Fuzzy nonparametric estimation of capability index \(\textit{C}_{pk}\)

  • Fereshteh MomeniEmail author
  • Bahram Sadeghpour Gildeh
  • Gholamreza Hesamian
Methodologies and Application


Process capability indices have been widely used in the manufacturing industry to measure the potential performance. This paper proposes a nonparametric approach for estimating the proportion of non-conforming items and capability index \(C_{pk}\), when sample observations and specification limits of a process are reported as imprecise numbers. In this approach, first the \(\alpha \)-pessimistic values of the imprecise observations were first applied to determine an unbiased estimator for population variance and optimal bandwidth. Thereafter, the fuzzy proportion of non-conforming items based on kernel distribution function was obtained. Finally, the fuzzy proportion of non-conforming items was applied to obtain the membership function of fuzzy nonparametric capability index \({\widetilde{C}}_{pk}\) . Moreover, the proposed nonparametric methods are examined to compare with some other existing parametric methods and their performance will be cleared via some numerical examples and some comparison studies.


Process capability index Fuzzy specification limits Optimal bandwidth Nonparametric estimator Kernel distribution function 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statisticss, Behshahr BranchIslamic Azad UniversityBehshahrIran
  2. 2.Department of Statistics, Faculty of Mathematical SciencesFerdowsi University of MashhadMashhadIran
  3. 3.Department of StatisticsUniversity of PayamenoorTehranIran

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