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

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## Abstract

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.

## Keywords

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

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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