R. Calabria

Point estimation under asymmetric loss functions for left-truncated exponential samples

In this paper, Bayes estimates of the parameters and functions thereof in the left-truncated exponential distribution are derived. Asymmetric loss functions are used to reflect that, in most situations of interest, overestimation of a parameter does not produce the same economic consequence than underestimation. Both the non-informative prior and an informative prior on the reliability level at a prefixed time value are considered, and the statistical performances of the Bayes estimates are compared to those of the maximum likelihood ones through the risk function.

Bayes inference for a non-homogeneous Poisson process with power intensity 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 alpha , 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 beta mean is interesting. >

An engineering approach to Bayes estimation for the Weibull distribution

Abstract In this paper an engineering approach to Bayes reliability analysis of Weibull failure data collected under a randomly censored sampling is proposed. The posterior distribution of several decision variables, such as the meanlife, the reliability function, the reliable life, and the hazard rate, are derived, when either a prior information on the reliability or a prior information on the hazard rate is available. Point estimates of the selected decision variables are given, by assuming both symmetric and asymmetric loss functions. Finally, numerical examples are presented to illustrate the proposed estimation procedures.

Bayes 2-sample prediction for the inverse weibull distribution

This paper deals with the problem of predicting, on the base of censored sampling, the ordered lifetimes in a future sample when samples are assumed to follow the inverse weibull distribution. Bayes prediction intervals are derived, both when no prior information is available and when prior informtion on the unreliability level at a fixed time is introduced in the predictive procedure. A Monte Carlo simulation study has shown that the the use of the prior information leads to a more accurate prediction, also when the choice of the informative prior density is quite wrong.

Inference and test in modeling the failure/repair process of repairable mechanical equipments

Abstract In many practical situations, repairable mechanical equipments are subjected to repair actions which significantly depart from both the minimal repair and the perfect maintenance assumptions. In this paper, two point processes are proposed, which allows the failure pattern of repairable equipments subjected to imperfect or hazardous maintenance and experiencing reliability improvement or worsening to be analyzed. Maximum likelihood estimators and testing procedures for the departure from minimal repair and perfect maintenance hypotheses are discussed. Numerical examples are also given to illustrate the use of the proposed models and procedures. Finally, results of a large simulation study are shown in order to assess the accuracy of the proposed inference and testing procedures for the small and moderate sample sizes encountered in practice.

Confidence limits for reliability and tolerance limits in the inverse Weibull distribution

Abstract The inverse Weibull distribution has been recently proposed as a suitable model to describe mechanical degradation phenomena. In this paper the statistical properties of the maximum likelihood estimator, R(t), of the inverse Weibull reliability, R(t), are investigated. Tables of lower confidence limits for R(t), for confidence levels of 0·90, 0·95 and 0·98, are also provided. Tolerance limits based upon maximum likelihood estimators of the parameters are derived, and it is shown that the same tables needed for obtaining confidence intervals for the reliability allow one to estimate tolerance limits, too. A numerical example is given to clarify the use of the tables.

Bayes estimation of prediction intervals for a power law process

Given the first n successive occurence times from a non-homogeneous Poisson process with a power-law intensity function, Bayes prediction intervals for future observations are derived. A Bayesian approach is compared, via Monte Carlo simulation, with a classical one, taking into account several factors, such as prior information, sample size and true values of process parameters. It is found that the Bayesian procedure generally attains sensibly better performances even when there is little prior information available.

A reliability-growth model in a Bayes-decision framework

Market requirements demand testing programs in the development phase of complex repairable systems to be planned in order to improve system reliability. Thus, reliability-growth models are important. This paper proposes a nonparametric reliability-growth model which analyzes, in a Bayes-decision framework, failure data from repairable systems undergoing a test-find-test growth program. The failure process in each stage of testing is assumed to follow a power-law process which can describe the failure pattern of systems subject to wear-out degradation during test. The mean number of failures in a prefixed time interval is used to measure the system reliability at each testing-stage, and the decision process is constructed around the posterior distribution of this quantity. A numerical example illustrates the decision process.

Bayes inference for the modulated power law process

The Modulated Power Law process has been recently proposed as a suitable model for describing the failure pattern of repairable systems when both renewal-type behaviour and time trend are present. Unfortunately, the maximum likelihood method provides neither accurate confidence intervals on the model parameters for small or moderate sample sizes nor predictive intervals on future observations. This paper proposes a Bayes approach, based on both non-informative and vague prior, as an alternative to the classical method. Point and interval estimation of the parameters, as well as point and interval prediction of future failure times, are given. Monte Carlo simulation studies show that the Bayes estimation and prediction possess good statistical properties in a frequentist context and, thus, are a valid alternative to the maximum likelihood approach. Numerical examples illustrate the estimation and prediction procedures.

Some modified maximum likelihood estimators for the weibull process

Abstract A non-homogeneous Poisson process with Weibull intensity function has been often used to analyze the failure pattern of repairable systems. Simple distributional results for the maximum likelihood estimator of the shape parameter β are available, so that an unbiased estimator for β can be introduced. In this paper the effect of using this unbiased estimator in the point estimation of quantities useful in performing reliability analyses, such as the expected number of failures in a given time interval and the failure intensity at a given time t, is analyzed. Moreover an approximately unbiased estimator of the failure intensity at the time of the failure n is introduced.