Narayanaswamy Balakrishnan

Best Constant-Stress Accelerated Life-Test Plans With Multiple Stress Factors for One-Shot Device Testing Under a Weibull Distribution

We discuss here the design of constant-stress accelerated life-tests for one-shot device testing by assuming a Weibull distribution as a lifetime model. Because there are no explicit expressions for the maximum likelihood estimators of the model parameters and their variances, we adopt the asymptotic approach here to develop an algorithm for the determination of optimal allocation of devices, inspection frequency, and the number of inspections at each stress level, by assuming a Weibull distribution with non-constant scale and shape parameters as the lifetime distribution. The asymptotic variance of the estimate of reliability of the device at a specified mission time is minimized subject to a pre-fixed experimental budget, and a termination time. Examples are provided to illustrate the proposed algorithm for the determination of the best test plan. A sensitivity analysis of the best test plan is also carried out to examine the effect of misspecification of the model parameters.

On usual multivariate stochastic ordering of order statistics from heterogeneous beta variables

Let Xi∼beta(αi,1) and Yi∼beta(γi,1), i=1,2, be all independent. We show that (α1,α2)⪰m(γ1,γ2) implies (Y1:2,Y2:2)≥st(X1:2,X2:2). We then extend this result to the general case of the proportional reversed hazard rates (PRHR) model.

Multivariate families of gamma-generated distributions with finite or infinite support above or below the diagonal

In this paper, we introduce two new families of multivariate distributions with finite or infinite support above or below the diagonal generated by McKays bivariate gamma distribution and show that their conditional distributions are univariate gamma- and beta-generated distributions. We derive the Shannon entropies of the introduced families of bivariate distributions. We then focus on the special cases of bivariate gamma-exponentiated exponential distributions, and discuss their properties. Finally, we illustrate the usefulness of the proposed bivariate gamma-exponentiated exponential distributions with a real dataset.

Orthogonal polynomials in the cumulative Ord family and its application to variance bounds

ABSTRACT This article presents and reviews several basic properties of the Cumulative Ord family of distributions; this family contains all the commonly used discrete distributions. A complete classification of the Ord family of probability mass functions is related to the orthogonality of the corresponding Rodrigues polynomials. Also, for any random variable X of this family and for any suitable function g in , the article provides useful relationships between the Fourier coefficients of g (with respect to the orthonormal polynomial system associated to X) and the Fourier coefficients of the forward difference of g (with respect to another system of polynomials, orthonormal with respect to another distribution of the system). Finally, using these properties, a class of bounds for the variance of is obtained, in terms of the forward differences of g. These bounds unify and improve several existing results.

Multivariate stochastic comparisons of multivariate mixture models and their applications

In this paper, we obtain some conditions to compare multivariate mixture models with respect to some well-known multivariate stochastic orders. We also utilize the established results in reliability theory to compare the vectors of residual life-lengths of live components of (n−k+1)-out-of-n systems in both one sample and two samples situations.

A note on the conditional residual lifetime of a coherent system under double monitoring

ABSTRACT In this note, we derive some mixture representations for the reliability function of the conditional residual lifetime of a coherent system with n independent and identically distributed (i.i.d.) components under the condition that at time t1 the jth failures has occurred and at time t2 the kth failures (j < k) have not occurred yet. Based on the mixture representations, we then discuss the stochastic comparisons of the conditional residual lifetimes of two coherent systems with i.i.d. components.

On the mean residual life of a generalized k-out-of-n system

ABSTRACT A generalized k-out-of-n system consists of N modules in which the i th module is composed of ni components in parallel. The system failswhen at least f components in the whole system or at least k consecutive modules have failed. In this article, we obtain the mean residual life function of such a generalized k-out-of-n system under different conditions, namely, when the number of components in each module is equal or unequal and when the components of the system are independent or exchangeable.

Maximum likelihood estimation of the parameters of a multiple step-stress model from the Birnbaum-Saunders distribution under time-constraint: A comparative study

ABSTRACT The cumulative exposure model (CEM) is a commonly used statistical model utilized to analyze data from a step-stress accelerated life testing which is a special class of accelerated life testing (ALT). In practice, researchers conduct ALT to: (1) determine the effects of extreme levels of stress factors (e.g., temperature) on the life distribution, and (2) to gain information on the parameters of the life distribution more rapidly than under normal operating (or environmental) conditions. In literature, researchers assume that the CEM is from well-known distributions, such as the Weibull family. This study, on the other hand, considers a p-step-stress model with q stress factors from the two-parameter Birnbaum-Saunders distribution when there is a time constraint on the duration of the experiment. In this comparison paper, we consider different frameworks to numerically compute the point estimation for the unknown parameters of the CEM using the maximum likelihood theory. Each framework implements at least one optimization method; therefore, numerical examples and extensive Monte Carlo simulations are considered to compare and numerically examine the performance of the considered estimation frameworks.