Testing goodness of fit to the increasing failure rate family

One branch of the reliability literature is concerned with devising statistical procedures with various nonparametric “restricted family” model assumptions because of the potential improved operating characteristics of such procedures over totally nonparametric ones. In the single-sample problem with unknown increasing failure rate (IFR) distribution F, (1) maximum-likelihood estimators of F have been calculated, (2) upper or lower tolerance limits for F have been determined, and (3) tests of the null hypothesis that F is exponential have been constructed. Barlow and Campo proposed graphical methods for assessing goodness of fit to the IFR model when the validity of this assumption is unknown. This article proposes several analytic tests of the IFR null hypothesis based on the maximum distance and area between the cumulative hazard function and its greatest convex minorant (GCM), and the maximum distance and area between the total time on test statistic and its GCM. A table of critical points is provided to implement a specific test having good overall power properties.     

Designing acceptance sampling schemes for life testing with mixed censoring

Mixed censoring is useful extension of Type I and Type II censoring and combines some advantages of both types of censoring. This paper proposes a general Bayesian framework for designing a variable acceptance sampling scheme with mixed censoring. A general loss function which includes the sampling cost, the time‐consuming cost, the salvage value, and the decision loss is employed to determine the Bayes risk and the corresponding optimal sampling plan. An explicit expression of the Bayes risk is derived. The new model can easily be adapted to create life testing models for different distributions. Specifically, two commonly used distributions including the exponential distribution and the Weibull distribution are considered with a special decision loss function. We demonstrate that the proposed model is superior to models with Type I or Type II censoring. Numerical examples are reported to illustrate the effectiveness of the method proposed. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Progressive Type‐II censoring and coherent systems

A new connection between the distribution of component failure times of a coherent system and (adaptive) progressively Type‐II censored order statistics is established. Utilizing this property, we develop inferential procedures when the data is given by all component failures until system failure in two scenarios: In the case of complete information, we assume that the failed component is also observed whereas in the case of incomplete information, we have only information about the failure times but not about the components which have failed. In the first setting, we show that inferential methods for adaptive progressively Type‐II censored data can directly be applied to the problem. For incomplete information, we face the problem that the corresponding censoring plan is not observed and that the available inferential procedures depend on the knowledge of the used censoring plan. To get estimates for distributional parameters, we propose maximum likelihood estimators which can be obtained by solving the likelihood equations directly or via an Expectation‐Maximization‐algorithm type procedure. For an exponential distribution, we discuss also a linear estimator to estimate the mean. Moreover, we establish exact distributions for some estimators in the exponential case which can be used, for example, to construct exact confidence intervals. The results are illustrated by a five component bridge system. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 512–530, 2015     

Statistical analysis of exponential lifetimes under an adaptive Type‐II progressive censoring scheme

In this article, a mixture of Type‐I censoring and Type‐II progressive censoring schemes, called an adaptive Type‐II progressive censoring scheme, is introduced for life testing or reliability experiments. For this censoring scheme, the effective sample size m is fixed in advance, and the progressive censoring scheme is provided but the number of items progressively removed from the experiment upon failure may change during the experiment. If the experimental time exceeds a prefixed time T but the number of observed failures does not reach m, we terminate the experiment as soon as possible by adjusting the number of items progressively removed from the experiment upon failure. Computational formulae for the expected total test time are provided. Point and interval estimation of the failure rate for exponentially distributed failure times are discussed for this censoring scheme. The various methods are compared using Monte Carlo simulation. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Designing an optimal life test with type I censoring

The present study considers the design of an optimal life test plan with Type I censoring in the case where only one testing machine is available and a machine tests items sequentially as seen in fatigue life tests of metallic materials. Under the condition that the life test is conducted in a fixed censoring time and with a fixed number of test items, a method for determining the censoring time and the number of test items is proposed from the Bayesian point of view. The proposed method is also applied to the Weibull life distribution with a known shape parameter.     

Geometry of the total time on test transform

Total time on test (TTT) plots provide a useful graphical method for tentative identification of failure distribution models. Identification is based on properties of the TTT transform. New properties of the TTT transform distribution are obtained. These results are useful to the user of TTT plots. Although IFR (DFR) distributions are particularly easy to identify from TTT plots, the user must exercise caution relative to identification of IFR A (DFRA) distributions.     

Bounding the optimal burn‐in time for a system with two types of failure

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we consider the problem of determining bounds to the optimal burn‐in time and optimal replacement policy maximizing the steady state availability of a repairable system. It is assumed that two types of system failures may occur: One is Type I failure (minor failure), which can be removed by a minimal repair, and the other is Type II failure (catastrophic failure), which can be removed only by a complete repair. Assuming that the underlying lifetime distribution of the system has a bathtub‐shaped failure rate function, upper and lower bounds for the optimal burn‐in time are provided. Furthermore, some other applications of optimal burn‐in are also considered. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Comparison between constant‐stress and step‐stress accelerated life tests under Time Constraint

By running life tests at higher stress levels than normal operating conditions, accelerated life testing (ALT) quickly yields information on the lifetime distribution of a test unit. The lifetime at the design stress is then estimated through extrapolation using a regression model. In constant‐stress testing, a unit is tested at a fixed stress level until failure or the termination time point of test, whereas step‐stress testing allows the experimenter to gradually increase the stress levels at some prefixed time points during the test. In this work, the optimal k‐level constant‐stress and step‐stress ALTs are compared for the exponential failure data under complete sampling and Type‐I censoring. The objective is to quantify the advantage of using the step‐stress testing relative to the constant‐stress one. Assuming a log‐linear life–stress relationship with the cumulative exposure model for the effect of changing stress in step‐stress testing, the optimal design points are determined under C/D/A‐optimality criteria. The efficiency of step‐stress testing to constant‐stress one is then discussed in terms of the ratio of optimal objective functions based on the information matrix. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 00: 000–000, 2013     

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     

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.     

Optimal design of accelerated life tests under periodic inspection

For the exponentially distributed lifetimes, optimal accelerated life test plans are determined under the assumptions of periodic inspection and Type I censoring. Computational results indicate that for the range of parameter values considered the asymptotic variance of the estimated mean or pth quantile at the use stress is not sensitive to the number of inspections at overstress levels. Senstivity analyses are also conducted to see how senstive the asymptotic variance of the estimated mean is with respect to the uncertainties involved in the guessed failure probabilities at the use and high stress levels. Computational results show that moderate deviations (several tens of percents) of the guessed failure probabilities from their true values are fairly tolerable in terms of the relative amount of increase in the asymptotic variance of the estimated mean. Procedures for selecting a sample size and for determining whether or not to conduct an accelerated life test are also discussed.     

Exact likelihood inference for the exponential distribution under generalized Type‐I and Type‐II hybrid censoring

Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Commun Statist Theory Methods 17 (1988), 1857–1870] considered a hybrid censoring scheme and obtained the exact distribution of the maximum likelihood estimator of the mean of an exponential distribution along with an exact lower confidence bound. Childs et al. [Exact likelihood inference based on Type‐I and Type‐II hybrid censored samples from the exponential distribution, Ann Inst Statist Math 55 (2003), 319–330] recently derived an alternative simpler expression for the distribution of the MLE. These authors also proposed a new hybrid censoring scheme and derived similar results for the exponential model. In this paper, we propose two generalized hybrid censoring schemes which have some advantages over the hybrid censoring schemes already discussed in the literature. We then derive the exact distribution of the maximum likelihood estimator as well as exact confidence intervals for the mean of the exponential distribution under these generalized hybrid censoring schemes. © 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     

Selecting among weibull,lognormal and gamma distributions using complete and censored smaples

In a recent paper, Kent and Quesenberry [19] considered using certain optimal invariant statistics to select the best fitting member of a collection of probability distributions using complete samples of life data. In the present work extensions of this approach in two directions are given. First, selection for complete samples based on scale and shape invariant statistics is considered. Next, the selection problem for type I censored samples is considered, and both scale invariant and maximum likelihood selection procedures are studied. The two-parameter (scale and shape) Weibull, lognormal, and gamma distributions are considered and applications to real data are given. Results from a (small) comparative simulation study are presented.     

Determining confidence bounds for highly reliable coherent systems based on a paucity of component failures

A computationally simple method for obtaining confidence bounds for highly reliable coherent systems, based on component tests which experience few or no failures, is given. Binomial and Type I censored exponential failure data are considered. Here unknown component unreliabilities are ordered by weighting factors, which are firstly presumed known then sensitivity of the confidence bounds to these assumed weights is examined and shown to be low.     

A note on two IFR systems

We present probabilistic proofs for the following two facts: (i) A k out of n system of i.i.d (independent identically distributed). IFR (increasing failure rate) components has an IFR life distribution. (ii) A compound Poisson process with nonnegative i.i.d jumps with PF₂ distribution is IFR.     

Multivariate meta analysis with potentially correlated marketing study results

A univariate meta analysis is often used to summarize various study results on the same research hypothesis concerning an effect of interest. When several marketing studies produce sets of more than one effect, multivariate meta analysis can be conducted. Problems one might have with such a multivariate meta analysis are: (1) Several effects estimated in one model could be correlated to each other but their correlation is seldom published and (2) an estimated effect in one model could be correlated to the corresponding effect in the other model due to similar model specification or the data set partly shared, but their correlation is not known. Situations like (2) happen often in military recruiting studies. We employ a Monte‐Carlo simulation to evaluate how neglecting such potential correlation affects the result of a multivariate meta analysis in terms of Type I, Type II errors, and MSE. Simulation results indicate that such effect is not significant. What matters is rather the size of the variance component due to random error in multivariate effects. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 500–510, 2000.     

Remarks on a univariate shock model with some bivariate generalizations

In a 1973 paper J. D. Esary, A. W. Marshall, and F. Proschan [5] considered a shock model giving rise to various nonparametric classes of life distributions of interest in reliability theory. A number of authors have extended these results in a variety of directions. In this paper, alternative proofs of the increasing failure rate (IFR) and decreasing mean residual life (DMRL) results are given which do not make use of the theory of total positivity. Some bivariate extensions are then obtained using a shock model similar to that originally used by H. W. Block, A. S. Paulson, and R. C. Kohberger [2] to unify various bivariate exponential distributions.     

Closure properties of the relevation transform*

Sufficient conditions under which the relevation of two probability distributions is (i) NBU, (ii) IFRA, (iii) IFR are derived. The result for case (iii) corrects an error in a previous article by Baxter.     

The logistic–exponential survival distribution

For various parameter combinations, the logistic–exponential survival distribution belongs to four common classes of survival distributions: increasing failure rate, decreasing failure rate, bathtub‐shaped failure rate, and upside‐down bathtub‐shaped failure rate. Graphical comparison of this new distribution with other common survival distributions is seen in a plot of the skewness versus the coefficient of variation. The distribution can be used as a survival model or as a device to determine the distribution class from which a particular data set is drawn. As the three‐parameter version is less mathematically tractable, our major results concern the two‐parameter version. Boundaries for the maximum likelihood estimators of the parameters are derived in this article. Also, a fixed‐point method to find the maximum likelihood estimators for complete and censored data sets has been developed. The two‐parameter and the three‐parameter versions of the logistic–exponential distribution are applied to two real‐life data sets. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008