The Use of the Extended Generalized Lambda Distribution for Controlling the Statistical Process in Individual Measurements
Abstract
In quality control (QC) of the individual pieces tested, the distribution of the intended variable is not normal in many cases. Therefore, the use of conventional statistical methods to design the control charts (X chart) can lead to misleading and inaccurate results, and hence, it can cause problems to users; because in such a situation it would not be expected that the failure in the production line will be detected in a timely manner. Therefore, in this study, the use of Extended Generalized Lambda distribution (XGLD) has been proposed for approximating the statistical distribution of data and finding a quality control chart based on it. For this purpose, the distribution function of data was firstly estimated by the XGLD, and then, based on this distribution, the control limits were calculated. The research data consists of the tensile strength of 149 aluminum plates categorized into 22 classes. For the goodness of fit test, Chi-square test was utilized and to evaluate the effectiveness of the proposed model, the average run length (ARL) method was used. The results showed that the ARL values in the proposed method are lower than that in the conventional method. Since this is true at all points, it can be concluded that the use of the new method can lead to the faster detection of the system’s failure and consequently, the subsequent costs can be reduced.In addition, since the XGLD is a highly flexible distribution and can estimate many conventional (and even unconventional) distributions with high precision, the use of the proposed method instead of using other cases which are based on distributions other than normal, will increase the accuracy of the system’s failure detection. References
W. Albers, W. C. Kallenberg, and S. Nurdiati, Parametric control charts, Journal of Statistical Planning and Inference, vol. 124, no.1, pp. 159–184, 2004.
C. Borror, D. C. Montgomery, and G. C. Runger, Robustness of the EWMA control chart to non-normality, Journal of Quality Technology, vol. 31, no. 3, pp. 309–316, 1999.
E. N. Cruthis, and S. E. Rigdon, Comparing Two Estimates of the Variance to determine the Stability of a Process, Quality Engineering, vol. 5, no. 1, pp. 67–74, 1992.
L. Devroye, Random variate generation in one line of code, Simulation Conference, 1996. Proceedings. Winter, 265–272, 1996.
M. G. Forbes, M. Guay, and J. F. Forbes, Control design for first-order processes: shaping the probability density of the process
state, Journal of process control, vol. 14, no. 4, pp. 399–410, 2004.
B. Fournier, L. Minguet, L. Niamien, and N. Rupin, Les lambda distributions appliquées au SPC, unpublished report, 2003.
M. Freimer, G. Kollia, G. S. Mudholkar, and C. T. Lin, A study of the generalized tukey lambda family, Communications in Statistics-Theory and Methods, vol. 17, no. 10, pp. 3547–3567, 1988.
W. Gilchrist, Statistical modelling with quantile functions, CRC Press, 2000.
D. C. Hoaglin, The small-sample variance of the Pitman location estimators, Journal of the American Statistical Association, vol. 70, no. 352, pp. 880–888, 1975.
B. L. Joiner, and J. R., Rosenblatt, Some properties of the range in samples from Tukey’s symmetric lambda distributions, Journal of the American Statistical Association, vol. 66, no. 334, pp. 394–399, 1971.
S. C. Kao, X and R control charts based on weighted variance with left-right tail-weighted ratio for skewed distributions, Communications in Statistics-Simulation and Computation, vol. 46, no. 4, pp. 2714–2732, 2017.
S. C. Kao, A control chart based on weighted bootstrap with strata, Communications in Statistics-Simulation and Computation, vol. 47, no.2, pp. 556–581, 2018.
Z. A. Karian, and E. J. Dudewicz, Fitting statistical distributions: the generalized lambda distribution and generalized bootstrap methods, CRC press, 2000.
H. Lam, K. O. Bowman,and L. R. Shenton, Remarks on the generalized Tukey’s lambda family of distributions, Chinese Univ. of Hong Kong, Shatin; Union Carbide Corp., Oak Ridge, TN (USA). Nuclear Div.; Georgia Univ., Athens (USA), 1980.
Y. C., Lin, and C. Y. Chou, On the design of variable sample size and sampling intervals X charts under non-normality, International Journal of Production Economics, vol. 96, no. 2, pp. 249–261, 2005.
D. C. Montgomery, Introduction to statistical quality control, John Wiley & Sons (New York), 2012.
D. C. Montgomery, C. L. Jennings, and M. E. Pfund, Managing controlling and improving quality, Philosophy, vol. 31, pp. 2–3, 2010.
M., Nili Ahmadabadi, Y. Farjami, and M. B. Moghadam, Approximating distributions by extended generalized lambda distribution (XGLD), Communications in Statistics-Simulation and Computation, vol. 41, no.1, pp. 1–23, 2012a.
M., Nili Ahmadabadi, Y. Farjami, and M. B. Moghadam, A process control method based on five-parameter generalized lambda distribution, Quality & Quantity, vol. 46, no. 4, pp. 1097–1111, 2012b.
S. Pal, Evaluation of nonnormal process capability indices using generalized lambda distribution, Quality Engineering, vol. 17, no.1, pp. 77–85, 2004.
J. S. Ramberg, and B. W. Schmeiser, An approximate method for generating asymmetric random variables, ommunications of the ACM, vol. 17, no. 2, pp. 78–82, 1974.
J. M. Sarabia, A hierarchy of lorenz curves based on the generalized tukey’s lambda distribution, Econometric Reviews, vol. 16, no. 3, pp. 305–320, 1997.
S. S. Shapiro, and M. B. Wilk, An analysis of variance test for normality (complete samples), Biometrika, vol. 52, no. 3/4, pp. 591–611, 1965.
T. A. Spedding, and P. Rawlings, Non-normality in statistical process control measurements, International Journal of Quality & Reliability Management, vol. 11, no. 6, pp. 27–37, 1994.
A. Tarsitano, Estimation of the generalized lambda distribution parameters for grouped data, Communications in Statistics, Theory and Methods, vol. 34, no. 8, pp. 1689–1709, 2005.
S. Teyarachakul, S. Chand, and J. Tang, Estimating the limits for statistical process control charts: A direct method improving upon the bootstrap, European Journal of Operational Research, vol. 178, no. 2, pp. 472–481, 2007.
J. W. Tukey, The future of data analysis, The Annals of Mathematical Statistics, vol. 33, no. 1, pp. 1–67, 1962.
H. Wang, Bounded dynamic stochastic systems: modelling and control, Springer Science & Business Media, 2012.
L. Wang, and F. Qian Shaping the PDF of the state variable based on piecewise linear control for non-linear stochastic systems, Science China Information Sciences, vol. 59, no. 11, pp. 102–112, 2016.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
