Partial orderings of life distributions with respect to their aging properties

New partial orderings of life distributions are given. The concepts of decreasing mean residual life, new better than used in expectation, harmonic new better than used in expectation, new better than used in failure rate, and new better than used in failure rate average are generalized, so as to compare the aging properties of two arbitrary life distributions.     

New partial orderings and applications

New concepts of partial stochastic orderings are introduced, and the relations among them and the classical partial orderings are shown. Relevance of these partial orderings in aging properties classification is discussed, and new classes of life distributions, based on them, are proposed. An application to stochastic comparison between Poisson shock models is proposed. © 1993 John Wiley & Sons, Inc.     

On preservation of classes of life distributions under reliability operations: Some complementary results

A counterexample of general character is constructed showing that the NWUE class is not preserved under mixing. Some general classes of life distributions are introduced and their behavior under mixing and convolution is studied. These classes have a theoretical character but coincide in many cases with well-known classes. It is proved that the DMRL class is not preserved under convolution.     

The hnbue and hnwue classes of life distributions

Different properties of the HNBUE (HNWUE) class of life distributions (i.e.), for which \documentclass{article}\pagestyle{empty}\begin{document}$\int_t^\infty {\,\,\,\mathop F\limits^-(x)\,dx\, \le \,(\ge)\,\mu }\]$\end{document} exp(?t/μ) for t ≥ 0, where μ = \documentclass{article}\pagestyle{empty}\begin{document}$\int_t^\infty {\,\,\,\mathop F\limits^-(x)\,dx}$\end{document} are presented. For instance we characterize the HNBUE (HNWUE) property by using the Laplace transform and present some bounds on the survival function of a HNBUE (HNWUE) life distribution. We also examine whether the HNBUE (HNWUE) property is preserved under the reliability operations (i) formation of coherent structure, (ii) convolution and (iii) mixture. The class of distributions with the discrete HNBUE (discrete HNWUE) property (i.e.), for which \documentclass{article}\pagestyle{empty}\begin{document}$\sum\limits_{j=k}^\infty {\mathop{\mathop P\limits^-_{j\,\,\,}\, \le(\ge)\,\mu(1 - 1/\mu)^{^k }}\limits^{}} $\end{document} for k = 0, 1, 2, ?, where μ =\documentclass{article}\pagestyle{empty}\begin{document}$\sum\limits_{j=0}^\infty {\mathop {\mathop P\limits^- _{j\,\,\,\,\,}and\mathop P\limits^ - _{j\,\,\,\,\,}=}\limits^{}}\,\,\sum\limits_{k=j+1}^\infty {P_k)}$\end{document} is also studied.     

寿命分布的参数Bootstrap拟合优度检验方法

拟合优度检验在统计和可靠性等领域具有非常重要的地位,基于参数Bootstrap重采样的思想,对未知参数的常用寿命分布进行拟合优度检验。数值仿真结果表明,相对于传统的经验分布函数检验,这种基于参数Bootstrap的拟合优度检验具有更高的功效,特别是在小样本的情况下,优势明显。     

On compounded bivariate poisson distributions

A unified treatment is given for a class of discrete distributions derived by compounding a bivariate Poisson with a bivariate discrete or continuous distribution. Using generating functions a number of interesting results are obtained for probabilities, moments, cumulants, factorial moments, and factorial cumulants. Conditional distributions and regression functions are also examined. Five illustrative examples are presented in detail. © 1994 John Wiley & Sons, Inc.     

Bivariate models from univariate life distributions: A characterization cum modeling approach

Bivariate life distribution models are of importance for studying interdependent components. We present a generic approach by introducing a new concept of characterized model in stead of a characterized distribution. It strikes a balance between characterization and modeling approaches to eliminate their individual limitations and incorporate their respective strengths. The proposed model, being a characterized one, admits many important properties irrespective of the choice of marginal distributions. The retention of univariate IFR, DFR, IFRA, DFRA, NBU, and NWU class properties in the bivariate setup has been ensured along with some results on series combinations and convolution. No other models, available in the literature, can ensure simultaneous retention of these fundamental and extremely important class properties. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Planning accelerated life tests for censored two‐parameter exponential distributions

We consider optimal test plans involving life distributions with failure‐free life, i.e., where there is an unknown threshold parameter below which no failure will occur. These distributions do not satisfy the regularity conditions and thus the usual approach of using the Fisher information matrix to obtain an optimal accelerated life testing (ALT) plan cannot be applied. In this paper, we assume that lifetime follows a two‐parameter exponential distribution and the stress‐life relationship is given by the inverse power law model. Near‐optimal test plans for constant‐stress ALT under both failure‐censoring and time‐censoring are obtained. We first obtain unbiased estimates for the parameters and give the approximate variance of these estimates for both failure‐censored and time‐censored data. Using these results, the variance for the approximate unbiased estimate of a percentile at a design stress is computed and then minimized to produce the near‐optimal plan. Finally, a numerical example is presented together with simulation results to study the accuracy of the approximate variance given by the proposed plan and show that it outperforms the equal‐allocation plan. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 169–186, 1999     

Calculation of the cost of warranty policies as a function of estimated life distributions

Two types of warranties are analyzed. These are the free-replacement warranty, under which failed items are replaced free of charge until a specified total operating time has been achieved, and the pro rata warranty, under which items that fail prior to a specified time are replaced at pro rata cost to the buyer. Both the buyer's and seller's points of view are considered. The basis of the analysis is a comparison of warranted and unwarranted (otherwise identical) items with regard to long-run cost to the buyer and long-run profit to the seller. Application of the results requires knowledge of certain characteristics of the life distribution of the items in question. Parametric and nonparametric methods of estimation of these characteristics from incomplete data are discussed. Single and multiple failure-mode situations are considered. Some solutions to the problem are illustrated using incomplete data on failure times of an aircraft component.     

On estimation in weibull distributions with random scale parameters

Work by the present authors on life distributions derived from stochastic hazard functions [4] is related to certain articles that have appeared in this journal. This relationship is illustrated. The emphasis of this article is upon problems of parameter estimation.     

Equivalent accelerated life testing plans for log‐location‐scale distributions

Accelerated life testing (ALT) is widely used to determine the failure time distribution of a product and the associated life‐stress relationship in order to predict the product's reliability under normal operating conditions. Many types of stress loadings such as constant‐stress, step‐stress and cyclic‐stress can be utilized when conducting ALT. Extensive research has been conducted on the analysis of ALT data obtained under a specified stress loading. However, the equivalency of ALT experiments involving different stress loadings has not been investigated. In this article, a log‐location‐scale distribution under Type I censoring is considered in planning ALT. An idea is provided for the equivalency of various ALT plans involving different stress loadings. Based on this idea, general equivalent ALT plans and some special types of equivalent ALT plans are explored. For demonstration, a constant‐stress ALT and a ramp‐stress ALT for miniature lamps are presented and their equivalency is investigated. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Adjacent orderings in single-machine scheduling with earliness and tardiness penalties

This article deals with a single-machine n job earliness-tardiness model with job-independent penalties. It demonstrates that the arrangement of adjacent jobs in an optimal schedule depends on a critical value of the start times. Based on these precedence relations, the article develops criteria under which the problem can be decomposed into smaller subproblems. The branching scheme that used the developed results was tested on 70 examples of size n = 10. This scheme should be incorporated in any branch-and-bound method once a lower bound is found. The branching scheme can easily handle small problems without a lower bound. © 1993 John Wiley & Sons, Inc.     

The estimation of P(x < Y) for distributions useful in life testing

In this paper the reliability function K = P(X < Y) has been estimated when X and Y follow gamma, exponential or bivariate exponential distributions. The paper is partly expository.     

On excess-, current-, and total-life distributions of phase-type renewal processes

Consider a renewal process whose interrenewal-time distribution is phase type with representation (α, T). We show that the (time-dependent) excess-life distribution is phase type with representation (α′, T), where α′ is an appropriately modified initial probability vector. Using this result, we derive the (time-dependent) distributions for the current life and the total life of the phase-type renewal process. They in turn enable us to obtain the equilibrium distributions for the three random variables. These results simplify the computation of the respective distribution functions and consequently enhance the potential use of renewal theory in stochastic modeling—particularly in inventory, queueing, and reliability applications. © 1992 John Wiley & Sons, Inc.     

Optimal accelerated life test designs for Burr type XII distributions under periodic inspection and type I censoring

This article develops optimal accelerated life test designs for Burr Type XII distributions under periodic inspection and Type I censoring. It is assumed that the mean lifetime (the Burr XII scale parameter) is a log-linear function of stress and that the shape parameters are independent of stress. For given shape parameters, design stress and high test stress, the test design is optimized with respect to the low test stress and the proportion of test units allocated to the low stress. The optimality criterion is the asymptotic variance of the maximum-likelihood estimator of log mean life at the design stress with the use of equally spaced inspection times. Computational results for various values of the shape parameters show that this criterion is insensitive to the number of inspection times and to misspecification of imputed failure probabilities at the design and high test stresses. Procedures for planning an accelerated life test, including selection of sample size, are also discussed. It is shown that optimal designs previously obtained for exponential and Weibull distributions are similar to those obtained here for the appropriate special cases of the Burr XII distribution. Thus the Burr XII distribution is a useful and widely applicable family of reliability models for ALT design. © 1996 John Wiley & Sons, Inc.     

A class of distributions generated from distributions of exponential type

A class of exponential type distributions with special exponential parameters is defined. It is assumed that the exponential parameters vary according to some (known) probability law. It has been shown in this paper that the compound distribution can be easily represented in form involving moment generating function of the mixing distribution. The results obtained in this paper provide an efficient and simple method of obtaining compound failure time distribution with known mixing distributions (uniform, exponential, gamma).     

On testing for decreasing mean residual life ordering

We consider the problem of testing for the decreasing mean residual life ordering introduced in Kochar and Wiens [14]. We treat both the one-sample and two-sample problems. The limiting distributions of the proposed test statistics are derived. © 1993 John Wiley & Sons, Inc.     

Optimum step‐stress accelerated life test plans for log‐location‐scale distributions

This article presents new tools and methods for finding optimum step‐stress accelerated life test plans. First, we present an approach to calculate the large‐sample approximate variance of the maximum likelihood estimator of a quantile of the failure time distribution at use conditions from a step‐stress accelerated life test. The approach allows for multistep stress changes and censoring for general log‐location‐scale distributions based on a cumulative exposure model. As an application of this approach, the optimum variance is studied as a function of shape parameter for both Weibull and lognormal distributions. Graphical comparisons among test plans using step‐up, step‐down, and constant‐stress patterns are also presented. The results show that depending on the values of the model parameters and quantile of interest, each of the three test plans can be preferable in terms of optimum variance. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Bilateral phase-type distributions

In this article we define a class of distributions called bilateral phase type (BPH), and study its closure and computational properties. The class of BPH distributions is closed under convolution, negative convolution, and mixtures. The one-sided version of BPH, called generalized phase type (GPH), is also defined. The class of GPH distributions is strictly larger than the class of phase-type distributions introduced by Neuts, and is closed under convolution, negative convolution with nonnegativity condition, mixtures, and formation of coherent systems. We give computational schemes to compute the resulting distributions from the above operations and extend them to analyze queueing processes. In particular, we present efficient algorithms to compute the steady-state and transient waiting times in GPH/GPH/1 queues and a simple algorithm to compute the steady-state waiting time in M/GPH/1 queues.     

On the behavior and shape of mixture failure rates from a family of IFR Weibull distributions

Populations of many types of component are heterogeneous and often consist of a small number of different subpopulations. This is called a mixture and it arises in a number of situations. For example, a majority of products in industrial populations are mixtures of defective items with shorter lifetimes and standard items with longer lifetimes. It is a well‐known result that distributions with decreasing failure rates are closed under mixture. However, mixtures of distributions with increasing failure rates are not easily classifiable. If the subpopulations involved in the mixture have increasing failure rates, there might be some upward movement in the mixture and later a general downward pull towards the strongest component. Little work has been done in describing the shape of mixture failure rates when all subpopulations do not have decreasing failure rate. In this paper, we present general results that describe the shape and behavior of a failure rate of a mixture obtained from two Weibull subpopulations with strictly increasing failure rates. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004