Tail behaviour of a general family of control charts

Abstract In this paper we consider a general control scheme. The control statistic Z t is equal to an arbitrary weighted sum of the past observations X t ,..., X 1 . This approach covers most of the applied control schemes like for instance moving average, EWMA and ARMA(1,1) charts. The process { X t } 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: Z t − E ( Z t )> c √{Var( Z t )}} with respect to the autocorrelation of { X t }. It is shown under which conditions on the weights and on the autocorrelations of { X t } the correlation between Z t and Z t − 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 { X t }, too. The weight conditions are verified for several well-known charts.