Hierarchical Bayesian reliability analysis using Erlang families of priors and hyperpriors

An hierarchical Bayes approach to reliability estimation for the exponential model with an unknown scale parameter, based on life tests that are terminated after a preassigned number of failures, is considered under the assumptions of squared error loss and Erlang distributions as the prior and hyperprior. The Bayesian estimation of reliability for the case of ‘attribute testing’ is also discussed.     

Modified Method-of-Moments in Empirical Bayes Estimation

A unit is put on test for a fixed time and the number of failures is observed. The probability distribution of the number of failures is assumed to be Poisson, and the Poisson failure intensity is assumed to be a stochastic variable with gamma prior distribution. Schafer & Feduccia introduced an empirical procedure for estimating the parameters of the prior based on method of moments. We investigate the s-efficiencies of empirical Bayes estimates of Poisson failure intensity and reliability when the prior is estimated by the Schafer & Feduccia method. Mean square errors (MSEs) are compared for a range of parameters which typifies certain military equipment failure data. The empirical Bayes estimates have high s-efficiencies for sample size more than 40. A modification of the Schafer & Feduccia procedure substantially improves s-efficiencies for small sample sizes.     

Bayesian estimation of reliability function and hazard rate

The Bayes estimates of reliability function and hazard rate function of the finite range failure model have been developed based on life tests that are terminated at a preassigned time point or after a certain number of failures have occurred, taking the order of observations into consideration. For the prior distribution of the parameter involved, the uniform, exponential and inverted gamma densities have been considered. As an example, failure data for a V600 indicator tube used in aircraft radar sets, which fit well the finite range failure model, have been considered as the current distribution for obtaining the Bayes estimates of the reliability function.     

The Burr type XII distribution as a failure model under various loss functions

Bayesian estimates of the parameter p and the reliability function for the two-parameter Burr type XII failure model under three different loss functions, absolute difference, squared error and logarithmic are derived. It is assumed that the parameter p behaves as a random variable having (i) a gamma prior and (ii) a vague prior. Monte Carlo simulations are presented to compare the Bayesian estimators and the maximum likelihood estimators of the parameter p and the reliability function. The results show that the “popular” squared error loss function is not always the best, and that the other loss functions give comparable results.     

Bayes Interval Estimation for the Shape Parameter of the Power Distribution

The power distribution is considered as failure model and uses a square-error loss function. Bayes credibility interval estimators for the shape parameter have been obtained assuming 1) the following priors for the shape parameter: Jeffrey's invariant prior, gamma, and inverted gamma; 2) the following priors for reliability: beta and log gamma function. It is straightforward to obtain estimators for reliability when the estimators for the shape parameter are known.     

On Bayes Estimation of Reliability for the Birnbaum-Saunders Fatigue Life Model

Estimation of reliability for the Birnbaum-Saunders fatigue life distribution is considered. The scale parameter is also the median lifetime, and assuming that the scale parameter is known, Bayes estimators of the reliability function are obtained for a family of proper conjugate priors as well as for Jeffreys' vague prior for the shape parameter. When both parameters are unknown, a modified Bayes estimator of reliability is proposed using a moment estimator of the scale parameter. In addition to being computationally simpler than the MLE of reliability, Monte Carlo simulations for small samples show that the modified Bayes estimator is better than the MME for all values of the shape parameter and as good as the MLE for small values of the shape parameter in the sense of root mean squared errors.     

Bayesian analysis of reliability and hazard rate function of a mixture model

The present article deals with the Bayesian analysis for a modified (general) Mukherji-Islam model [A finite range distribution of failure times, Naval Research Logistics Quarterly, 1983, 30, 487–491] on the basis of failure time data x₁, x₂, …, x_r for a prefixed number of failures. Under the use of different prior densities for the parameter and the squared error loss function (SELF), the Bayesian estimates of hazard rate and the reliability functions are obtained, considering uniform, Gamma and inverted gamma densities as prior failure densities.     

A Bayes nonparametric framework for software-reliability analysis

This paper presents a Bayes nonparametric approach for tracking and predicting software reliability. We use the common assumptions on the software operational environment to get a stochastic model where the successive times between software failures are exponentially distributed; their failure rates have Markov priors. Under these general assumptions we give Bayes estimates of the parameters that assess and predict the software reliability. We give algorithms (based on Monte-Carlo methods) to compute these Bayes estimates. Our approach allows the reliability analyst to construct a personal software reliability model simply by specifying the available prior knowledge; afterwards the results in this paper can be used to get Bayes estimates of the useful reliability parameters. Examples of possible prior physical knowledge concerning the software testing and correction environments are given. The maximum-entropy principle is used to translate this knowledge to prior distributions on the failure-rate process. Our approach is used to study some simulated and real failure data sets     

Reliability analysis of a finite range failure model

The Bayes estimates of reliability and hazard rate functions of the finite range failure model (1) have been developed based on life tests that are terminated at a preassigned time point taking the order of observations into consideration. For the parameter involved, the priors gamma, Weibull and log normal densities have been considered. The importance of the distributions, considered here as priors, in the theory of reliability has led the authors to use them as prior distributions.     

Statistical Analysis of a Compound Power-Law Model for Repairable Systems

A compound (mixed) Poisson distribution is sometimes used as an alternative to the Poisson distribution for count data. Such a compound distribution, which has a negative binomial form, occurs when the population consists of Poisson distributed individuals, but with intensities which have a gamma distribution. A similar situation can occur with a repairable system when failure intensities of each system are different. A more general situation is considered where the system failures are distributed according to nonhomogeneous Poisson processes having Power Law intensity functions with gamma distributed intensity parameter. If the failures of each system in a population of repairable systems are distributed according to a Power Law process, but with different intensities, then a compound Power Law process provides a suitable model. A test, based on the ratio of the sample variance to the sample mean of count data from s-independent systems, provides a convenient way to determine if a compound model is appropriate. When a compound Power Law model is indicated, the maximum likelihood estimates of the shape parameters of the individual systems can be computed and homogeneity can be tested. If equality of the shape parameters is indicated, then it is possible to test whether the systems are homogeneous Poisson processes versus a nonhomogeneous alternative. If deterioration within systems is suspected, then the alternative in which the shape parameter exceeds unity would be appropriate, while if systems are undergoing reliability growth the alternative would be that the shape parameter is less than unity.     

Empirical Bayes Estimation in the Weibull Distribution

In part I empirical Bayes estimation procedures are introduced and employed to obtain an estimator for the unknown random scale parameter of a two-parameter Weibull distribution with known shape parameter. In part II, procedures are developed for estimating both the random scale and shape parameters. These estimators use a sequence of maximum likelihood estimates from related reliability experiments to form an empirical estimate of the appropriate unknown prior probability density function. Monte Carlo simulation is used to compare the performance of these estimators with the appropriate maximum likelihood estimator. Algorithms are presented for sequentially obtaining the reduced sample sizes required by the estimators while still providing mean squared error accuracy compatible with the use of the maximum likelihood estimators. In some cases whenever the prior pdf is a member of the Pearson family of distributions, as much as a 60% reduction in total test units is obtained. A numerical example is presented to illustrate the procedures.     

The zero-truncated negative binomial distribution as a failure model from the Bayesian approach

The zero-truncated negative binomial distribution is considered as a failure model from the Bayesian point of view. It is assumed that the location parameter is a stochastic variable with beta as its prior distribution. Under this assumption Bayes estimators are derived for the location parameter and reliability function. By means of Monte Carlo Simulation the minimum variance unbiased estimators (MVUEs) for the parameter and reliability function are compared to the corresponding Bayes estimators.     

Bayes inference for a non-homogeneous Poisson process with powerintensity law [reliability]

Monte Carlo simulation is used to assess the statistical properties of some Bayes procedures in situations where only a few data on a system governed by a NHPP (nonhomogeneous Poisson process) can be collected and where there is little or imprecise prior information available. In particular, in the case of failure truncated data, two Bayes procedures are analyzed. The first uses a uniform prior PDF (probability distribution function) for the power law and a noninformative prior PDF for α, while the other uses a uniform PDF for the power law while assuming an informative PDF for the scale parameter obtained by using a gamma distribution for the prior knowledge of the mean number of failures in a given time interval. For both cases, point and interval estimation of the power law and point estimation of the scale parameter are discussed. Comparisons are given with the corresponding point and interval maximum-likelihood estimates for sample sizes of 5 and 10. The Bayes procedures are computationally much more onerous than the corresponding maximum-likelihood ones, since they in general require a numerical integration. In the case of small sample sizes, however, their use may be justified by the exceptionally favorable statistical properties shown when compared with the classical ones. In particular, their robustness with respect to a wrong assumption on the prior β mean is interesting     

A Monte Carlo Technique for Obtaining System Reliability Confidence Limits from Component Test Data

A digital computer technique is developed, using a Monte Carlo simulation based on common probability models, with which component test data may be translated into approximate system reliability limits at any confidence level. The probability distributions from which the component failures are assumed to come are the exponential, Weibull (shape parameter K known), gamma (shape parameter ? known), normal, and lognormal. The components can be arranged in any system configuration, series, parallel, or both. Since reliability prediction is meaningful only when expressed with an associated confidence leve, this method provides a valuable and economical tool for the reliability analyst.     

A Bayesian Reliability Growth Model

A model is presented for the change (growth) in reliability of a system during a test program. Parameters of the model are assumed to be random variables with appropriate prior density functions. Expressions are then derived that enable estimates (in the form of expectations) and precision statements (in the form of variances) to be made of: 1) projected system reliability at time ? after the start of the test program, and 2) system reliability after the observation of failure data. Numerical examples are presented, and extension to multimode failures is indicated.     

Inspection Intervals for Fail-Safe Structure

The Fail-Safe principle as applied to aircraft structural design implies that there is insufficient knowledge of the life capability of the design. Control of inspection intervals is not supported by risk calculations, yet only a sample of aircraft is inspected, at intervals whose duration is rapidly increased. This paper provides risk estimates based on a simple mathematical model. Catastrophic failure is treated in two stages modeled respectively by 2-parameter and 3-parameter Weibull distributions. Bayes inferences are made about the scale parameter using in-service survivor times. Only those cases are treated for which no failures have occurred. This results in a suggested form of inspection policy. A separate non-Bayes analysis confirms the Bayes risk estimate; thus the assumed improper prior is interesting. This prior, the only simple one which is tractable for the case of no failures, transforms, for the exponential distribution, to the uniform prior, in contrast to the hyperbolic one usually used. The analysis is simplistic but provides a ball-park estimate which would otherwise be unavailable. It can be used with caution as a check on inspection programs already derived by other means. It can also serve in tutorial demonstration of the statistical effects of the various parameters, to airworthiness managers. Possibly it might form the basis of a more sophisticated analysis.     

An Empirical Bayes Approach for the Poisson Life Distribution

A smooth empirical Bayes estimator is derived for the intensity parameter (hazard rate) in the Poisson distribution as used in life testing. The reliability function is also estimated either by using the empirical Bayes estimate of the parameter, or by obtaining the expectation of the reliability function. The behavior of the empirical Bayes procedure is studied through Monte Carlo simulation in which estimates of mean-squared errors of the empirical Bayes estimators are compared with those of conventional estimators such as minimum variance unbiased or maximum likelihood. Results indicate a significant reduction in mean-squared error of the empirical Bayes estimators over the conventional variety.     

A nonparametric nonstationary procedure for failure prediction

The time between failures is a very useful measurement to analyze reliability models for time-dependent systems. In many cases, the failure-generation process is assumed to be stationary, even though the process changes its statistics as time elapses. This paper presents a new estimation procedure for the probabilities of failures; it is based on estimating time-between-failures. The main characteristics of this procedure are that no probability distribution function is assumed for the failure process, and that the failure process is not assumed to be stationary. The model classifies the failures in Q different types, and estimates the probability of each type of failure s-independently from the others. This method does not use histogram techniques to estimate the probabilities of occurrence of each failure-type; rather it estimates the probabilities directly from the values of the time-instants at which the failures occur. The method assumes quasistationarity only in the interval of time between the last 2 occurrences of the same failure-type. An inherent characteristic of this method is that it assigns different sizes for the time-windows used to estimate the probabilities of each failure-type. For the failure-types with low probability, the estimator uses wide windows, while for those with high probability the estimator uses narrow windows. As an example, the model is applied to software reliability data.     

A Bayesian Approach to Parameter and Reliability Estimation in the Poisson Distribution

For life testing procedures, a Bayesian analysis is developed with respect to a random intensity parameter in the Poisson distribution. Bayes estimators are derived for the Poisson parameter and the reliability function based on uniform and gamma prior distributions of that parameter. A Monte Carlo procedure is implemented to make possible an empirical mean-squared error comparison between Bayes and existing minimum variance unbiased, as well as maximum likelihood, estimators. As expected, the Bayes estimators have mean-squared errors that are appreciably smaller than those of the other two.     

Bayesian estimation and prediction for Burr‐Rayleigh mixture model using censored data

In this study, Burr‐XII and Rayleigh distributions are combined to form a new mixture model that is considered to model heterogeneous data. Our objective is to estimate parameters of the proposed mixture model using Bayesian technique under type‐I censoring. Bayesian parameter estimation for the said mixture model is conducted by using informative priors, ie, gamma and squared root inverted gamma (SRIG) as well as noninformative prior, ie, Jeffrey's prior. Squared error loss function (SELF) and quadratic loss function (QLF) are employed to obtain and Bayes estimators. Properties of the proposed Bayes estimators are highlighted through a simulation study. When prior distributions and loss functions utilized in the study are compared in terms of posterior risks, informative prior found to be more suitable and decision turns out to be in favor of QLF. Prediction limits for the single sample case and two sample case are obtained to provide an insight into future sample data. Application of the proposed model is also elaborated using a real‐life example.