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Wolfgang Schmid, Yarema Okhrin

Tail behaviour of a general family of control charts

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, ..., X₁. 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∈mathbb{N}:Z_t−E(Z_t)>csqrt{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.

Statistics & Decisions, Oldenbourg Wissenschaftsverlag

Print ISSN: 0721-2631
Volume: 21, 01/2003
Pages: 079

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