Evaluation of the behavior of robust process capability estimators based on their bootstrap confidence intervals for Gamma distribution

Subject Areas : Statistical Quality Control

hosein ebrahimi ¹ ( Department of Industrial Engineering, Yazd University )
Davood Shishebori ² ( Department of Industrial Engineering, Yazd University, Yazd, Iran )

Received: 2022-02-24 Accepted : 2024-09-15 Published : 2024-09-05

Keywords: Robust estimators, Gamma distribution, process capability indices, bootstrap confidence intervals.,

Abstract :

Today, producing a product with high quality, according to the customer needs requires a clear strategy of the manufacturers in the market. To produce a good product, measures are taken to measure and control products at all production levels among, which the analysis of process capability indices is of great importance in the industry. In this context, the usability indicators can be effective when the data follow a normal distribution. On the other hand, if the data aren't standard normal, evaluation of the process's capability based on these indices will typically be confronted with the problem. In this paper, after investigating the behavior and characteristics of the median absolute deviation (MAD) and interquartile range (IQR) and (Q_n), their analysis is conducted for the Gamma distribution. Then, the bias errors and standard errors are obtained using the jackknife method. Three estimators are evaluated in three different modes according to the bootstrap methods and based on their confidence intervals. Finally, by analyzing the results of this research, the reliability, and performance of the estimators are evaluated in different states.

References:

[1] Shishebori, D., & Hamadani, A. Z. (2010). Properties of multivariate process capability in the presence of gauge measurement errors and dependency measure of process variables. Journal of manufacturing systems, 29(1), 10-18. DOI: https://doi.org/10.1016/j.jmsy.2010.06.005
[2] Besseris, G.J. (2019), Evaluation of robust scale estimators for modified Weibull process capability indices and their bootstrap confidence intervals. Computers & Industrial Engineering, 128: p. 135-149. Doi:/10.1016/j.cie.2018.12.037. DOI: https://doi.org/10.1016/j.cie.2018.12.037
[3] Piao, C., & Zhi-Sheng, Y. (2018). A systematic look at the Gamma process capability indices. European Journal of Operational Research, 265(2), 589-597. DOI: https://doi.org/10.1016/j.ejor.2017.08.024
[4] Shishebori, D., Akhgari, M. J., Noorossana, R., & Khaleghi, G. H. (2015). An efficient integrated approach to reduce scraps of industrial manufacturing processes: A case study from gauge measurement tool production firm. The International Journal of Advanced Manufacturing Technology, 76, 831-855. DOI: https://doi.org/10.1007/s00170-014-6282-8
[5] Balamurali, S. (2012). Bootstrap confidence limits for the process capability index C_pmk. International Journal of Quality Engineering and Technology, 3(1), 79-90. DOI: https://doi.org/10.1504/IJQET.2012.046156
[6] Chou, Y.-M., et al. (1990). Lower confidence limits on process capability indices. Journal of Quality Technology, 22(3), 223-229. DOI: https://doi.org/10.1080/00224065.1990.11979242
[7] Ouyang, L., Dey, S., & Park, C. (2024). Development of robust confidence intervals for the cost-based process capability index. Computers & Industrial Engineering, 190, 110048. DOI: https://doi.org/10.1016/j.cie.2024.110048
[8] Alotaibi, R., Dey, S., & Saha, M. (2022). Estimation and Confidence Intervals of a New PCI 〖CN〗_pmc for Logistic‐Exponential Process Distribution. Journal of Mathematics, 3135264. DOI: https://doi.org/10.1155/2022/3135264
[9] Shishebori, D., & Zeinal Hamadani, A. (2010). The effect of gauge measurement capability and dependency measure of process variables on the MCp. Journal of Industrial and Systems Engineering, 4(1), 59-76. DOI: https://doi.org/ 20.1001.1.17358272.2010.4.1.5.2
[10] Ouyang, L., Dey, S., & Park, C. (2024). Development of robust confidence intervals for the cost-based process capability index. Computers & Industrial Engineering, 190, 110048. DOI: https://doi.org/10.1016/j.cie.2024.110048
[11] Kashif, M., Ullah, S., Aslam, M., & Iqbal, M. Z. (2023). Robust process capability indices C_pm and C_pmk using Weibull process. Scientific Reports, 13(1), 16977. DOI: https://doi.org/10.1038/s41598-023-44267-4
[12] Kashif, M., et al. (2017). Evaluation of modified non-normal process capability index and its bootstrap confidence intervals. IEEE Access, 5, 12135-12142. DOI: https://doi.org/10.1109/ACCESS.2017.2713884
[13] Kashif, M., et al. (2017). The efficacy of process capability indices using median absolute deviation and their bootstrap confidence intervals. Arabian Journal for Science and Engineering, 42(11), 4941-4955. DOI: https://doi.org/10.1007/s13369-017-2699-4
[14] Efron, B. (1992). Bootstrap methods: another look at the jackknife, in Breakthroughs in Statistics. Springer, 569-593.
[15] Tibshirani, R. J., & Efron, B. (1993). An introduction to the bootstrap. Monographs on Statistics and Applied Probability, 57, 1-436. DOI: https://doi.org/10.1007/978-1-4612-4380-9_41
[16] Jafarian-Namin, S., Shishebori, D., & Goli, A. (2024). Analyzing and predicting the monthly temperature of Tehran using ARIMA model, artificial neural network, and its improved variant. Journal of Applied Research on Industrial Engineering, 11(1), 76-92. DOI: https://doi.org/10.22105/jarie.2023.356297.1502
[17] Shishebori, D., & Zeinal Hamadani, A. (2009). The effect of gauge measurement capability on MCp and its statistical properties. International Journal of Quality & Reliability Management, 26(6), 564-582. DOI: https://doi.org/10.1108/02656710910966138
[18] Wang, S., Chiang, J. Y., Tsai, T. R., & Qin, Y. (2021). Robust process capability indices and statistical inference based on model selection. Computers & Industrial Engineering, 156, 107265. DOI: https://doi.org/10.1016/j.cie.2021.107265
[19] Lee, A.H., Wu, C.W., Liu, S.W., & Liu, C.H. (2021). Designing acceptance sampling plans based on the lifetime performance index under gamma distribution. The International Journal of Advanced Manufacturing Technology, 1-14. https://doi.org/10.1007/s00170-021-07299-6
[20] Piao, C., & Zhi-Sheng, Y. (2018). A systematic look at the gamma process capability indices. European Journal of Operational Research, 265(2), 589-597. DOI: https://doi.org/10.1016/j.ejor.2017.08.024
[21] Mehri, S., Ahmadi, M. M., Shahriari, H., & Aghaie, A. (2021). Robust process capability indices for multiple linear profiles. Quality and Reliability Engineering International, 37(8), 3568-3579. DOI: https://doi.org/10.1002/qre.2934
[22] Yalcin, S., & Kaya, I. (2022). Design and analysis of process capability indices c_pm and c_pmk by neutrosophic sets. Iranian Journal of Fuzzy Systems, 19(1), 13-30. DOI: https://doi.org/10.22111/ijfs.2022.6548
[23] Park, H.-I. (2017). A New Definition on the Process Capability Index Based on Quantiles. Applied Mathematical Sciences, 11, 173-183. DOI: https://doi.org/10.12988/ams.2017.610266
[24] Haque, M.E., & Khan, J.A. (2012). Globally robust confidence intervals for location. Dhaka University Journal of Science, 60(1), 109-113. DOI:10.3329/dujs.v60i1.10347
[25] Pham-Gia, T., & Hung, T. (2001). The mean and median absolute deviations. Mathematical and Computer Modelling, 34(7-8), 921-936. DOI: https://doi.org/10.1016/S0895-7177(01)00109-1
[26] Niknam, A. R. R., Sabaghzadeh, M., Barzkar, A., & Shishebori, D. (2024). Comparing ARIMA and various deep learning models for long-term water quality index forecasting in Dez River, Iran. Environmental Science and Pollution Research, 1-17. DOI: https://doi.org/10.1007/s11356-024-32228-x
[27] Bidabadi, H. S., Shishebori, D., & Yazdi, A. A. (2021). Multivariate incapability index for high technology manufacturing processes in presence of the measurement errors: A case study in electronic industry. Journal of Industrial Engineering International, 17(1), 14. DOI: https://doi.org/10.30495/jiei.2021.1923885.1101
[28] Besseris, G. (2014). Robust process capability performance: An interpretation of key indices from a non-parametric viewpoint. The TQM Journal, 26(5), 445-462. DOI: https://doi.org/10.1108/TQM-03-2013-0036
[29] Goodarzi, M. R., Niknam, A. R. R., Barzkar, A., & Shishebori, D. (2023). River water flow prediction rate based on machine learning algorithms: A case study of Dez River, Iran. In River, Sediment and Hydrological Extremes: Causes, Impacts and Management (pp. 203-219). Singapore: Springer Nature Singapore. DOI: https://doi.org/10.1007/978-981-99-4811-6_11
[30] Efron, B. (1979). Bootstrap methods: another look at the jackknife. The Annals of Statistics, 7(1), 1–26. URL: http://www.jstor.org/stable/2958830. DOI: https://doi.org/10.1007/978-1-4612-4380-9_41
[31] Efron, B. (1987). Better bootstrap confidence intervals. Journal of the American Statistical Association, 82(397), 171-185. https://doi.org/10.2307/2289144
[32] Efron, B., & Tibshirani, R. (1986). Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science, 54-75. DOI: 10.1214/ss/1177013815