar t ment of Stat istics, Western M ichigan Universit y, K alamazoo, M ichigan, U SA SYNOPTIC ABSTR AC T Th is ar ticle p roposes the u se of component-wise m eans and m edians fo r t he purpose o f estimation and inference in a multivariate m ixed design co ntex t. Th at is, for data that is essentially a combination of paired and independent samples. A general limiting d istribution result is derived and this result leads to the joint asymptotic distribution of the estimates. Subsequently, t his result yields Wald-t ype mean and median-based procedures that can b e u sed for hypothesis testing and confience regions. The proposed methodology is illustrated with an example. Fu rthermore, the mean - based procedure is compared against its m edian-based analogue in a simulation study. Results indicate that the mean - based procedure is superior to t he median - based procedure in small neighborhoods of the multivariate normal distribution , while the median - based procedure is superior to the mean – based procedure for heavy - tailed distributions. KEY WORDS AND PHR ASES A s y m p to t i c d i s t r i b u t i o n ; mean; m edian; mixed design; multivariate On Conditional Lower Partial Moments and Its Applicat ions S. M. Sunoj and Vipin N. Department of Statistics, Cochin U niversity of Science and Technology, Cochin, Kerala, India SYNOPTIC ABSTR AC T Lower partial moments ( LPM) are predominantly used in portfolio theory and inareas involving financial risk , which addresses the problem of managing risky investment policies with the object of maximizing returns. LPM also plays an important role in the analysis of risks in income (poverty) studies. In this article, we extend the notion of LPM to the conditional s et up and s tudy its u sefulness in the contex t of stochastic modeling. Th e relationship between various measures in reliability studies, income (poverty ) studies, and risk analysis areal soproved. Finally, a non-parametric estimator for conditional LPM is proposed and has been validated through simulated and real data sets. KEY WO RDS AND PHR ASES Lower p ar tial moments; bivariate distributions; charac terizations; co nditional expec ted shor t fall ; incomegapratio Tests for Scale Parameter of Ske Log Laplace Distribution V. U. Dixita and Pradnya P. Khandeparkarb aDepartment of Statistics, University of Mumbai, Mumbai, India; bDepartment of Statistics, SIES College of Arts Science and Commerce, Mumbai, India SYNOPTIC ABSTR AC T Kozubowski and Podgorski (2003) have discussed properties, characterizations, and estimation of parameters of skew log Laplace distribution (SLLD).In this article, classical optimum tests for scale parameter of SLLD are derived. The most powerful (MP) test is obtained for scale parameter when shape parameters are known. Wald’s sequential probability ratio test (SPRT) is obtained, and its properties are studied. The likelihood ratio tests (LRT) for scale parameter are derived when the shape parameters are known and unknown. Finally, the SPRT and LRT are illustrated to the real life data. KEY WO RDS AND PHR ASES Skew log Laplace distribution; most powerful test; likelihood ratio test; sequential probability ratio test Weibull Step -Stress M odel with a L agged Effe c t Nandini K annana and Debasis Kundub aDivision of Mathematical Sciences, National Science Foundation, Arlington, Virginia, USA; bDepartment of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India SYNOPTIC ABSTR AC T In survival analysis and reliability, researchers are often interested in assessing the effcts of diffrent stress factors on the lifetime of experimental units. The model introduced in this article is motivated by a study of the effcts of altitude and other risk factors on decompression sickness, a condition encountered when individuals are exposed to signifiant changes in environmental pressure. Unlike standard life-testing experiments, in this study, the levels of the stress factor, viz. altitude, are changed during the exposure duration. This is known as a step-stress test, a class of accelerated testing, widely used in material testing. Recently Kannan, Kundu, and Balakrishnan (2010) introduced the cumulative risk model as an alternative to the widely used cumulative exposure model. The new model allows for the inclusion of a lag period in the hazard function, a more realistic assumption in most applications. In this article, we consider the cumulative risk model assuming that the lifetime distributions of the experimental units follow Weibull distributions at the diffrent levels of the risk factor. It is assumed that the level of the stress factor is changed only once during the exposure duration at a pre-fied time τ1. The maximum likelihood and the least squares methods have been used to estimate the unknown parameters. Monte Carlo simulations are performed to compare the performances of the two diffrent methods. We further propose the Bayes estimators of the unknown parameters of the model. To evaluate the performance of the model, one data set from the altitude decompression sickness experiment has been analyzed, and it is observed that the proposed model fis the data quite well. KEY WO RDS AND PHR ASES Cumulative exposure model; hazard function; least squares estimators; maximum likelihood estimators; Step-stress model