Approximate Bayes estimation of the parameters and reliability function of a mixture of two inverse Weibull distributions under Type-2 censoring

The main goal of this paper is to develop the approximate Bayes estimation of the five-dimensional vector of the parameters and reliability function of a mixture of two inverse Weibull distributions (MTIWD) under Type-2 censoring. Usually, the posterior distribution is complicated under the scheme of Type-2 censoring and the integrals that are involved cannot be obtained in a simple explicit form. In this study, we use Lindley's [Approximate Bayesian method, Trabajos Estadist. 31 (1980), pp. 223–237] approximate form of Bayes estimation in the case of an MTIWD under Type-2 censoring. Later, we calculate the estimated risks (ERs) of the Bayes estimates and compare them with the corresponding ERs of the maximum-likelihood estimates through Monte Carlo simulation. Finally, we analyse a real data set using the findings.     

The umvue and bayes estimate of reliability of mixed failure time distribution

We study the reliability estimates of the non-standard mixture of degenerate (degenerated at zero) and exponential distributions. The Uniformly Minimum Variance Unbiased Estimator (UMVUE) and Bayes estimator of the reliability for some selective prior when the mixing proportion is known and unknown are derived. The Bayes risk is computed for each Bayes estimator of the reliability. A simulated study is carried out to assess the performance of the estimators alongwith the true and Maximum Likelihood Estimate (MLE) of the reliability. An example from Vannman (1991) is also discussed at the end of the paper.     

Posterior analysis,prediction and reliability in three-parameter weibull distributions

The paper develops Bayesian analysis in the context of samples from three-parameter Weibull distributions and shows how to tackle the problems of prediction and estimation of reliability curves. As Johnson, Kotz and Balakrishnan ( 1994 ) mentioned, the prediction problems for the three-parameter Weibull model seem to be unresolved and is certainly worth looking into (p.671). Posterior analysis organized around Gibbs sampling is shown to perform well. An application to stock returns is used to illustrate the potential of the approach.     

Reliability estimation based on ranked set sampling

In this paper we consider the problem of estimating the reliability of an exponential component based on a Ranked Set Sample (RSS) of size n. Given the first r observations of that sample, 1≤r≤n, we construct an unbiased estimator for this reliability and we show that these n unbiased estimators are the only ones in a certain class of estimators. The variances of some of these estimators are compared. By viewing the observations of the RSS of size n as the lifetimes of n independent k-out-of-n systems, 1≤k≤n, we are able to utilize known properties of these systems in conjunction with the powerful tools of majorization and Schur functions to derive our results.     

Modeling of covariance structures of random effects and random errors in linear mixed models

Abstract

In this paper, we discuss how to model the mean and covariancestructures in linear mixed models (LMMs) simultaneously. We propose a data-driven method to modelcovariance structures of the random effects and random errors in the LMMs. Parameter estimation in the mean and covariances is considered by using EM algorithm, and standard errors of the parameter estimates are calculated through Louis’ (1982 Louis, T.A. (1982). Finding observed information using the EM algorithm. J. Royal Stat. Soc. B 44:98–130. [Google Scholar]) information principle. Kenward’s (1987 Kenward, M.G. (1987). A method for comparing profiles of repeated measurements. Appl. Stat. 36:296–308.[Crossref], [Web of Science ®] , [Google Scholar]) cattle data sets are analyzed for illustration,and comparison to the literature work is made through simulation studies. Our numerical analysis confirms the superiority of the proposed method to existing approaches in terms of Akaike information criterion.     

Generalized Laplacian Distributions and Autoregressive Processes

Generalized Laplacian distribution is considered. A new distribution called geometric generalized Laplacian distribution is introduced and its properties are studied. First- and higher-order autoregressive processes with these stationary marginal distributions are developed and studied. Simulation studies are conducted and trajectories of the process are obtained for selected values of the parameters. Various areas of application of these models are discussed.     

A General Model of Random Variation

A statistical distribution of a random variable is uniquely represented by its normal-based quantile function. For a symmetrical distribution it is S-shaped (for negative kurtosis) and inverted S-shaped (otherwise). As skewness departs from zero, the quantile function gradually transforms into a monotone convex function (positive skewness) or concave function (otherwise). Recently, a new general modeling platform has been introduced, response modeling methodology, which delivers good representation to monotone convex relationships due to its unique “continuous monotone convexity” property. In this article, this property is exploited to model the normal-based quantile function, and explored using a set of 27 distributions.     

Identifiability of distributions under competing risks and complementary risks model

Let X₁,X₂,…,X_p be p random variables with cdf's F₁(x),F₂(x),…,F_p(x)respectively. Let U = min(X₁,X₂,…,X_p) and V = max(X₁,X₂,…,X_p).In this paper we study the problem of uniquely determining and estimating the marginal distributions F₁,F₂,…,F_p given the distribution of U or of V.

First the problem of competing and complementary risks are introduced with examples and the corresponding identification problems are considered when the X₁'s are independently distributed and U(V) is identified, as well as the case when U(V) is not identified. The case when the X₁'s are dependent is considered next. Finally the problem of estimation is considered.     

Hazard functions and life distributions in discrete time

The problem of studying lifelength distributions in discrete time is considered for certain forms of hazard functions. A class of life distributions that consists of the geometric, the Waring and the negative hypergeometric distributions is shown to result when the hazard function is inversely proportional to some linear function of time.     

Reliability models for categorical data

As assumed hypothetical consensus category corresponding to a case being classified provides a basis for assessment of reliability of judges. Equivalent judges are characterised by the joint probability distribution of the judge assignment and the consensus category. Estimates of the conditional probabilities of judge assignment given consensus category and of consensus category given judge assignments are indices of reliability. All parameters can be estimated if data include classifications of a number of cases by 3 or more judges. Restrictive assumptions are imposed to obtain models for data from classifications by two judges. Maximum likelihood estimation is discussed and illustrated by example for the 3 or more judges case.