CUSUM with Laplace distributed scores

Zucca Cristina, University of Torino

In many situations that arise in applications it is important to detect an anomalous behaviour of a measured item. Often the anomaly is a mean shift in the process. Mathematically the observations can be described as a sequence $\{X_n\}_{n \ge 1}$ of independent random variables that changes its distribution at an unknown time $m$ \begin{equation} X_n \sim \left\{ \begin{array}{lr} f(x),\qquad \text{if } 1\le n < m \\ g(x),\qquad \text{if } n \ge m \end{array} \right. \end{equation} Here we consider a particular situation when the pre-change density is Laplace distributed with mean $\mu$, $X_n\sim Laplace(\mu, \sigma)$ while, after time $m$, it corresponds to a skewed Laplace distribution with density \begin{equation} g(x)=f(x)e^{\theta x-b(\theta)} \end{equation} with $b(\theta)=\mu \theta -\log(1-\sigma^2\theta^2)$ and $\theta$ positive parameter such that $0<\theta< 1/\sigma$. One of the classical methods to deal with this detection problem is the cumulative sum control chart (CUSUM) \cite{Page}. Hence, we study the loglikelihood ratio process (LLR) of the $n$-th observation and we build an exact CUSUM test. For this aim it is necessary the knowledge of the distribution of the position of the LLR and of the first exit time of the LLR from the domain $[0, h]$. After recognizing the LLR as a Lindley process \begin{eqnarray} \label{Lindley} W_n&=&\max(0,W_{n-1}+Z_{n})\\ W_0&=&x>0.\nonumber \end{eqnarray} with Laplace distributed space increments $\{Z_n\}_{n=1}^{\infty}$, we obtain closed form recursive expressions for such quantities. We illustrate the results in terms of the parameters of the process \cite{LSZ}. Joint work with Emanuele Lucrezia (University of Torino) and Laura Sacerdote (University of Torino) \bigskip \begin{thebibliography}{99} \bibitem{Page}E.S. Page, Continuous inspection schemes, Biometrika, \textbf{41}, 1/2 (1954) 100 -- 115. \bibitem{LSZ} E. Lucrezia, L. Sacerdote, C. Zucca, Some exact results on Lindley process with Laplace jumps. (2023) \em{submitted}. \end{thebibliography}

Area: CS51 - Optimality: theoretical and applied results (Cristina Zucca)

Keywords: CUSUM, Anomalies detection, Lindley process

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