On the ARL of CUSUM in multinomial models

Year 2024, Volume: 53 Issue: 5, 1484 - 1496, 15.10.2024

Shangchen Yao , Mohammad Khan

https://doi.org/10.15672/hujms.1428934

Abstract

There is no known closed form expression for the average sample number, also known as average run length, of a multivariate CUSUM procedure $N = \min\{ M_1, M_2,\cdots, M_m\}$ for $m\geq 3$, where $M_i$ are univariate CUSUM procedures. The problem is generally considered to be hopelessly complicated for any model. In this paper, for the multinomial model we show, however, that there is a rather simple closed form expression for the average run length of $N$ with an elementary proof. A bit surprisingly, we further show that the average run length of $N$ is related to the average run lengths of $M_i$ the same way as the capacitance of a series network of capacitors is related to the capacitances of its own components.

Keywords

Average run length, multivariate CUSUM, Markov chain, sequential change point detection

Thanks

Thank you for considering our article for publication in Hacettepe Journal of Mathematics and Statistics.

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