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In this work we propose a representation of a bivariate density corresponding to the given geometrical behavior of the marginals. A continuous density with compact support can be approximated by the exponential of an infinite polynomial. We find intervals for the possible values of its coefficients in the simplest cases upon the available information about the marginals.

For predicting a future development in a reliable and precise way, one must necessarily develop a stochastic model given by a Bernoulli-Space introduced by von Collani in 2004. The core of the stochastic model is a set of probability distributions, each describing one potential future development. Any probability distribution for a real-world phenomenon is uniquely determined by the range of variability and the values of a certain number of moments of the considered random variable. Thus, the problems arise how to determine the range of variability on the one hand and the values of the necessary moments on the other. While there are a number of statistical methods to deal with the later problem, the former one is hardly mentioned in any statistical textbook. This is a surprising fact as the range of variability of a random variable is most important and in the special case of a uniform distribution it even determines uniquely the probability distribution and, hence, the values of all moments. Besides the principle importance of the bounds of the range of variability, they are often of great significance, if the extreme values of a random variable are of interest, e.g. for natural phenomena as floods or weather conditions or for technical threshold values. Moreover, the extreme values of a relevant random variable are in general of special interest in the design, maintenance and quality control of systems and facilities. The reason, why there are almost no statistical methods available for determining the range of variability of a given random variable, is the frequently made unrealistic assumption that the range of variability is unbounded. This paper deals with developing a method for improving the knowledge about the range of variability of a given random variable based on the Bernoulli Space as stochastic model of uncertainty. Stübner et al. (2004) derived a general method for improving the variability function χ of a given Bernoulli Space . Subsequently, this method has been applied to the case of uniform distributions by Sans et al. (Economic Quality Control 20: 121-142, 2005) and to the case of monotonic probability distributions by Sans et al. (Economic Quality Control 20: 121-142, 2005). In this paper the method is extended to the important case of the family of uni-modal probability distributions.

In previous papers, a new adaptive sampling method was presented for application in the economic statistical control context. By means of an economic model, an optimal solution can be defined. The results obtained previously indicate that the approach might be more useful than both the fixed-sampling-intervals and variable-sampling-intervals approaches. In this paper, further results together with new simple approximations are presented and analyzed. In the light of previous results, the main conclusion is that the new approach seems to be useful for practical purposes as it is comparatively simple and also allows for some reductions in terms of the total expected costs per cycle.

The reliability of a system is often referred to as a quality index reflecting the capability of the system to complete the specified function within specified time in reasonably satisfactory manner under the specified conditions (including hard conditions: say, operating environments, input materials and soft conditions: say, system management and operator team, and computerized monitoring atomization subsystems etc.) The uncertainty or vagueness of system reliability is an intrinsic and inherent feature of any system and its operating environment. Vagueness has been described by fuzzy mathematics and, therefore, in this paper we propose a credibility hazard concept associated with fuzzy lifetimes. The proposed approach is based on the conditional distribution of a fuzzy variable under Liu's non-classical credibility measure theory.

This work presents a Bayesian approach for estimating the reliability of a parallel multi-component system. It is assumed that the strengths of the components are independent random variables, which are subjected to a common stress with the same distribution. It is supposed that the failure times follow exponential and Weibull distributions, respectively. The Bayesian analysis is developed assuming a highly informative prior and a less informative prior distribution, respectively. A simulation based on certain data sets is used to study the performance of the Bayesian solutions. The solutions are computed by Markov Chain Monte Carlo (MCMC) methods. Finally, some observations are made in relation to the maximum likelihood method and some extensions are discussed.

Statistical process control procedures are usually implemented under the assumption that the observations from a process are independent over time. However, this assumption is often violated. Therefore, we propose a single Shewhart-type control chart for autocorrelated process by fitting a time series model into the process and monitoring the residuals from the forecast values of a fitted time series model. Numerical results illustrate the ARL of the AR (1) plus random error model, for the cases of step changes in the mean and/or standard deviation. Compared to other charts that monitor autocorrelated processes, this chart quickly detects shifts in the process location and spread particularly for large shifts.

Some classes of lifetime distributions based on the failure rate are considered, namely new better than used failure rate (NBUFR), new better than average renewal failure rate (NBARFR) and generalized new better than average renewal failure rate (GNBARFR). Moment inequalities and moment generating functions of these classes under mixture and convolution are studied. In addition, the relationship between the scaled TTT-transform and the NBARFR class is investigated.

This paper introduces modified producer's risk and consumer's risk in the case of chain sampling plans obtained by modelling the defective probability p not as a constant, but as a random variable that follows a beta distribution.

Imprecise measurements may seriously affect the efficiency of process control and production cost. For two dependent processes, the in-coming quality variable in the first process may influence the out-going quality variable in the second process. To effectively monitor and distinguish the state of the two dependent process steps with imprecise measurements, the EWMA control chart and cause-selecting control chart are constructed by taking account of the measurement errors. The EWMA control chart for imprecise measurements is constructed to monitor small shifts in the process mean on the first step, and the cause-selecting control chart for imprecise measurements, is constructed to monitor small shifts in the process mean in the second step. The effects of imprecise measurements on the performance of the two proposed control charts are examined for the case, where the mean of each process may be changed by the occurrence of a special cause. The performance of the proposed control charts is measured by the average run length. A sensitivity analysis indicates that a larger variation of imprecise measurements leads to a larger average run length for the out-of-control process and a larger average loss.

In this paper the reliability analysis of two new models generalizing the m -consecutive- k -out-of- n : F system is carried out using GERT: Model I: m -consecutive- k -out-of- n : F system with ( k -1)-step Markov dependence, and Model II: m -consecutive- k -out-of- n : F system with Block- k dependence. For both the models, the system consists of n linearly ordered components. In Model I, the system fails, if and only if there are at least m non-overlapping runs of k consecutive failed components having ( k -1)-step Markov dependence. We call such a system m -consecutive- k -out-of- n : F with ( k -1)-step Markov dependence. Model II represents an m -consecutive- k -out-of- n : F system, in which each subsequent occurrence of a block of k -consecutive failures increases the failure probability of the remaining components. We define such a system as m -consecutive- k -out-of- n : F with Block- k dependence. GERT provides a visual picture of the system and helps to analyze the system in a less inductive manner. Mathematica Software is used for systematic computations. Illustrative numerical examples for reliability evaluation of these systems showing the time efficiency of GERT analysis are also provided.