A DOUBLE-INTEGRAL EQUATION FOR THE AVERAGE RUN-LENGTH OF A MULTIVARIATE EXPONENTIALLY WEIGHTED MOVING AVERAGE CONTROL CHART

The multivariate exponentially weighted moving average control chart is a control charting scheme that uses weighted averages of previously observed random vectors. This scheme, which is defined using Z(0) = mu(0), Z(i) = rX(i) + (1 - r)(i-1) (i greater than or equal to 1), where X(1), X(2),... denote the vector-valued output of a process, can be used to detect shifts in the process mean vector more quickly, on the average, than the usual Hotelling T-2 chart. We prove that for the special case mu = 0, Sigma = I, the average run length (ARL) depends on the initial value z(0) for the MEWMA statistic only through its magnitude and the angle it makes with the mean vector. This theorem is then used to derive an integral equation of the ARL. This integral equation involves a double integral, and the. unknown function is a function of two variables. ARLs can be obtained by approximating the solution to the integral equation. Previously, simulation was needed to approximate the ARLs.