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{M1, M2, · · · , Mm} for m ≥ 3, where Mi 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 ARL of N with an elementary proof. A bit surprisingly, we further show that the ARL of N is related to the ARLs of Mi the same way as the capacitance of a series network of capacitors is related to the capacitances of its own components.
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