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Univariate Lévy processes have become quite common in the reliability literature for modeling accumulative deterioration. In case of correlated deterioration indicators, several possibilities have been suggested for modeling their dependence. The point of this article is the study of three different dependence models: use of a regular copula, superposition of independent univariate Lévy processes and use of a Lévy copula. The three methods are first presented and analyzed. In this way, it is shown that the multivariate process constructed through an ordinary copula cannot have independent increments in general, that is, it is not a Lévy process. The impact of a wrong choice for the model is next explored, based on data simulated from one model and next adjusted to all three models. It is shown that a wrong model can lead to either overestimate or underestimate the reliability function, which could be problematic in an application context. [ABSTRACT FROM AUTHOR]

The well-known cumulative sum (CUSUM) sequential rule for abrupt model change detection in stochastic dynamic systems relies on the knowledge of the probability density functions of the system output variables conditional on their past values and on the system functioning mode at each time step. This paper shows how to build an asymptotically optimal detection rule under the common average run length (ARL) constraint when these densities are not available but can be consistently estimated. This is the case for nonlinear state--space systems observed through output variables: for such systems, a new class of particle filters based on convolution kernels allows to get consistent estimates of the conditional densities, leading to an optimal CUSUM-like filter detection rule (FDR). Index Terms--Average run length (ARL) constraint, convolution kernel filter, cumulative sum (CUSUM) rule, model change detection, particle filter, state--space systems.

Extended gamma processes have been seen as a flexible extension of standard gamma processes in the recent reliability literature, for the purpose of cumulative deterioration modeling. The probabilistic properties of the standard gamma process have been well explored since the 1970s, whereas those of its extension remain largely unexplored. In particular, stochastic comparisons between degradation levels modeled by standard gamma processes and ageing properties for the corresponding level-crossing times are now well understood. The aim of this paper is to explore similar properties for extended gamma processes and see which ones can be broadened to this new context. As a by-product, new stochastic comparisons for convolutions of gamma random variables are also obtained. [ABSTRACT FROM AUTHOR]

In change detection problem, the distribution of a series of observations can change at some unknown instant. The aim of on-line change detection rule is to detect this change, as rapidly as possible, while ensuring a low rate of false alarm. The most popular rule to treat this problem is the Page's CUSUM rule. The use of this rule supposes that the two distributions, before and after the change, are known, which is often restrictive in practice. In this article, a nonparametric rule is proposed. Only two learning samples, characterizing the in-control and the out-of-control functioning modes of the system, are needed to implement the rule. The new detection approach is based on the use of a well-known nonparametric method, Empirical Likelihood. Some numerical studies show the relevance of our approach, especially when the size of the learning samples are quite small. [ABSTRACT FROM AUTHOR]

The on-line monitoring of water quality is of crucial interest. Control charts are well suited to perform this monitoring. However, these statistical methods need to be adapted to the particularity of the environmental data studies. In this paper, new control charts are developed to treat the case of a French river for which the parameter of interest, the Dissolved Oxygen Concentration (DOC), is characterized by a non-stationary and seasonal time evolution. The principle is to construct a test statistic, not directly based on the variable of interest, but rather on its seasonal regularity when the system is under control. The methods are studied through numerical simulation and is applied on real data. The ability of the control chart to be used as a classifier in a retrospective way is also studied. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In recent years, fault detection has become a crucial issue in semiconductor manufacturing. Indeed, it is necessary to constantly improve equipment productivity. Rapid detection of abnormal behavior is one of the primary objectives. Statistical methods such as control charts are the most widely used approaches for fault detection. Due to the number of variables and the possible correlations between them, these control charts need to be multivariate. Among them, the most popular is probably the Hotelling T^2 rule. However, this rule only makes sense when the variables are Gaussian, which is rarely true in practice. A possible alternative is to use nonparametric control charts, such as the k-nearest neighbor detection rule by He and Wang, in 2007, only constructed from the learning sample and without assumption on the variables distribution. This approach consists in evaluating the distance of an observation to its nearest neighbors in the learning sample constituted of observations under control. A fault is declared if this distance is too large. In this paper, a new adaptive Mahalanobis distance, which takes into account the local structure of dependence of the variables, is proposed. Simulation trials are performed to study the benefit of the new distance against the Euclidean distance. The method is applied on the photolithography step of the manufacture of an integrated circuit. [ABSTRACT FROM AUTHOR]