### Abstract

Bissell (1990) proposed an estimator Ĉ′_{pk} for the process capability index C_{pk} assuming that P(μ ≥ m) = 0, or 1, where μ is the process mean, and m is the midpoint between the upper and lower specification limits. Pearn and Chen (1996) considered a new estimator C̃_{pk}, which relaxes Bissell's assumption on the process mean. The evaluation of C̃_{pk} only requires the knowledge of P(μ ≥ m) = p, where 0 ≤ p ≤ 1. The new estimator C̃_{pk} is unbiased, and the variance is smaller than that of Bissell's. In this paper, we investigated the asymptotic properties of the estimator C̃_{pk} under general conditions. We derived the limiting distribution of C̃_{pk} for arbitrary population assuming the fourth moment exists. The asymptotic distribution provides some insight into the properties of C̃_{pk} which may not be evident from its original definition.

Original language | English |
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Pages (from-to) | 2489-2497 |

Number of pages | 9 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 26 |

Issue number | 10 |

DOIs | |

State | Published - 1 Jan 1997 |

### Keywords

- Asymptotic distribution
- Process capability index
- Process mean
- Process standard deviation