Control statistics, originally developed by Walter A. Shewhart, are widely used to monitor the quality of processes in various fields, including industry, healthcare, and machine learning. These statistics give an alarm when observed data exceed a threshold, traditionally set as a constant value to maintain a desired false alarm rate. However, one area of research in control statistics remains under-explored: When monitoring a sequence of observations, there may be additional information that potentially affects the law of the observations and should be considered by the design of the monitoring process. Precisely, we would like to change the design by using adapted thresholds, which are functions of the additional information.

So far, we have introduced several classes of adaptive threshold functions for continuous observations, including constant, proportional, and dominated classes. Our focus is on the proportional class, which adjusts sensitivity based on the external information, making it particularly effective in detecting rare but critical cases. We derive an estimator for this threshold function using kernel density estimation and establish its consistency. Further, we prove the asymptotic normality of the estimator using the functional delta method, which allows for deriving confidence intervals and error variances. Finally, we conduct simulations to evaluate the performance of the estimator. This work provides a foundation for future research on adaptive monitoring methods in various applications.

The inherent simplicity of the exponential distribution allows explicit solutions of the average run length (ARL) integral equation for various control charts (here EWMA charts), cf. to some handy methods published in the last 30 years. On the other hand, the omnipresent Markov chain approximation method works feebly (an example will be given, for illustration). Nonetheless, it was used, for example, in prominent papers and elsewhere. Deploying the link between the exponential and the chi-square distribution with two degrees of freedom allows the usage of results (collocation

applied to the ARL integral equation) for EWMA S^2 charts, where an R package is available. Finally, Monte Carlo studies can be utilized. Thus, one goal of the talk is comparing all these numerical algorithms and promoting efficient ones.

Having all these methods, we can judge whether transformations of the exponential proposed in dozen papers, in particular X^1/3.6 for achieving a bell-shape density function, which mimics the normal case, or other ones like e^-X and Phi^-1(e^-X) to get (exact) beta (uniform in the in-control case) and normal distribution, respectively, are appropriate convenience wrappers or simply diminish the detection performance. Eventually, some recommendations regarding the usage of these transformations will be given.

In this talk we are interested in detecting change-points in multivariate nonstationary time series in a nonparametric setting. Firstly, we construct and discuss statistics which depend on the pairwise Wasserstein distance between the empirical marginal distributions of the vectors and from these, we construct CUSUM-statistics to detect change-points. The latter statistics depend on the partial sum process of the Wasserstein statistics and we show that this weakly converges to a Gaussian limit. A Bahadur representation result allows us to consider the asymptotic behavior of the empirical distribution function instead of the quantile function, which characterizes the one dimensional Wasserstein distance. A simulation study shows how well the significance level is retained under the null hypothesis of no change. Lastly, an outlook towards the power of the tests will be given.

In this talk we are monitoring the correlation between two normally distributed random variables through Exponentially Weighted Moving Average (EWMA) control charts. For this control chart, we calculate symmetric control limits around the initial value for a given Average Run Length (ARL), as well as ARL unbiased control limits. We determine these control limits not only through Monte Carlo simulation but also by numerically solving an integral equation for the ARL. In this context, we consider four different numerical methods: Markov chain approximation, Gauss-Legendre Nyström method, and plain and piecewise collocation methods. Afterwards, we examine the out-of-control behavior of the control chart with symmetric control limits compared to those with unbiased control limits for various underlying correlations. The results show that the closer the underlying correlation is to zero or the larger the sample size becomes, the more symmetric the resulting ARL curves are and the less we need the elaborated unbiased ARL control limits design. Lastly, an outlook will be given towards an application of the control chart to data from a bridge project.