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 ${\widehat{C}}_{pk}$ distribution even under the normal distribution. Thus, traditional methods of the hypothesis testing of $C_{pk}$ based on the natural estimator ${\widehat{C}}_{pk}$ 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.

中文翻译：

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广义基准推断视角下的过程能力指数Cpk假设检验

假设检验 $C_{pķ}$是决策过程能力的重要组成部分。检验假设的难度$C_{pķ}$ 是自然估计量的复杂度 ${\widehat{C}}_{pķ}$分布甚至在正态分布下。因此，传统的假设检验方法$C_{pķ}$ 根据自然估计 ${\widehat{C}}_{pķ}$仅给出正态分布下的近似检验方法，很少讨论一些常用但复杂的非正态分布下的假设检验。近年来出现的广义基准推理（GFI）是对复杂统计数据进行统计推理的有效方法。因此，我们首先提出一种新颖的假设检验方法$C_{pķ}$基于广义p值。对于应用，详细推导了所提出的常用正态分布，伽玛分布，威布尔分布和两参数指数分布的方法的数学表达式。接下来，为了研究所提出的方法在频率特性方面的性能，通过仿真计算了I型误差和II型误差的真实概率。计算结果表明，该方法在不同分布情况下具有令人满意的性能。最后，通过两个真实的例子来说明所提出的方法的实现。