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Critical values for the significance levels α =0.0100, 0.0500, and 0.1000 and sample sizes from n = 3 to n = 30 for Dixon's outlier test have been calculated using Monte Carlo simulation. For the one-sided and two-sided case critical values are determined, which are accurate to within ±0.00001 to ±0.00007.

Consider a Γ-distributed random variable X and a random variable Y with a probability distribution of exponential, Γ, Weibull, Pareto or log-normal type. In this paper the exact distributions of the products XY for the different cases with respect to Y are derived. For illustration an application is outlined, which involves modelling the claim amount of an insurance company over a given period.

The so-called confidence interval for the first moment (expectation) of a random variable X based on the t -distribution and the confidence interval for the variance based on the χ 2 -distribution are two very important statistical methods. They allow to determine the value μ of E [ X ] and the value σ 2 of V [ X ] for given confidence level β . These two statistical methods fit perfectly well into the stochastic approach and, therefore, they are used here to illustrate stochastic measurement procedures. The first moment E [ X ] and the variance V [ X ] of a random variable X are of utmost importance and, therefore, stochastic β -measurement procedures for them are crucial for developing a stochastic model for a given random variable. In this paper the two statistical methods are used for deriving stochastic procedures.

In this paper, a maintenance strategy is proposed for a multi-state deteriorating system subject to both aging and fatal shocks under continuous inspection. The strategy allows one of the three actions “do nothing”, “maintenance” or “replacement” to be taken at each state of the system. A two-phase maintenance is applied and an optimal state-dependent maintenance/replacement policy for the system is derived. At each state, only a given number of successive minimal repairs can be performed, which is followed by one minimal maintenance. The overall number of allowable minimal maintenances for all states may not exceed a given limit. An iteration algorithm is used for determining the optimal policy, which minimizes the long-run cost per time of the system. Numerical examples are given to illustrate and evaluate the performance of the proposed policy.

This paper introduces two techniques for computing bounds for several quantile-based risk measures based on distortion functions. Knowledge about the marginal distribution of the involved random variables is assumed with the optional assumption of some partial information about the structure of dependence. The aim is to derive bounds for risk measures of functions of dependent random variables. Several examples taken from an insurance context are given. We use Embrechts et al. (2003) methodology and the stochastic ordering approach to derive bounds for various risk measures in the bi-dimensional and the multidimensional cases.

The profile is an important quality characteristic of hot rolled aluminium coil. It is referred to as a measure of pattern of thickness variation along the width of the coil. The aim of this article is to study some aspects of the distribution of profile of hot rolled aluminium coils. A probability distribution of the profile is derived under the normal approximation. The parameters of the distribution are estimated by the maximum likelihood method. Next, the derived probability distribution is compared with the real observations and a goodness of fit test is carried out. It shows that the derived distribution fits reasonably well. Finally, a large sample distribution of the profile is also derived.

Nonparametric control chart techniques represent a rather neglected area in Statistical Process Control, in particular with respect to monitoring process variability. There are only very few relevant papers published. The reason for not regarding nonparametric techniques for controlling process variability are investigated in this paper by means of a control chart based on the two sample rank-sum test of Ansari and Bradley. Using the popular 3 σ -control limits, the in-control state performance ( ARL ) is compared by simulation with that of a Shewhart S -chart for different distributions and different sample sizes. Moreover, the efficiency to detect shifts in the variability is investigated. It is shown that the in-control performance is excellent, however, the performance in the out-of-control state is rather poor.

Particles such as dust have serious negative effects on yield, performance and reliability in the semiconductor manufacturing process. As a result, there is the need to control particles in the manufacturing process, which often is done by means of control charts. The c -chart is a control chart based on the Poisson distribution generally used to monitor the number of defects in statistical process control. However, if the c -chart is used in a semiconductor manufacturing process, the process will often be declared out-of-control although the process is in-control, because the number of particles on a wafer does not follow a Poisson distribution. In this study, we assume that the particle distribution follows a negative binomial distribution, and propose a procedure for the use of a control chart that includes outlier removal. The usefulness of this proposal is then shown by simulation, numerical analysis, and application of the proposed steps to real data.

More and more often, procedures for profile/surface monitoring and inspection refer to geometric specifications. In fact, the number of form tolerance requirements (such as straightness, roundness, flatness, cylindricity, free-form profiles and surfaces) are increasingly appearing in technical drawings. Usually, Coordinate Measuring Machines (CMMs) are used to measure points on the specific feature under study in order to decide whether the machined item is conforming to requirements. In fact, measured points are used to compute the form error which has to be smaller than the tolerance value to avoid scrapping or reworking. The estimated form error is usually affected by uncertainty, which depends on both the sample size (number of points measured on the profile/surface) and the sampling strategy (location of the points on the profile/surface). The present research work focuses on the second issue and investigates the performance of several existing sampling strategies with the ones achieved by two newly presented approaches. These new approaches explore advantages arising by including a statistical analysis of a set of machined features in the sampling strategy design. Throughout the paper, a real case study concerning flatness tolerance is used as reference.

In 1809 Carl Friedrich Gauss' masterpiece “Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium” on the motion of celestial bodies was published. This book contains also a first version of the famous error propagation formula, which is still the basis of modern metrology and of any relevant national and international standard. Gauss' ideas are reflected in the GUM, the “Guide to the Expression of Uncertainty in Measurement”, published in 1993 by ISO, which is a key document for national measurement institutes and industrial calibration laboratories for evaluating uncertainty in the output of a measurement system. Uncertainty constitutes the main problem of any measurement theory. Because uncertainty is investigated in stochastics, any serious measurement theory should necessarily take into account the state of the art in stochastics. Stochastics was initiated by Jakob Bernoulli in his masterpiece “Ars conjectandi”, which appeared posthumously in 1713, i. e., about 100 years before Gauss established the basis of metrology. The difference between Jakob Bernoulli and Carl Friedrich Gauss is the following: Bernoulli aimed at quantifying and measuring uncertainty, while Gauss aimed at quantifying and measuring a measurement error. In this paper a modern measurement theory based on stochastics is briefly outlined. Subsequently, the antiquated concepts and methods as recommended in the GUM, which go back to the early 19th century, are discussed and their weaknesses are listed.