ASSESING THE GENERALIZED PROCESS CAPABILITY INDEX Cpyk FOR LOMAX DISTRIBUTION USING DIFFERENT ESTIMATION METHODS AND BOOTSTRAP CONFIDENCE INTERVAL

İklim Gedik Balay

DOI Number
https://doi.org/10.22190/FUMI220407036B
First page
489
Last page
508

Abstract


Process capability index is an important tool to assess the process performance used mostly in industrial areas. Many process capability indices have been proposed in the literature. In this article, we consider different estimation methods to estimate generalized process capability index, Cpyk, introduced by Maiti et al. [32] for the Lomax distribution. Maximum likelihood (ML), least squares (LS), weighted least squares (WLS), Cram´er-von Mises (CVM), Anderson Darling (AD), right-tail Anderson Darling (RAD) and maximum product of spacings (MPS) methods are used during the estimation process. Next, bootstrap confidence intervals, namely, standard bootstrap (SB), percentile bootstrap (PB), and bias-corrected percentile (BCPB) are considered to obtain 95% confidence intervals for the proposed estimators of Cpyk. The performances of proposed estimators are compared via Monte-Carlo simulation study for different parameter settings. Furthermore, we perform a simulation study to compare the coverage probabilities (CP) and average lengths (AL) of bootstrap confidence intervals. Finally, two real data sets are analyzed for an illustrative purposes.

Keywords

Process capability index, bootstrap confidence intervals, Lomax distribution, Estimation, Monte Carlo simulation.

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References


T. W. ANDERSON : Asymptotic theory of certain goodness-of-fit criteria based on stochastic processes. Ann Math Stat. 23 (1952), 193–212.

M. C. BRYSON : Heavy-tailed distributions: properties and tests. Technometrics. 16 (1974), 61–68.

L. L. CHAN, F. SPIRING, and H. XIAO : An OC curve approach for analyzing the process capability index Cpk. Technical Report, Canada, Department of Statistics, University of Manitoba. (1988).

R. C. H. CHENG, and N. A. K. AMIN : Maximum product-of spacings estimation with applications to the log-normal distribution. Math Report, Department of Mathematics, UWIST, Cardiff. (1979).

P. CHEN, B. X. WAN, and Y. E. ZHI-SHENG : Yield-based process capability indices for nonnormal continuous data. J Qual Technol. 51 (2019), 171–180.

I. S. CHOI, and D. S. BAI : Process capability indices for skewed distributions. Proceedings of 20th International Conference on Computer and Industrial Engineering. Kyongju, Korea. (1996), 1211—1214.

B. C. CHOI, and D. B. OWEN : A study of a new process capabily index. Communications in Statistics-Theory and Methods. 19 (2019), 1231–1245.

Y. M. CHOU, D. B. OWEN and S. A. BORREGO : Lower Confidence Limits on Process Capability Indices. Journal of Quality Technology. 22 (1990). 223–229.

J. A. CLEMENTS : Process capability calculations for non-normal distributions. Qual Prog. 22 (1989), 95–97.

G. K. CONSTABLE and J. R. HOBBS : Small samples and nonnormal capability. In: ASQC Quality Congress Transactions, New York, NY. (1992). 37–43.

S. DEY and M. SAHA : Bootstrap confidence intervals of generalized process capability index C pyk using different methods of estimation. Journal of Applied Statistics. 46 (2019). 1843-1869.

S. DEY, M. SAHA, S. S. MAITI and C. H. JUN : Bootstrap confidence intervals of generalized process capability index Cpyk for Lindley and power Lindley distributions. Commun Stat Simul Comput. 47 (2017). 249–262.

B. EFRON : The Jackknife, the bootstrap and other re-sampling plans. Philadelphia, PA: SIAM;1982. (SIAM, CBMS-NSF Monograph; 38).

B. EFRON and R. TIBSHIRANI : Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statist Sci. 1 (1986). 54–75.

L. A. FRANKLIN and W. GARY : Bootstrap confidence interval estimates of Cpk: an introduction. Commun Stat Simul Comput. 20 (1986). 231–242.

I. GEDIK BALAY : Estimation of the generalized process capability index C pyk based on bias-corrected maximum-likelihood estimators for the generalized inverse Lindley distribution and bootstrap confidence intervals. Journal of Statistical Computation and Simulation. 91 (2021). 1960–1979.

D. E. GILES, H. FENG and R. T. GODWIN : On the bias of the maximum likelihood estimator for the two-parameter Lomax distribution. Communications in Statistics-Theory and Methods. 42 (2013). 1934–1950.

B. H. GUNTER : The use and abuse of Cpk. Qual Prog. 22 (1989). 108–109.

A. S. HASSAN and A. S. AL-GHAMDI : Optimum step stress accelerated life testing for Lomax distribution. J. Appl. Sci. Res. 5 (2009). 2153–2164.

T. C. HSIANG and G. A. TAGUCHI : A tutorial on quality control and assurance. Annual Meeting on the American Statistical Association. Las Vegas (NV) (unpublished presentation). 1985.

O. HOLLAND, A. GOLAUP and A. H. AGHVAMI : Traffic characteristics of aggregated module downloads for mobile terminal reconfiguration. IEE Proceedings-Communications. 135 (2006). 683–690.

C. W. HONG, J. W. WU and C. H. CHENG : Implementing lifetime performance index for the pareto lifetime businesses of the service industries. Quality & quantity. 43 (2009). 291–304.

X. HU, and W. GUI : Assessing the lifetime performance index with Lomax distribution based on progressive type I interval censored sample. Journal of Applied Statistics. 47 (2020). 1757-1775.

J. M. JURAN. : Juran’s Quality Control Handbook. 3rd. New York, N.Y.: McGraw-Hill. (1974).

V. E. KANE. : Process capability indices. J Qual Technol. 18 (1986). 41–52.

M. KASHIF, M. ASLAM, A. H. AL-MARSHADI and C. H. JUN : Capability indices for non-normal distribution using Gini’s mean difference as measure of variability. IEEE Access. (2010). 7322–7330.

M. KASHIF, M. ASLAM, G. S. RAO, A. H. AL-MARSHADI and C. H. JUN. : Bootstrap confidence intervals of the modified process capability index for Weibull distribution. J Qual TechnolArabian J. Sci. Eng. (2017). 4565–4573.

H. S. LOMAX. : Business failures: Another example of the analysis of failure data. J. Amer. Statist. Assoc. 49 (1954). 847–852.

V. LEIVA, C. MARCHANT, H. SAULO. and F. ROJAS : Capability indices for Birnbaum-Saunders processes applied to electronic and food industries. J Appl Stat. 19 (2014). 1881–1902.

P. D. M. MACDONALD : Comment on an estimation procedure for mixtures of distributions by Choi and Bulgren. J R Stat Soc Ser B. 33 (1971). 326–329.

M. A. W. MAHMOUD, R. M. EL-SAGHEER, A. A. E. SOLIMAN. and A. H. ABD ELLAH : Inferences of the lifetime performance index with Lomax distribution based on progressive type-II censored data. Economic Quality Control. 29 (2014). 39–51.

S. S. MAITI, S. MAHENDRA, and K. N. ASOK : On generalizing process capability indices. Quality Technology & Quantitative Management. 7 (2010). 279–300.

S. P. MUKHERJEE and N. K. SINGH : Sampling properties of an estimator of a new process capability index for Weibull distributed quality characteristics. Qual Eng. 10 (1997). 291–294.

A. M. NIGM and H. I. HAMDY : Bayesian prediction bounds for the Pareto lifetime model. Commun. Stat. Theory Met. 16 (1987). 1761–1772.

M. JABBARI NOOGHABI : Process capability indices in normal distribution with the presence of outliers. Journal of Applied Statistics. 47 (2020). 2443–2478.

C. PARK, S. DEY, L. OUYANG, J. H. BYUN and M. LEEDS : Improved bootstrap confidence intervals for the process capability index Cpk. Communications in Statistics-Simulation and Computation. 49 (2020). 2583–2603.

W. L. PEARN and K. S. CHEN : One sided capability indices Cpu and Cpl: decision making with sample information. International journal of quality & reliability management. (2002).

W. L. PEARN,and K. S. CHEN : Capability indices for non-normal distributions with an application in electrolytic capacitor manufacturing. Microelectron. Reliab. 37 (1997). 1853–1858.

W. L. PEARN, S. KOTZ and N. L. JOHNSON : Distributional and inferential properties of process capability indices.J Qual Technol. 24 (1992). 216–231.

W. L. PEARN, Y. T. TAI, I. F. HSIAO and Y. P. AO : Approximately unbiased estimator for non-normal process capability index Cnpk. J Test Eval. 42 (2014). 1408—1417.

W. L. PEARN, G. H. LIN, and K. S. CHEN : Distributional and Inferential Properties of the Process Accuracy and Process Precision Indices. Communications in Statistics – Theory and Methods. 27 (1998). 985–1000.

C. PENG : Parametric lower confidence limits of quantile-based process capability indices. J. Qual. Technol. Quant. Manage. (2010a). 199–214.

C. PENG : Estimating and testing quantile-based process capability indices for processes with skewed distributions. J. Data. Sci. (2010b). 253–268.

B. RANNEBY : The maximum spacing method. an estimation method related to the maximum likelihood Method. Scand J Statist. 11 (1984). 93–112.

G. S. RAO, M. ASLAM and R. R. L. KANTAM : Bootstrap confidence intervals of CNpk for inverse Rayleigh and log-logistic distributions. J Stat Comput Simul. 86 (2016). 862–873.

M. SAHA, S. DEY, A. S. YADAV and S. KUMAR : Classical and Bayesian inference of C py for generalized Lindley distributed quality characteristic. Quality And Reliability Engineering International. 25 (2019). 2593–2611.

M. SAHA, S. DEY, and S. S. MAITI : Parametric and nonparametric bootstrap confidence intervals of CNpk for exponential power distribution. J Ind Prod Eng. 35 (2018). 160–169.

J. SWAIN, S. VENKATRAMAN, and J. WILSON : Least squares estimation of distribution function in Johnsons translation system. J Stat Comput Simul. 29 (1988). 271–297.

L. A. FRANKLIN and G. S. WASSERMAN : A Note on the Conservative Nature of the Tables of Lower Confidence Limits for Cpk with a Suggested Correction. Communications in Statistics – Simulation and Computation. 21 (1992). 1165–1169.

S. WEBER, T. RESSURREICAO and C. DUARTE : Yield prediction with a new generalized process capability index applicable to nonnormal data. IEEE Trans Comput Aided Design Integrated Circuits Syst. 35 (2016). 931–942.

A. WONGE : Approximate studentization for Pareto distribution with application to censored data. Stat.Papers. 39 (1998). 189—201.

L. S. ZIMMER and N. F. HUBELE : Quantiles of the sampling

distribution of Cpm. Qual Eng. 10 (1998). 309–329.




DOI: https://doi.org/10.22190/FUMI220407036B

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