Hypothesis testing of process capability index C_pk from the perspective of generalized fiducial inference

Hypothesis testing of C_pk is an essential part for decision making on process capability. The difficulty to test the hypothesis of C_pk is the complexity of the natural estimator C_pk distribution even under the normal distribution. Thus, traditional methods of the hypothesis testing of C_pk based on the natural estimator C_pk only give the approximate test method under the normal distribution and seldom discuss the hypothesis testing under some commonly used but complex non-normal distributions. The emerging generalized fiducial inference (GFI) in recent years is an effective method for statistical inference on complex statistics. Thus, we first propose a novel hypothesis testing method for C_pk based on generalized p-value. For application, the mathematic expression of the proposed method for the commonly used normal distribution, Gamma distribution, Weibull distribution, and two-parameter exponential distribution is derived in detail. Next, to study the performance of the proposed method in terms of the frequency property, the real probabilities of Type I error and Type II error are calculated through simulation. The calculated results show that the proposed method has satisfactory performance for different distributions. Finally, the implementation of the proposed method is illustrated by two real examples.