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Analytical Expression of Average Run Length for Exponential CUSUM

Y. Areepong

Abstract


The Cumulative SUM chart (CUSUM) is widely used in a great variety of practical applications such as finance and economics, medicine, engineering, psychology, signal processing, and in other areas. The Average Run Length (ARL) is the most common characteristic used to design CUSUM as well as other control charts (e.g. EWMA, Shiryayev-Roberts). The ARL is usually computed via Markov chain, Monte Carlo simulations or numerical integral equations approaches. In not many cases the solution for the ARL can be found in explicit formula. In this paper we use the integral equation method to derive analytical solutions for the ARL when CUSUM is employed. We derive the ARL for CUSUM chart assuming that the random observations are independent and identically distributed (i.i.d.) exponentially distributed. Checking the accuracy of results, we found an excellent agreement between numerical solutions and the closed form expressions. In addition, the performance comparison of CUSUM and EWMA charts are presented for detecting changes in process mean.

Keywords


Cumulative SUM chart, Average Run Length, Integral Equation.

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