Distributional and Inferential Properties of the Process Loss Indices

Johnson (1992) developed the process loss index Le, which is defined as the ratio of the expected quadratic loss to the square of half specification width. Tsui (1997) expressed the index LeasLe=Lpe+Lot, which provides an uncontaminated separation between information concerning the potential relative expected loss (Lpe) and the relative off-target squared (Lot), as the ratio of the process variance and the square of the half specification width, and the square of the ratio of the deviation of mean from the target and the half specification width, respectively. In this paper, we consider these three loss function indices, and investigate the statistical properties of their natural estimators. For the three indices, we obtain their UMVUEs and MLEs, and compare the reliability of the two estimators based on the relative mean squared errors. In addition, we construct 90%, 95%, and 99% upper confidence limits, and the maximum values of L^e for which the process is capable, 90%, 95%, and 99% of the time. The results obtained in this paper are useful to the practitioners in choosing good estimators and making reliable decisions on judging process capability.