Tail behaviour of a general family of control charts

Abstract

In this paper we consider a general control scheme. The control statistic Zt is equal to an arbitrary weighted sum of the past observations Xt,...,X1. This approach covers most of the applied control schemes like for instance moving average, EWMA and ARMA(1,1) charts. The process {Xt} is assumed to be a stationary Gaussian process. The aim of the work is to analyze the behaviour of the tail probability of the run length N=inf{t∈ℕ:Zt−E(Zt)>c√{Var(Zt)}} with respect to the autocorrelation of {Xt}. It is shown under which conditions on the weights and on the autocorrelations of {Xt} the correlation between Zt and Zt−i is a nondecreasing function in the autocorrelations of the observed process. Using this result it can be proved that the probability of a false alarm is a nondecreasing function of the autocorrelations of {Xt}, too. The weight conditions are verified for several well-known charts.