Nonparametric procedures for process control

Abstract

Three nonparametric control chart procedures are developed. The procedures are designed to detect any shift in the median of a sequence of observations from a specified control value. The first two procedures require that groups of g ≥ 1 observations be made sequentially on the output of the process. Then the Wilcoxon signed-ranks of the observations are computed within each group separately. The Wilcoxon signed-rank statistic SRig for the ith group is computed as the sum of the signed ranks of the th g observations in the ith group. One of the two procedures employs a cumulative sum control chart-type stopping rule and it signals indicating a shift in the median of the process at the first n for which Σi=1n (SRig - k) - min0≤m≤n Σi=1m (SRig - k) ≥ h, where k ≥ 0 and h > 0 are parameters of the procedures, and where Σi=10 ≡ 0. The other procedure employs a linear barrier-type stopping rule and it signals at the first n for which Σi=1n (SRig & (-a, a), where a > 0 is a parameter of the procedure. Based on the fact that { Σi=1n (SRig ; n = 1, 2, ...} forms a discrete time Markov chain, a method for determining the exact properties (average run lengths) of the procedures was developed. The third procedure is proposed for situations where single observations, rather than grouped observations, are made on the output of the process. The procedure requires that an integer M > 1 be fixed apriori and the rank of an observation be computed only with respect to the preceding (M-1) observations. The procedure employs the sum of the signed ranks as a test statistic and a cumulative sum control chart-type stopping rule. It was not possible to determine the exact properties of the procedure through a Markov chain approach. All the proposed procedures are simple to apply in practice since they require little effort in computing the ranks of the observations. Their application does not require that the distribution or the variance of the observations be known. Several comparisons of the proposed procedures were made with other parametric control chart procedures. For normal observations and when a small shift in the mean is considered, there is indication that the proposed procedures perform nearly as good as the parametric procedures. For double exponential observations, some of the proposed nonparametric procedures perform better than the parametric procedures when a small shift in the mean is considered.