Nonparametric smooth estimation of the expected inactivity time function

The expected inactivity time (EIT) function (also known as the mean past lifetime function) is a well known reliability function which has application in many disciplines such as survival analysis, actuarial studies and forensic science, to name but a few. In this paper, we use a fixed design local polynomial fitting technique to obtain estimators for the EIT function when the lifetime random variable has an unknown distribution. It will be shown that the proposed estimators are asymptotically unbiased, consistent and also, when standardized, has an asymptotic normal distribution. An optimal bandwidth, which minimizes the AMISE (asymptotic mean integrated squared error) of the estimator, is derived. Numerical examples based on simulated samples from various lifetime distributions common in reliability studies will be presented to evaluate the performances of these estimators. Finally, three real life applications will also be presented to further illustrate the wide applicability of these estimators.     

Minimum average risk estimators for coefficients in linear models

We consider the problem of estimating the coefficient vector β of a linear regression model with quadratic loss function. Some biased estimators which utilize the prior information about β are considered. Also studied is the problem of estimating the parameters of an over-identified structural equation from undersized samples.     

Notes on likelihood intervals and profiling

In many applications, decisions are made on the basis of function of parameters g(θ). When the value of g(theta;) is calculated using estimated values for te parameters, its is important to have a measure of the uncertainty associated with that value of g(theta;). Likelihood ratio approaches to finding likelihood intervals for functions of parameters have been shown to be more reliable, in terms of coverage probability, than the linearization approach. Two approaches to the generalization of the profiling algorithm have been proposed in the literature to enable construction of likelihood intervals for a function of parameters (Chen and Jennrich, 1996; Bates and Watts, 1988). In this paper we show the equivalence of these two methods. We also provide and analysis of cases in which neither profiling algorithm is appropriate. For one of these cases an alternate approach is suggested Whereas generalized profiling is based on maximizing the likelihood function given a constraint on the value of g(θ), the alternative algorithm is based on optimizing g(θ) given a constraint on the value of the likelihood function.     

Are the Order Statistics Ordered? A Survey of Recent Results

The distributions of coherent systems with components with exchangeable lifetimes can be represented as mixtures of distributions of order statistics (k-out-of-n systems) from possibly dependent samples by using the concept of the signature of Samaniego (1985 Samaniego , F. ( 1985 ). On closure of the IFR class under formation of coherent systems . IEEE Trans. Reliabil. R-34 : 69 – 72 .[Crossref], [Web of Science ®] , [Google Scholar]). This representation, together with Rychlik's (1993 Rychlik , T. ( 1993 ). Bounds for expectation of L-estimates for dependent samples . Statistics 24 : 1 – 7 .[Taylor & Francis Online] , [Google Scholar]) results, can be used to obtain sharp bounds on the distribution (or the reliability) function and on the expected lifetime of the system. Also, this representation can be used to determine the asymptotic behavior of the hazard rate of the system when the order statistics are ordered in the hazard rate order. Moreover, the lifetime distributions of coherent systems (and in particular, of order statistics) can also be represented as generalized mixtures (that is, mixtures with some negative weights) of distributions of series system lifetimes by using the concept of the minimal signature defined by Navarro et al. (2007a Navarro , J. , Ruiz , J. M. , Sandoval , C. J. ( 2007a ). Properties of coherent systems with dependent components . Commun. Statist. Theory Methods 36 : 175 – 191 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). This representation can also be used to determine the final behavior of the hazard rate of the system through the behavior of the hazard rate of the series systems. In particular, it can be used to show that the order statistics are, under some conditions, asymptotically hazard rate ordered. However, in general, this result is not true, that is, the order statistics need not be hazard rate ordered.     

A Bivariate Distribution for the Reliability Analysis of Failure Data in Presence of a Forewarning or Primer Event

This article presents a bivariate distribution for analyzing the failure data of mechanical and electrical components in presence of a forewarning or primer event whose occurrence denotes the inception of the failure mechanism that will cause the component failure after an additional random time. The characteristics of the proposed distribution are discussed and several point estimators of parameters are illustrated and compared, in case of complete sampling, via a large Monte Carlo simulation study. Confidence intervals based on asymptotic results are derived, as well as procedures are given for testing the independence between the occurrence time of the forewarning event and the additional time to failure. Numerical applications based on failure data of cable insulation specimens and of two-component parallel systems are illustrated.     

Non-negative estimation of variance components in heteroscedastic one-way random-effects ANOVA models

There is a considerable amount of literature dealing with inference about the parameters in a heteroscedastic one-way random-effects ANOVA model. In this paper, we primarily address the problem of improved quadratic estimation of the random-effect variance component. It turns out that such estimators with a smaller mean squared error compared with some standard unbiased quadratic estimators exist under quite general conditions. Improved estimators of the error variance components are also established.     

Transient Analysis of Large Markov Models with Absorbing States Using Regenerative Randomization

ABSTRACT

In this article, we develop a new method, called regenerative randomization, for the transient analysis of continuous time Markov models with absorbing states. The method has the same good properties as standard randomization: numerical stability, well-controlled computation error, and ability to specify the computation error in advance. The method has a benign behavior for large t and is significantly less costly than standard randomization for large enough models and large enough t. For a class of models, class C, including typical failure/repair reliability models with exponential failure and repair time distributions and repair in every state with failed components, stronger theoretical results are available assessing the efficiency of the method in terms of “visible” model characteristics. A large example belonging to that class is used to illustrate the performance of the method and to show that it can indeed be much faster than standard randomization.     

Water cycle algorithm for solving the reliability-redundancy allocation problem with a choice of redundancy strategies

Abstract

This paper proposes a new mathematical model for the reliability-redundancy allocation problem (RRAP) with a choice of redundancy strategies. To maximize the reliability of a system, this model chooses the best redundancy strategy from among both active and standby ones for each subsystem. For those with a standby strategy, a continuous time Markov chain model is used to calculate the exact reliability values. In order to solve the proposed mixed-integer non-linear programing model, a powerful evolutionary algorithm, called water cycle algorithm (WCA), is developed and implemented on three famous benchmark problems. Finally, the results of different benchmark problems are compared with those previously reported to show the superiority of the proposed model and the efficiency of WCA.     

On approximate likelihood inference in a poisson mixed model

A two-step estimation approach is proposed for the fixed-effect parameters, random effects and their variance σ² of a Poisson mixed model. In the first step, it is proposed to construct a small σ²-based approximate likelihood function of the data and utilize this function to estimate the fixed-effect parameters and σ². In the second step, the random effects are estimated by minimizing their posterior mean squared error. Methods of Waclawiw and Liang (1993) based on so-called Stein-type estimating functions and of Breslow and Clayton (1993) based on penalized quasilikelihood are compared with the proposed likelihood method. The results of a simulation study on the performance of all three approaches are reported.     

Reliability Analysis for Hazardous Waste Treatment Processes

The reliability of a treatment process is addressed in terms of achieving a regulatory effluent concentration standard and the design safety factors associated with the treatment process. This methodology was then applied to two aqueous hazardous waste treatment processes: packed tower aeration and activated sludge (aerobic) biological treatment. The designs achieving 95 percent reliability were compared with those designs based on conventional practice to determine their patterns of conservatism. Scoping-level treatment costs were also related to reliability levels for these treatment processes. The results indicate that the reliability levels for the physical/chemical treatment process (packed tower aeration) based on the deterministic safety factors range from 80 percent to over 99 percent, whereas those for the biological treatment process range from near 0 percent to over 99 percent, depending on the compound evaluated. Increases in reliability per unit increase in treatment costs are most pronounced at lower reliability levels (less than about 80 percent) than at the higher reliability levels (greater than 90 percent, indicating a point of diminishing returns. Additional research focused on process parameters that presently contain large uncertainties may reduce those uncertainties, with attending increases in the reliability levels of the treatment processes.