Abstract
Control charts are commonly used for monitoring the mean of processes. However, there are practical applications in which asymmetric data are the standard. In these scenarios, the use of robust statistics, such as the median, is advantageous over the mean. Based on this, we propose an empirical control chart for monitoring the median of a wide class of distributions, known as the log-symmetric class. Closed-form estimators, which perform better than the maximum likelihood estimator, are considered. Simulation studies are carried out with the following objectives: to evaluate the in-control and the out-control average run length; to evaluate the behavior of the control limits; and to compare the proposed method with a naive method based on the asymptotic distribution of the three estimators. The results indicate that the proposed approach presents better in-control average run length than the naive method and better power of detection for negative shifts in the median. A practical use of the proposed approach is illustrated with a real engineering problem, followed by a goodness of fit based on AIC and BIC, considering the most common asymmetric distributions. We also perform a residual analysis with the chosen distribution to verify its fit. Finally, based on the chosen distribution the proposed method indicates that there is an out-of-control point in phase II, which is not detected by the naive approach. Therefore, showing a gain in using the proposed method.
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Availability of data and material
The dataset used in the application section is available in the follow web page: https://github.com/lucasdofs/Control-Chart-Log-Symmetric-class
Code availability
The code is available in the follow web page: https://github.com/lucasdofs/Control-Chart-Log-Symmetric-class.
References
Balakrishnan N, Saulo H, Bourguignon M, Zhu X (2017) On moment-type estimators for a class of log-symmetric distributions. Comput Stat 32(4):1339–1355
Bilal M, Mohsin M, Aslam M (2021) Weibull-exponential distribution and its application in monitoring industrial process. Mathematical Problems in Engineering 2021
Bourguignon M, Ho LL, Fernandes FH (2020) Control charts for monitoring the median parameter of Birnbaum-Saunders distribution. Qual Reliab Eng Int
Burkhalter R, Lio Y (2021) Bootstrap control charts for the generalized pareto distribution percentiles. Chilean J Stat 12(1):37–52
Davison AC, Hinkley DV (1997) Bootstrap methods and their application, vol 1. Cambridge University Press, Cambridge
Dunn PK, Smyth GK (1996) Randomized quantile residuals. J Comput Graph Stat 5(3):236–244
Efron B, Tibshirani RJ (1994) An introduction to the bootstrap. CRC Press, Cambridge
Gandy A, Kvaløy JT (2013) Guaranteed conditional performance of control charts via bootstrap methods. Scand J Stat 40(4):647–668
Graham MA, Chakraborti S, Mukherjee A (2014) Design and implementation of cusum exceedance control charts for unknown location. Int J Prod Res 52(18):5546–5564
Grimshaw SD, Alt FB (1997) Control charts for quantile function values. J Qual Technol 29(1):1–7
Hodges JL, Lehmann EL (1963) Estimates of location based on rank tests. Ann Math Stat 34:598–611
Huang WH, Wang H, Yeh AB (2016) Control charts for the lognormal mean. Qual Reliab Eng Int 32(4):1407–1416
Janacek G, Meikle S (1997) Control charts based on medians. J R Stat Soc Ser D 46(1):19–31
John B, Subhani S (2020) A modified control chart for monitoring non-normal characteristics. Int J Product Qual Manage 29(3):309–328
Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2. Wiley Online Library
Kanji G, Arif OH (2000) Median rankit control chart by the quantile approach. J Appl Stat 27(6):757–770
Karagöz D (2018) Asymmetric control limits for range chart with simple robust estimator under the non-normal distributed process. Math Sci 12(4):249–262
Khoo M, Atta AMA, Chen C (2009) Proposed X and S control charts for skewed distributions. In: 2009 IEEE international conference on industrial engineering and engineering management, IEEE, pp 389–393
Khoo MB (2005) A control chart based on sample median for the detection of a permanent shift in the process mean. Qual Eng 17(2):243–257
Kozubowski TJ, Podgórski K (2003) Log-laplace distributions. Int Math J 3:467–495
Lahcene B (2015) Control charts for skewed distributions: Johnson’s distributions. Int J Stat Med Res 4(2):217–223
Lio YL, Park C (2008) A bootstrap control chart for Birnbaum-Saunders percentiles. Qual Reliab Eng Int 24(5):585–600
Nadarajah S (2004) Reliability for laplace distributions. Math Prob Eng
Nelson LS (1982) Control chart for medians. J Qual Technol 14(4):226–227
Page ES (1954) Continuous inspection schemes. Biometrika 41(1/2):100–115
Park HI (2009) Median control charts based on bootstrap method. Commun Stat Simul Comput 38(3):558–570
Puig P (2008) A note on the harmonic law: a two-parameter family of distributions for ratios. Stat Prob Lett 78(3):320–326
R Core Team (2018) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, https://www.R-project.org/
Roberts S (1959) Control chart tests based on geometric moving averages. Technometrics 1(3):239–250
Sales LOF, Pinho ALS, Medeiros FMC, Bourguignon M (2021) Control chart for monitoring the mean in symmetric data. Chilean J Stat 12(1):37–52
Shen H, Brown LD, Zhi H (2006) Efficient estimation of log-normal means with application to pharmacokinetic data. Stat Med 25(17):3023–3038
Shewhart WA (1931) Economic control of quality of manufactured product. ASQ Quality Press, Milwaukee
Shimizu K (1988) Point estimation, chap. 2 in lognormal distributions: Theory and applications, el crow and k. shimizu eds
Stasinopoulos DM, Rigby RA et al (2007) Generalized additive models for location scale and shape (gamlss) in r. J Stat Softw 23(7):1–46
Van Buren M, Watt W, Marsalek J (1997) Application of the log-normal and normal distributions to stormwater quality parameters. Water Res 31(1):95–104
Vanegas LH, Paula GA (2016) Log-symmetric distributions: statistical properties and parameter estimation. Brazil J Probab Stat 30(2):196–220
Zhang L, Chen G, Castagliola P (2009) On t and EWMA t charts for monitoring changes in the process mean. Qual Reliab Eng Int 25(8):933–945
Acknowledgements
To the reviewers for valuable contributions to this paper. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) - Finance Code 001 and CNPq-Brazil (421656/2018-2).
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This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brazil (CAPES) - Finance Code 001 and CNPq-Brazil (421656/2018-2).
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Sales, L.O.F., Pinho, A.L.S., Bourguignon, M. et al. Control charts for monitoring the median in non-negative asymmetric data. Stat Methods Appl 31, 1037–1068 (2022). https://doi.org/10.1007/s10260-022-00624-7
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DOI: https://doi.org/10.1007/s10260-022-00624-7