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Regressing sample quantiles to perform nonparametric capability analysis

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Abstract

Capability analysis corresponds to a set of methods used to estimate and test the ability of an in-control process to provide a specific output. When there is only one quality characteristic that behaves as a continuous random variable, indices like C p and C pk can be used to measure how well requirements are met. Under normality, variation is indicated using 3−s i g m a limits; otherwise, the corresponding quantiles are used. Distribution fitting and transformations to normality can be used to estimate quantiles by finding an overall fit to the data available. However, by giving the same weight to all observations, the best possible fit of extreme values can be lost. To address this issue, a regression approach is proposed to fit functions over maximum likelihood estimates of probabilities of extreme values. A case study from the automotive industry is used to illustrate the proposed approach. To evaluate the performance, extensive Monte Carlo simulation is used, and the results are compared with the corresponding approach using the Clements method. The proposed nonparametric technique shows smaller MAD when high levels of skewness exist. Practitioners with basic knowledge of regression analysis may find the approach useful to estimate capability indices without the need of a specific probability distribution.

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References

  1. Ahmad S, Abdollahian M, Zeephongsekul P (2007) Process capability analysis for non-normal quality characteristics using gamma distribution. In: Fourth International Conference on Information Technology, 2007. ITNG’07. IEEE, pp 425–430

  2. Bittanti S, Lovera M, Moiraghi L (1998) Application of non-normal process capability indices to semiconductor quality control. IEEE Trans Semicond Manuf 11(2):296–303

    Article  Google Scholar 

  3. Box GE, Cox DR (1964) An analysis of transformations. J R Stat Soc Ser B (Methodological):211–252

  4. Brodin E (2006) On quantile estimation by bootstrap. Comput Stat Data Anal 50(6):1398–1406

    Article  MathSciNet  MATH  Google Scholar 

  5. Chang PL, Lu KH (1994) Pci calculations for any shape of distribution with percentile. Quality World Technical Supplement:110–114

  6. Chen JP (2000) Re-evaluating the process capability indices for non-normal distributions. Int J Protein Res 38(6):1311– 1324

    Article  MATH  Google Scholar 

  7. Clements JA (1989) Process capability calculations for non-normal distributions. Qual Prog 22(9):95–97

    Google Scholar 

  8. Conover W (1999) Practical nonparametric statistics. Wiley, New York

    Google Scholar 

  9. Czarski A (2008) Estimation of process capability indices in case of distribution unlike the normal one. Archives of Materials Science and Engineering 34(1):39–42

    Google Scholar 

  10. De Haan L, Ferreira A (2007) Extreme value theory: an introduction. Springer

  11. Hosseinifard S, Abbasi B, Ahmad S, Abdollahian M (2009) A transformation technique to estimate the process capability index for non-normal processes. Int J Adv Manuf Technol 40(5–6):512–517

    Article  Google Scholar 

  12. Hu J, Su Z (2008) Bootstrap quantile estimation via importance resampling. Comput Stat Data Anal 52 (12):5136– 5142

    Article  MathSciNet  MATH  Google Scholar 

  13. Johnson NL (1949) Systems of frequency curves generated by methods of translation. Biometrika 36(1–2):149–176

    Article  MathSciNet  MATH  Google Scholar 

  14. Kotz S, Lovelace CR (1998) Process capability indices in theory and practice. Hodder Education

  15. Liu PH, Chen FL (2006) Process capability analysis of non-normal process data using the burr xii distribution. Int J Adv Manuf Technol 27(9–10):975–984

    Article  Google Scholar 

  16. Montgomery DC (1985) Statistical quality control. Wiley

  17. Montgomery DC (2007) Introduction to statistical quality control. Wiley

  18. Moon YI, Lall U (1993) A kernel quantile function estimator for flood frequency analysis

  19. Reiss RD, Thomas M (2001) Statistical analysis of extreme values. Springer

  20. Sheather SJ, Marron JS (1990) Kernel quantile estimators. J Am Stat Assoc 85(410):410–416

    Article  MathSciNet  MATH  Google Scholar 

  21. Tang LC, Than SE (1999) Computing process capability indices for non-normal data: a review and comparative study. Qual Reliab Eng Int 15(5):339–353

    Article  Google Scholar 

  22. Vishnyakov BV, Kibzun AI (2007) Application of the bootstrap method for estimation of the quantile function. Autom Remote Control 68(11):1931–1944

    Article  MathSciNet  MATH  Google Scholar 

  23. Zelterman D (1990) Smooth nonparametric estimation of the quantile function. Journal of statistical planning and inference 26(3):339–352

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Industrial Engineering, Texas Tech University, Lubbock, TX, USA

    María I. Salazar-Alvarez & Mario G. Beruvides

  2. Dirección de Vinculación y Extensión de la Rectoria Zona Norte, Tecnológico de Monterrey, Monterrey, México

    Carmen Temblador-Pérez

  3. Department of Mathematics & Statistics, Texas Tech University, Lubbock, TX, USA

    William J. Conover

  4. Escuela de Ingeniería y Ciencias, Tecnológico de Monterrey, Monterrey, México

    Víctor G. Tercero-Gómez

  5. Facultad de Ciencias Físico Matemáticas, Universidad Autónoma de Nuevo León, Monterrey, México

    Alvaro E. Cordero-Franco

Authors
  1. María I. Salazar-Alvarez
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  2. Carmen Temblador-Pérez
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  3. William J. Conover
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  4. Víctor G. Tercero-Gómez
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  5. Alvaro E. Cordero-Franco
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  6. Mario G. Beruvides
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Corresponding author

Correspondence to María I. Salazar-Alvarez.

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Salazar-Alvarez, M.I., Temblador-Pérez, C., Conover, W.J. et al. Regressing sample quantiles to perform nonparametric capability analysis. Int J Adv Manuf Technol 86, 1347–1356 (2016). https://doi.org/10.1007/s00170-015-8285-6

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  • DOI: https://doi.org/10.1007/s00170-015-8285-6

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