On Optimality of Q—Charts for Outliers in Statistical Process Control

Abstract

In a sequence of papers, Quesenberry (1991a, 1991b, 1991c) proposed using certain normalized statistics, called Q—statistics, to construct Shewhart type control charts for individual observations from normal, binomial, and Poisson process distributions when the process parameters are unknown in advance of process start up. These are cases of the so-called “short runs” charting problem, i.e., the problem of charting when estimates of process parameters are not available before a production run is begun. In this work it is shown that the tests obtained on these Q-charts on individual points are uniformly most powerful unbiased tests for a single outlier. That is, the probability of detecting a mean shift for any of these distributions on the next point after it occurs is the maximum possible for an unbiased test.