A Powerful and Robust Nonparametric Statistic for Joint Mean-Variance Quality Control

For statistical process control, a number of single charts that jointly monitor both process mean and variability recently have been developed. For quality control-related hypothesis testing, however, there has been little analogous development of joint mean-variance tests: only one two-sample statistic that is not computationally intensive has been designed specifically for the one-sided test of Ho: Mean2 <= Mean1 and Variance2 <= Variance1 vs. Ha: Mean2 > Mean1 OR Variance2 > Variance1 (see Opdyke (2005)). For these joint hypotheses, under many conditions tests of stochastic dominance (e.g., one-sided Kolmogorov-Smirnov) can severely violate the nominal level, and exceedance tests (e.g., Rosenbaum (1954)) and tests of distributional equality (e.g., most permutation tests) can have virtually no power. This paper further develops the maximum test proposed in Opdyke (2005) and demonstrates via thorough simulation that under typical quality control conditions a) it always maintains good level control; b) it has good power under symmetry and modest power under asymmetry; and c) it often has dramatically more power and much better level control than the only widely endorsed competitor. The statistic - OBMax2 - is not computationally intensive, and although initially designed for quality control testing in regulatory telecommunications, its range of application is as broad as the number of quality control settings requiring a one-sided test of the first two moments.