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Hypothesis testing of Cpk is an essential part for decision making on process capability. The difficulty to test the hypothesis of Cpk is the complexity of the natural estimator Ĉpk distribution even under the normal distribution. Thus, traditional methods of the hypothesis testing of Cpk based on the natural estimator Ĉpk 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 Cpk 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.

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Product quality is an essential issue to promote product sale, and process capability analysis plays an important role to assess and guarantee the process quality of products. Taguchi index is widely adopted to measure the process capability, because it essentially quantifies the quality loss from the actual process quality to the customer need. Large sample approximation and the bootstrap method are often used to test the hypothesis of index , but they do not have satisfactory performance under small or medium sample size. To address this issue, we propose a novel hypothesis testing method of index based on generalized p‐value theory. For application, its mathematical expression is derived in detail for the normal, gamma, and Weibull distributions, since these distributions are commonly used in engineering practice. Simulation results show that the supremum of type I error probability of the proposed testing method is closer to the significance level than the existing methods, and its type II error probability is lower as well. Based on the proposed hypothesis testing method, a new procedure of process capability analysis is designed for applications. Finally, two real cases are carried out to show the effectiveness of the newly designed procedure.