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The Improved Estimation of σ in Quality Control, Revisited

Recently, Derman and Ross (1995, An improved estimator of a in quality control, Probability in the Engineering and Informational Sciences 9: 411–415) derived an estimator of the standard deviation in the standard quality control model and showed that it had smaller mean squared error than the usual estimator. The new estimator was also shown to be consistent even when the underlying distribution deviates from normality, unlike the usual estimator. In this note, the mean squared error is further improved via shrinkage of the Derman-Ross estimator, and a consistent minimum variance unbiased estimator is presented. Finally, by making use of additional subgroup statistics, a minimum variance unbiased estimator is derived and further improved via shrinkage.