Gianpaolo Pulcini

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.

A mixed-Weibull regression model for the analysis of automotive warranty data

Abstract This paper presents a case study regarding the reliability analysis of some automotive components based on field failure warranty data. The components exhibit two different failure modes, namely early and wearout failures, and are mounted on different vehicles, which differ among themselves for car model and engine type, thus involving different operating conditions. Hence, the failure time of each component is a random variable with a bimodal pdf which also depends upon a vector of covariates that indexes the specific operating condition. Then, a mixed-Weibull distribution, where the pdf of each subpopulation (namely the ‘weak’ and ‘strong’ subpopulation) depends on the covariates through the scale parameter, is used to analyze the component lifetime. A Fortran algorithm for the maximum likelihood estimation of model parameters has been implemented and a stepwise procedure, in its backwards version, has been used to test the significance of covariates and to construct the regression model. The presence of a weak subpopulation has been verified and the fraction of weak units in the population has also been estimated. Finally, the adequacy of the proposed model to fit the observed data has been assessed.

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.

An age- and state-dependent Markov model for degradation processes

Many technological units are subjected during their operating life to a gradual deterioration process that progressively degrades their characteristics until a failure occurs. Statisticians and engineers have almost always modeled degradation phenomena using independent increments processes, which imply that the degradation growth depends, at most, on the unit age. Only a few models have been proposed in which the degradation growth is assumed to depend on the current unit state. In many cases, however, both the current age and the current state of a unit can affect the degradation process. As such, this article proposes a degradation model in which the transition probabilities between unit states depend on both the current age and the current degradation level. Two applications based on real data sets are analyzed and discussed.

Modeling the failure data of a repairable equipment with bathtub type failure intensity

Abstract The paper deals with the reliability modeling of the failure process of large and complex repairable equipment whose failure intensity shows a bathtub type non-monotonic behavior. A non-homogeneous Poisson process arising from the superposition of two power law processes is proposed, and the characteristics and mathematical details of the proposed model are illustrated. A graphical approach is also presented, which allows to determine whether the proposed model can adequately describe a given failure data. A graphical method for obtaining crude but easy estimates of the model parameters is then illustrated, as well as more accurate estimates based on the maximum likelihood method are provided. Finally, two numerical applications are given to illustrate the proposed model and the 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.

A competing risk model for the reliability of cylinder liners in marine Diesel engines

In this paper, a competing risk model is proposed to describe the reliability of the cylinder liners of a marine Diesel engine. Cylinder liners presents two dominant failure modes: wear degradation and thermal cracking. The wear process is described through a stochastic process, whereas the failure time due to the thermal cracking is described by the Weibull distribution. The use of the proposed model allows performing goodness-of-fit test and parameters estimation on the basis of both wear and failure data. Moreover, it enables reliability estimates of the state of the liners to be obtained and the hierarchy of the failure mechanisms to be determined for any given age and wear level of the liner. The model has been applied to a real data set: 33 cylinder liners of Sulzer RTA 58 engines, which equip twin ships of the Grimaldi Group. Estimates of the liner reliability and of other quantities of interest under the competing risk model are obtained, as well as the conditional failure probability and mean residual lifetime, given the survival age and the accumulated wear. Furthermore, the model has been used to estimate the probability that a liner fails due to one of the failure modes when both of these modes act.

A Bounded Intensity Process for the Reliability of Repairable Equipment

In the failure pattern of repairable equipment subjected to reliability deterioration with operating time, the repeated application of the repair actions sometimes produces a finite bound for the increasing failure intensity. In this paper, a non-homogeneous Poisson process for which the failure intensity is an increasing bounded function is proposed. The characteristics of the proposed model and the physical meaning of its parameters are discussed. It is shown that the proposed model evolves initially as the power law process with shape parameter equal to 2, and then converges asymptotically to the homogeneous Poisson process, the latter being a limiting form of the proposed bounded intensity model. Maximum likelihood estimates and approximate confidence intervals for the model parameters are given as well as a testing procedure for a time trend. Percentile points of the test statistic are computed by simulation for failure truncated samples, and the power of the proposed testing procedure is evaluated and compared to that of two commonly used tests. Finally, numerical examples are given to illustrate the proposed model and related inference and testing procedures.

A State-Dependent Wear Model With an Application to Marine Engine Cylinder Liners

In this paper a new wear model is proposed in which the transition probabilities between process states, unlike models with independent increments, depend on the current system state. The model is used to describe the wear process of the cylinder liners of some identical heavy-duty diesel engines for marine propulsion. The application is developed on the basis of a real dataset of wear measures obtained via staggered inspections. A time and state space discretization is introduced to obtain the likelihood function of the observed data. The model parameters and reliability characteristics of the liners are then estimated and the wear growth during future inspection intervals is predicted. The homogeneity of wear data and the goodness of fit of the proposed model are tested. A simplified maintenance scenario is also considered to show the need for accurate modeling of the wear process for planning condition-based maintenance activities. Finally, inferential, predictive, and decision-making results derived within the proposed model are compared to those obtained within one of the most widely used age-dependent wear models. Fortran codes and executable programs, as well as the cylinder liner data, are available online as supplemental material.