Volume 22, Issue 2

Because of the increasing number of interrelated processes continuous monitoring and controlling of processes get more and more important. Time series constitute one possibility of modelling processes in order to determine an appropriate monitoring policy. One major problem when deriving a time series consists of estimating the values of the relevant parameters. This paper deals with the estimation of the parameters of a first order autoregressive gamma process by means of the EM algorithm. The formulae of the EM sequence are derived, the convergence of the procedure is established and the results of a simulation study are presented. The EM algorithm proves to be an appropriate estimation procedure in the case of the complex statistical model represented by a GAR(1) process.

This work considers point and interval estimation for the lifetime distribution of a k -unit parallel system assuming exponential distribution for the components lifetime. The parameter of the distribution is estimated by the maximum likelihood method based on samples which are progressively Type-II censored. The expression for Fisher's information is derived. Approximate confidence intervals are constructed based on the asymptotic distribution of the maximum likelihood estimator and log-transformed maximum likelihood estimator. The confidence levels of the approximate confidence intervals are examined via simulation. They turn out to be quite satisfactory. An approximate β -expectation tolerance interval for the life distribution is also discussed.

Weibull distributions are popular models for quality and reliability. In this note, we investigate a distribution proposed by Bousquet et al. as an alternative to the Weibull distribution. We derive – for the first time – explicit and general expressions for the moments and the moment generating function associated with the distribution. We also point out that the distribution has been known before in different contexts.

This paper deals with a Markov process used for stochastic modeling of the international banking supervision related to the Basel's accords. A quality index reflecting the performance of a bank is proposed, which is based on the asymptotic behavior of the Markov process.

With the motivation of developing more realistic test plans, an attempt is made in this paper to construct reliability test plans for the components of series systems under the assumptions that the component lifetimes are independent, identically distributed exponential random variables, the failure rates of these exponential random variables depend on covariates, such as room temperature, humidity etc. through linear and exponential relationships. Our test plans differ from the test plans proposed by Altinel and Özekici (1997) and Alitnel et al. (IIE Transactions 33: 1093-1108, 2001, Reliability Engineering and System Safety 78: 33-43, 2002) in the sense that the former assume dependence on covariates through explicit relationships, while the latter make a very general assumption concerning dependence on covariates. It should be noted that as far as the construction of reliability test plans on component level is concerned, this is the first instance of the use of the penalty function methodology. It should also be highlighted that the numerical results given in this paper illustrate that the use of covariates-related information in the linear case leads to more than 90% savings in total testing cost.

Statistical Process Control (SPC) has been established in many industries over the last decades. Particularly, the automotive industry including its suppliers of components is responsible for the spread of SPC supported by different norms. By means of process capability indices the fulfilment of requirements with respect to a product is examined, while control charts are used to monitor a process over time by measuring and analyzing continuously the product variables characterizing product quality. This may lead to high costs, which arise mainly as a result of sampling the data. The objective is to recognize and correct changes in quality as soon as possible. This can be achieved by shorter sampling intervals or by an improved analysis of the data. Conditional control charts are a new way of analyzing sample data by using multivariate information provided in classical control charts, which increase the sensibility of control charts.

Any random variable X describing a real phenomenon has necessarily a bounded range of variability implying that the values of the moments determine the probability distribution uniquely. In fact, the range of variability of a random variable restricts the range of the first moment; the value of the first moment limits considerably the range of the second moment; etc. Thus, any knowledge about the values of lower moments may be used without a further sample for drawing inference on the higher moments. In this paper we assume without loss of generality that the range of variability of the random variable X is given by the unit interval. Subsequently, the arising restrictions for the three first moments are derived and the implications with respect to variance and skewness are identified. Moreover, the situation with respect to unimodal random variables is investigated and it is shown that for unimodal probability distributions the third moment yields only marginal additional information.

This paper proposes an efficient heuristic approach to solving the constrained redundancy optimization problem in multi-state systems (MSS) with multi-state components using minimal path vectors. A discrete multi-state model is considered, where the system state depends on the discipline of the elements' interaction in the system. When the multi-state nature of the system is considered, exact solution methodologies e.g. Dynamic, Integer Programming are no longer valid. The proposed heuristic offers an efficient and straightforward analysis. To illustrate the simplicity and ease of the application of the algorithm, solutions of a flow network problem with linear constraint and, bridge structure with linear as well as nonlinear constraints are obtained. The results would be applicable to multi-state design problems in real life.

The Jelinski-Moranda (JM) model for software failures was one of the first models used for analyzing software reliability. Later Moranda proposed a modification of the JM model, labeled Geometric de-Eutrophication model. In the Moranda Geometric de-Eutrophication model, N ( t ) is defined as the number of faults detected in the time interval (0, t ]. In this paper, N ( t ) is assumed to be a pure stochastic birth process, where failure rates decrease geometrically with a detection and rectifying of a fault. In this paper, a recursive scheme is proposed for studying the probability of detecting n bugs in the time (0, t ]. The method uses a constructed table, which makes the method easier compared to other existing methods for computing P n ( t ), the intensity function and the reliability R τ ( t ). In the proposed procedure P n ( t ) is the sum of ( n +1) terms and each term is based on a factor, which can be from the above mentioned table.

A comprehensive economic cost function is derived for a modified variable sample size and sampling interval (MVSSI) chart. As a variable sample size and sampling interval, a variable sample size, a variable sampling interval, and a static charts are particular cases of an MVSSI chart, the model is directly applicable to these charts, too. The cost function can be used to obtain optimal design parameters of these charts that minimize the process control related cost per hour for given process and cost parameters. It can also be used to compare the economic performances of these charts. A numerical study shows that the optimal adaptive charts differ in their performance only marginally from each other. However, both the statistical and economic performances of an optimal MVSSI chart are superior to that of the other adaptive charts. A sensitivity analysis for a particular example reveals that the optimal design obtained using the cost function is relatively insensitive to errors in specifying various process and cost parameters.

We propose frailty regression models in mixture distributions assuming the distribution of frailty to be a gamma distribution. We consider Weibull mixture as an example. There are some interesting situations like survival times in genetic epidemiology, dental implants of patients and twin births (both monozygotic and dizygotic) where genetic behavior (which is unknown and random) of patients follows a known frailty distribution. These are the situations, which motivate to study this particular model.

A truncated version of the bivariate t distribution is introduced. Unlike the t distribution, this possesses finite moments of all orders and could, therefore, be a better model for real-world situations, as it becomes possible to incorporate always available knowledge about the range of variability of the considered aspect into the model. Using the usual t distribution, this knowledge has to be abandoned and additionally the purely mathematical problems of infinite moments occurs. One such situation is discussed. Explicit expressions for the moments of the truncated distribution are also derived.