Constructing BCa bootstrap confidence interval for the difference between two non-normal process capability indices CNpmk

Lee-Ing Tong*, Hsi Tien Chen, Yu Fang Tai

*Corresponding author for this work

Research output: Contribution to journalArticle

6 Scopus citations


Process capability index is a highly effective means of assessing product quality and process performance. Among many developed process capability indices, Cp, Cpk, Cpm, and Cpmk are the four most popular indices under normally distributed processes. Engineers always emphasize applicability and accuracy when a capability index is used to measure how a process performs. However, using these traditional indices to evaluate a non-normally distributed process often leads to inaccurate results. Thus, CNp, CNpk, CNpm, and CNpmk were proposed to overcome this shortcoming under non-normally distributed processes. Pearn and Kotz (1994) compared the index CNpmk to CNp, CNpk, and CNpm as well as found CNpmk is more restrictive and sensitive with regard to process median deviation from the target value than the other indices. Thus, this study employed an appropriate index CNpmk to evaluate non-normally and normally distributed processes. However, the exact probability distribution of CNpmk is too complicated to be derived. Consequently, the related hypotheses testing and confidence interval cannot be developed. For this reason, the applicability of CNpmk is limited. The main purpose of this study is to utilize bootstrap simulation method to construct a 100(1 - 2α)% BCa confidence interval for the difference between two indices, CNpmk1 - CNpmk2. The proposed bootstrap interval can be effectively employed to determine which one of the two processes or suppliers has a better process capability. Moreover, engineers without much statistics background can also easily adopt the proposed index and related procedures to compare processes or select suppliers. If this research procedure performs effectively, the industries can use it to analyze the capabilities of any process distributions in the future.

Original languageEnglish
Pages (from-to)209-220
Number of pages12
JournalQuality Engineering
Issue number2
StatePublished - 1 Apr 2008


  • Bootstrap simulation method
  • Confidence interval
  • Non-normal distribution
  • Process capability index

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