Determining the process efficiency index by nonparametric predictive inference method and investigating it in fuzzy environment

Document Type : Original Article

Abstract

This paper presents a novel statistical approach to process capability index. With this
end in view, we present a nonparametric predictive approach to process capability index.
Attention is restricted to the approach to process capability index for discrete processes.
This approach is a statistical method based on fewer assumptions about probability dis-
tributions, together with inferences based on factual data. The nonparametric predictive
approach assumes exchangeability of random variables, both related to observed data and
future observations, and uncertainty is quantified through using interval probabilities. Fi-
nally, process capability indexes that are introduced in this study, are analyzed under the
fuzzy environment.
This paper presents a novel statistical approach to process capability index. With this
end in view, we present a nonparametric predictive approach to process capability index.
Attention is restricted to the approach to process capability index for discrete processes.
This approach is a statistical method based on fewer assumptions about probability dis-
tributions, together with inferences based on factual data. The nonparametric predictive
approach assumes exchangeability of random variables, both related to observed data and
future observations, and uncertainty is quantified through using interval probabilities. Fi-
nally, process capability indexes that are introduced in this study, are analyzed under the
fuzzy environment.

Keywords


[3] Augustin, T. and Coolen, F.P.A. (2004). Nonparametric predictive inference and
 [4] Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances.
Philosophical Transactions of the Royal Society London, 53, 370-418; 54, 296-
325.
[5] Clements, J.A. (1989). Process capability calculations for non-normal distributions. Quality Progress, 22(9), 95–100.
[6] Coolen, F.P.A. (1998). Low structure imprecise predictive inference for Bayes’
problem. Statistics & Probability Letters, 36, 349-357.
[7] Coolen, F.P.A. (2004). On the use of imprecise probabilities in reliability. Quality
and Reliability Engineering International, 20, 193–202.
[8] Coolen, F.P.A. (2006). On nonparametric predictive inference and objective
Bayesianism. Journal of Logic, Language and Information, 15, 21-47.
[9] Coolen, F.P.A. (2011). Nonparametric Predictive Inference. In International encyclopedia of statistical science, ed. M. Lovric, 968–970. Berlin, Springer.
[10] Coolen, F.P.A., Ahmadini, A. and Coolen-Maturi, T. (2021). Imprecise inference
based on the log-rank test for accelerated life testing. Metrika, 84, 913–925.
[11] Coolen, F.P.A. and Augustin, T. (2009). A nonparametric predictive alternative
to the Imprecise Dirichlet Model: the case of a known number of categories. International Journal of Approximate Reasoning, 50, 217-230.
[12] Coolen, F.P.A. and Bin Himd, S. (2020). Nonparametric predictive inference
bootstrap with application to reproducibility of the two-sample Kolmogorov–
Smirnov test. Journal of Statistical Theory and Practice, 14(2), 1-13.
[13] Coolen, F.P.A. and Elsaeiti, M.A. (2009). Nonparametric Predictive methods for
acceptance sampling. Journal of Statistical Theory and Practice, 3, 907-921.interval probability. Journal of Statistical [14] Coolen, F.P.A. and Marques, F.J. (2020). Nonparametric Predictive Inference for
Test Reproducibility by Sampling Furture Data Orderings. Journal of Statistical
Theoty and Practice, 14(4), 1-22.
[15] Coolen, F.P.A. and Utkin, L.V. (2011). Imprecise reliability. In: International
Encyclopedia of Statistical Science, M. Lovric (Ed.). Springer, 649-650.
[16] Coolen, F.P.A. and Van Der Laan, P. (2001). Imprecise predictive selection based
on low structure assumptions. Journal of Statistical Planning and Inference, 98,
259-277.
[17] Coolen, F.P.A. and Yan, K.J. (2004). Nonparametric predictive inference with
right-censored data. Journal of Statistical Planning and Inference, 126, 25-54.
[18] De Finetti, B. (1974). Theory of Probability, 2 vols. Wiley, London.
[19] Geisser, S. (1993). Predictive Inference: An Introduction. Chapman & Hall, London.
[20] Hill, B.M. (1968). Posterior distribution of percentiles: Bayes’ theorem for sampling from a population. Journal of the American Statistical Association, 63, 677-
691.
[21] Hill, B.M. (1988). De Finetti’s theorem, induction, and A(n) or Bayesian nonparametric predictive inference (with discussion). In J.M. Bernardo, et al. (Eds.),
Bayesian Statistics 3, 211-241. Oxford University Press.
[22] Kane, V.E. (1986). Process capability indices. Journal of Quality Technology,
18(1), 41–52 (Corrigenda, 18(4), 265).
[23] Pearn, W.L. and Kotz, S. (1994–1995). Application of Clements’ method for calculating second- and thirdgeneration process capability indices for non-normal
[24] Pearn, W.L. and Chen, K.S. (1995). Estimating process capability indices for nonnormal pearsonian populations. Quality and Reliability Engineering International,
11, 389–391.
[25] Perakis, M. and Xekalaki, E. (2002). A process capability index that is based on
the proportion of conformance. Journal of Statistical Computation and Simulation, 72(9), 707–718.
[26] Perakis, M. and Xekalaki, E. (2005). A process capability index for discrete processes. Journal of Statistical Computation and Simulation, 75(3), 175–187.
[27] Weichselberger, K. (1995). Axiomatic foundations of the theory of intervalprobability. In: Mammitzsch, V., SchneeweiU, H. (Eds.), Proceedings of the Second GauU Symposion, Section B. De Gruyter, Berlin, 47–64.
[28] Weichselberger, K. (2000). The theory of interval-probability as a unifying concept for uncertainty. International Journal of Approximate, 24, 149–170.
[29] Zadeh, L.A. (1965). Fuzzy sets. Information and Control 8, 338–353.
[30] Zadeh, L.A. (1975). The concept of a linguistic variable and its application to
approximate reasoning. Information Sciences, 3(8), 199-249.
[31] Zimmermann, H.J. (1991). Fuzzy set theory and its applications. Kluwer Academic Publishers, Dordrecht.pearsonian