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Gamma processes are commonly used for modelling the accumulative deterioration of systems, in many fields. However, given a series of observations, it is not always easy to affirm that the choice of a gamma process modelling is a good choice. In particular, it would be of great interest to have a statistical test, i.e. a goodness-of-fit test, to answer this question. In this paper, a practical procedure combining three statistical tests is firstly proposed, whose aim is to reject the gamma process modelling as soon as the observations are clearly in contradiction with the basic properties of a homogeneous gamma process, observed with periodic inspections: stationarity, independence and gamma distribution for the increments. The procedure is then extended to non-homogeneous gamma process and aperiodic inspection times. The efficiency of the approach is investigated through numerical simulations, and on real data. [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]

Abstract: The most popular multivariate control chart for monitoring the mean of a distribution is probably the Hotelling T 2 rule. Unfortunately, this rule relies on the assumption that the distribution under control is Gaussian, which is rarely true in practice. The objective of this paper is to propose a new approach for the non-normal multivariate case. It consists in the construction of a tolerance region obtained from a density level set estimation. The method follows a “plug-in” approach in which the density of the observations is previously estimated. This estimation is conducted using copulas modeling, an increasingly popular tool in multivariate modeling. [Copyright &y& Elsevier]

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]

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). [ABSTRACT FROM AUTHOR]

Let X = (X t) t ≥ 0 be a process behaving as a general increasing Lévy process (subordinator) prior to hitting a given unknown level m 0 , then behaving as another different subordinator once this threshold is crossed. This paper addresses the detection of this unknown threshold m 0 ∈ [ 0 , + ∞ ] from an observed trajectory of the process. These kind of model and issue are encountered in many areas such as reliability and quality control in degradation problems. More precisely, we construct, from a sample path and for each ε > 0 , a so-called detection level L ε by considering a CUSUM inspired procedure. Under mild assumptions, this level is such that, while m 0 is infinite (i.e. when no changes occur), its expectation E ∞ (L ε) tends to + ∞ as ε tends to 0, and the expected overshoot E m 0 ( [ L ε − m 0 ] +) , while the threshold m 0 is finite, is negligible compared to E ∞ (L ε) as ε tends to 0. Numerical illustrations are provided when the Lévy processes are gamma processes with different shape parameters. [ABSTRACT FROM AUTHOR]