On the Statistical Modeling and Analysis of Repairable Systems

We review basic modeling approaches for failure and maintenance data from repairable systems. In particular we consider imperfect repair models, defined in terms of virtual age processes, and the trend-renewal process which extends the nonhomogeneous Poisson process and the renewal process. In the case where several systems of the same kind are observed, we show how observed covariates and unobserved heterogeneity can be included in the models. We also consider various approaches to trend testing. Modern reliability data bases usually contain information on the type of failure, the type of maintenance and so forth in addition to the failure times themselves. Basing our work on recent literature we present a framework where the observed events are modeled as marked point processes, with marks labeling the types of events. Throughout the paper the emphasis is more on modeling than on statistical inference.Comment: Published at http://dx.doi.org/10.1214/088342306000000448 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

A New Class of Life Distribution based on Laplace Transform and It’s Applications

Based on the approach of Laplace transform, a new class of life distributions called used better than aged in increasing concave denoted by (UBAC(2)L) is introduced. The implication of our proposed class of life distribution with other classes is given. Some properties of UBAC(2)L class of life distribution are studied. By using the goodness of fit methodology, a new test statistic is proposed for testing exponentiality versus UBAC(2)L class of life distribution. Critical values of our test are calculated for complete and censored data. The power of the test and pitman’s asymptotic efficiency (PAE) for some commonly used distributions in reliability are calculated. Finally, a set of real data is used as an example to elucidate the use of the proposed test statistic for practical reliability analysis

A New Class of Life Distribution based on Laplace Transform and It’s Applications

Based on the approach of Laplace transform, a new class of life distributions called used better than aged in increasing concave denoted by (UBAC(2)L) is introduced. The implication of our proposed class of life distribution with other classes is given. Some properties of UBAC(2)L class of life distribution are studied. By using the goodness of fit methodology, a new test statistic is proposed for testing exponentiality versus UBAC(2)L class of life distribution. Critical values of our test are calculated for complete and censored data. The power of the test and pitman’s asymptotic efficiency (PAE) for some commonly used distributions in reliability are calculated. Finally, a set of real data is used as an example to elucidate the use of the proposed test statistic for practical reliability analysis

NONPARAMETRIC TEST FOR UBACT CLASS OF LIFE DISTRIBUTION BASED ON U-STATISTIC

Based on U-statistic, testing exponentially versus used better than aged in convex tail ordering (UBACT) class of life distribution is introduced for complete and cen-sored data. Convergence of the proposed statistic to the normal distribution is proved. Selected critical values are tabulated for sample sizes 5(5)80 for complete data, and (61)(10)(201) for censored data: The Pitman asymptotic relative e¢ ciency of the pro- posed tests to the other classes is studied. A numerical examples in medical science demonstrates practical application of the proposed test

Nonparametric approach to reliability and its applications

Reliability concepts are used by reliability engineers in the industry to perform systematic reliability studies for the identification and possible elimination of failure causes, quantification of failure occurrences and for the reduction of failure consequences. Apart from applications to mechanical, electronic systems and software, reliability concepts are heavily used in biomedicine to model and understand biological processes such as aging. The standard approach in estimating reliability measures is to assume that the underlying lifetime distribution is known, even if only approximately. When the assumed parametric model is valid, the accuracy of corresponding inferences made based on the estimated function is usually sufficient. However, when this is in doubt, use of a parametric approach could lead to inaccurate inferences. In the literature, this issue has been studied extensively. In such circumstances, estimating these reliability measures using nonparametric techniques has the advantage of flexibility as they generally impose less restriction on the underlying distribution of the life time variable. This thesis considers three popular reliability measures, namely, Reversed Hazard Rate (RHR), Expected Inactivity Time (EIT) and Mean Residual Life (MRL) functions and introduces new estimation methods based on a nonparametric technique called the fixed-design local polynomial regression method. Investigations were undertaken on the theoretical properties of these estimators such as their asymptotic bias, variance and distribution. Extensive simulations were carried out to investigate their performances. The thesis also introduces some novel hypothesis testing procedures for comparing between reliability measures based on samples from two populations using nonparametric techniques. Finally, these methods were applied to address various interesting problems in biomedicine and reliability engineering to demonstrate their practical utility

Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing