Band 21, Heft 2

The reduction of run length variability has received little attention in the statistical process control literature. This note proposes a modification of the signal rule for the EWMA control chart resulting in a reasonable reduction of run length variability.

Stochastic modelling has become a scientific catchword in many branches of science and in particular in quality theory. However, there are many misunderstandings concerning the term, and in many cases it is used wrongly. Therefore, this paper aims at introducing stochastic models, their nature and their significance. On any level of society decisions have to be made about actions aiming at controlling future developments. In particular this is true in the realm of quality control and quality assurance. The foundation for a good decision is necessarily a reliable prediction and, therefore, the development of appropriate prediction procedures constitutes the major task in quality control. Any reliable prediction procedure is necessarily based on a stochastic model implying that the development of stochastic models is of fundamental importance not only for the control of quality.

In the present paper we discuss the problem of estimating the survival function R ( x ) = P ( X > x ) of the Pareto distribution, when the sample contains discordant observations. Bayes point estimates and credible intervals are obtained by assuming exchangeable and identifiable models and by discarding the outlying observations under a class of loss functions. We illustrate the estimators with a real data-set and then compare the three estimators using a simulation study.

When evaluating point estimators by means of general loss functions, the expected loss is not always minimal, similar to the case of mean-biased estimators, whose mean squared error can be reduced by accounting for the mean-bias. Depending on the loss function, the socalled Lehmann-bias can be significantly more important than the mean-bias of an estimator. Although a simple decomposition does not hold for expected losses as it does for the mean squared error, the expected loss can still be reduced by correcting for the Lehmann-bias. An asymptotic and a bootstrap-based correction are suggested and compared in small samples for the exponential distribution by means of Monte Carlo simulation.

Most of the component level reliability test plans that are available in literature have been developed for series and parallel systems under the assumption that components have constant failure rates. Here an attempt is made to construct a test plan for a parallel system with the failure rates of the corresponding exponential distributions depending upon covariates such as room temperature, pressure etc. In this note, unlike in the works of Altinel and others [Computational Statistics & Data Analysis 14: 281-292, 1992, European Journal of Operational Research 78: 318-333, 1994, Naval Research Logistics 44: 187-197, 1997, European Journal of Operational Research 111: 175-186, 1998, IIE Transactions 33: 1093-1108, 2001, Reliability Engineering and System Safety 78: 33-43, 2002], component failure rates are explicitly expressed in terms of covariates via exponential relationship. The development of this note parallels that of Yan and Mazumdar [IEEE Transactions on Reliability 36: 419-424, 1987], except that it involves the use of Arithmetic-Geometric Mean inequality. The obtained numerical results highlight the importance of this note by distinctly showing economic advantages in using covariates related information.

This paper presents a Bayesian approach to the estimation of the reliability of a parallel system with k components. It is assumed that the strengths of the k components are subjected to a common stress which is independent of the strengths of the k components. The reliability of the parallel system or system reliability is estimated assuming that the strengths and the stress are exponentially distributed random variables. The performed Bayesian analysis is based on conjugated prior and non-informative prior distributions, respectively. A simulation data set is considered in order to study the performance of the Bayesian solutions. Some observations are discussed concerning alternative methods and possible extensions.

The purpose of this paper is to derive an approximation of the reliability of a system with doubly bounded performance functions. The problem is illustrated through the probability of an n dimensional hyper cube of the multivariate normal distribution. An approximation method is presented to evaluate this probability based on n + 2 computations of the CDF of the multi-normal distribution.

A new control chart namely S pw -Chart for monitoring the changes in the process variability is proposed and is based on Probability Weighted Moments (PWMs) and assuming that the quality characteristic follows a normal distribution. The coefficients r 2 and r 3 (similar as the d 2 and d 3 coefficients used for R -Charts) are derived for sample sizes n = 2, 3, . . . , 20, 25, 30, 35, 50, 100 by means of a simulation study. The quantiles of which are used for determining the values of the control limits and the power of the S pw -Chart to detect shifts in process variability, are also derived for n = 2, 3, . . . , 20, 25, 30, 35, 50, 100 by simulation. Each of the simulation studies is based on 10,000 random samples from the corresponding normal distribution. The performance of S pw -Chart is investigated by comparing its power curves with those of R and S Charts. It is observed that the power curves of the S pw -Chart are above those of the R -Chart, while slightly below those of the S -Chart in detecting shifts in the process variability. The effect of non-normality on the designs of S , R , and S pw Charts, is studied by simulating random samples from the exponential and the t distributions. The simulations reveal superiority of the S pw -Chart over both R and S Charts in the sense that the power curve of S pw -Chart is least affected by non-normality among all the three charts under study.

A maximum likelihood estimation procedure is developed for a bivariate frailty regression model, in which dependence is generated by a gamma distribution. It is assumed that the random lifetimes follow a bivariate Weibull distribution proposed in Hanagal [Economic Quality Control 11: 193-200, 1996] and that censoring is independent of the two lifetimes. The proposed model may be applied for survival times in genetic epidemiology, survival times of dental implants of patients and survival times of twin births (both monozygotic and dizygotic), with genetic frailty behavior (which is unknown and random) of patients.

This paper extends the decomposition of marginal effects suggested by McDonald and Moffitt [The Review of Economics and Statistics 62: 318-321, 1980] to a nonlinear tobit model, considering the increasing use of tobit analysis and substantive economic implications of the decomposition. The decomposition provides more information than is commonly realized based on the coefficients obtained from fitting a tobit model. In this paper the generalized decomposition of marginal effects is derived and the effects in the decomposition are expressed in explicit and closed form. A simulation study illustrates the application of the nonlinear decomposition using a simple nonlinear consumption function, along with maximum likelihood estimation on the parameters of the nonlinear tobit regression model.

A generalization of the log-logistic distribution called exponentiated log-logistic distribution (in lines of exponentiated Weibull distribution suggested by Mudholkar and Srivastava [Technometrics 37: 436-445, 1995]) is considered. In this paper the operating characteristic for a sampling plan is determined for the case that a lot of products is submitted for inspection with lifetimes specified by an exponentiated log-logistic distribution (ELLD). The results are illustrated by a numerical example.

A finite capacity Markovian queueing system with a number of heterogeneous channels depending on the system size is considered. The system consists of one regular channel and a maximum of c additional channels. When all the channels are in operation, the arrival rate is controlled depending on the queue length in order to minimize the waiting time of the customers. Expressing the Laplace transforms of the system of governing equations in matrix form and using the properties of symmetric tridiagonal matrices, the transient and steady state probabilities are derived. Some numerical results are presented.