Gouri K. Bhattacharyya

A tampered failure rate model for step-stress accelerated life test

A statistical model for step-stress accelerated life test is motivated from the point of view that a change of the stress has a multiplicative effect on the failure rate function over the remaining life.Properties of the proposed model, including an interpretation in terms of the conditional reliability, and relationships with the existing models are discussed. For the parametric setting of a Weibull family representing the life distribution under a constant stress, maximum likelihood estimation of the parameters is investigated and the Fisher information matrix is derived. The proposed model is found to have certain analytical advantages over the cumulative exposure model that is commonly used in step-stress analysis. An extension of the model to include a regression structure and inferences for life under the use condition stress are briefly discussed.

The Asymptotics of Maximum Likelihood and Related Estimators Based on Type II Censored Data

Abstract Some simple procedures are provided for establishing the asymptotic normality and uniform strong convergence of a class of functions that arise in the context of estimating parameters from a Type II censored sample. These lead to an elementary yet rigorous treatment of the asymptotic properties of maximum likelihood estimators based on Type II censored data. Further applications include the treatment of asymptotics of some modified maximum likelihood (MML) estimators. In particular, conditions are provided for the consistency and limiting normality of the MML estimators of Mehrotra and Nanda, and the asymptotic efficiencies of these estimators are evaluated.

Some new constructions of bivariate Weibull models

In this article, several approaches are advanced towards the construction of bivariate Weibull models from the consideration of failure behaviors of the components of a two-component system. First, a general method of construction of bivariate life models is developed in the setting of random environmental effects. Some new bivariate Weibull models are derived as special cases and added insights are provided for some of the existing ones. In the course of model formulation in terms of the dependence structure, a new bivariate family of life distributions is constructed so as to incorporate both positive and negative quadrant dependence in the same parametric setting, and a bivariate Weibull model is obtained as a special case. Finally, some distributional properties are presented for a bivariate Weibull model derived from the consideration of random hazards.

Fatigue Failure Models ߝ Birnbaum-Saunders vs. Inverse Gaussian

The assumption of a common underlying failure process leads to both the inverse Gaussian distribution and the fatigue life distribution of Birnbaum & Saunders. The first is obtained by an exact derivation while the second involves certain approximations. Although both of these distributions fit many failure data well, the inverse Gaussian distribution enjoys a distinct advantage in regard to the availability of procedures for a sound statistical analysis and tractability of the sampling distributions.

Estimation of Reliability in a Multicomponent Stress-Strength Model

Abstract A stress-strength model is formulated for s of k systems consisting of identical components. We consider minimum variance unbiased estimation of system reliability for data consisting of a random sample from the stress distribution and one from the strength distribution when the two distributions are exponential with unknown scale parameters. The asymptotic distribution is obtained by expanding the unbiased estimate about the maximum likelihood value and establishing their equivalence. Performance of the two estimates for moderate samples is studied by Monte Carlo simulation. Uniformly most accurate unbiased confidence intervals are also obtained for system reliability.

Exact confidence bounds for an exponential parameter under hybrid censoring

Consider a life testing experiment in which n units are put on test, successive failure times are recorded, and the observation is terminated either at a specified number r of failures or a specified time T whichever is reached first. This mixture of type I and type II censoring schemes, called hybrid censoring, is of wide use. Under this censoring scheme and the assumption of an exponential life distribution, the distribution of the maximum likelihood estimator of the mean life θ is derived. It is then used to construct an exact lower confidence bound for θ.

A Purchase Incidence Model with Inverse Gaussian interpurchase Times

Abstract This paper deals with a new purchase incidence model where the interpurchase time of an individual household is described by a two-parameter inverse Gaussian distribution, and the population heterogeneity is modeled by the natural conjugate family which has truncated t and modified gamma marginals. The model, more flexible than the exponential and one-parameter gamma models previously used for purchase incidence, is applied to consumer panel data on toothpaste purchases and an excellent fit is obtained. A more logical approach is employed for the assessment of consumer heterogeneity than the methods in existing literature.

Bayesian Results for the Inverse Gaussian Distribution with an Application

Some Bayesian results are derived for the inverse Gaussian family of distributions with noninformative reference prior as well as the natural conjugate prior. With a particular parameterization, the posterior distributions are found to have remarkable similarities with the corresponding results for the normal model. Finally, an application of the Bayesian results is given toward analyzing some equipment failure data.

On a Test of Independence in a Bivariate Exponential Distribution

Abstract The problem of testing independence in the bivariate exponential distribution of Marshall and Olkin is considered here with the assumption of identical marginal distributions. It is shown that in spite of the presence of a nuisance parameter, a uniformly most powerful test exists. The test turns out to be the same as the one proposed by Bernis, et al., on a heuristic basis. In addition to demonstrating the optimality property of the test, an easily computable expression is provided for the exact power function.

Confidence limits for stress-strength models with explanatory variables

A lower confidence bound is obtained for Pr(Y > X|z 1, z 2), where X and Y are independent normal variables, with explanatory variables z 1 and z 2, respectively. For equal residual variances, an exact solution is obtained, but for the unequal variance case, an approximate lower confidence bound is developed. Examples of the use of these procedures are given.