Tail behavior of a general family of control charts

摘要

In this paper we consider a general control scheme. The control statistic Z{sub}t is equal to an arbitrary weighted sum of the past observations K{sub}t,...,X{sub}1. This approach covers most of the applied control schemes like for instance moving average, EWMA and ARMA(1,1) charts. The process {X{sub}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{sub}t - E(Z{sub}t) ＞ c(Var(Z{sub}t)){sup}(1/2)} with respect to the autocorrelation of {X{sub}t}. It is shown under which conditions on the weights and on the autocorrelations of {X{sub}t} the correlation between Z{sub}t and Z{sub}(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{sub}t}, too. The weight conditions are verified for several well-known charts.