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On ARL-unbiased c-charts for INAR(1) Poisson counts

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Abstract

Counts of nonconformities are frequently assumed to have a Poisson distribution. The integer and asymmetrical character of this distribution and the value of its target mean may prevent the quality control operator to deal with a chart with a pre-specified in-control average run length (ARL) and the ability to promptly detect both increases and decreases in the mean of those counts. Moreover, as far as we know, the c-chart proposed to monitor the mean of first-order integer-valued autoregressive [INAR(1)] Poisson counts tends to be ARL-biased, in the sense that it takes longer, in average, to detect some shifts in the process mean than to trigger a false alarm. In this paper, we capitalize on the randomization of the emission of a signal and on a nested secant rule search procedure not only to eliminate the bias of the ARL function of the c-chart for the mean of INAR(1) Poisson counts, but also to bring its in-control ARL exactly to a pre-specified and desired value. Striking illustrations of the resulting ARL-unbiased c-chart are provided.

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References

  • Acosta-Mejía CA (1999) Improved p-charts to monitor process quality. IIE Trans 31:509–516

    Google Scholar 

  • Acosta-Mejía CA Jr, Pignatiello JJ (2000) Monitoring process dispersion without subgrouping. J Qual Technol 32:89–102

    Article  Google Scholar 

  • Boucher J-P, Denuit M, Guillén M (2008) Models of insurance claim counts with time dependence based on generalization of Poisson and negative binomial distributions. Variance 2:135–162

    Google Scholar 

  • Brook D, Evans DA (1972) An approach to the probability distribution of cusum run length. Biometrika 59:539–549

    Article  MathSciNet  MATH  Google Scholar 

  • Cheng C-S, Chen P-W (2011) An ARL-unbiased design of time-between-events control charts with runs rules. J Stat Comput Simul 81:857–871

    Article  MathSciNet  MATH  Google Scholar 

  • Du J, Li Y (1991) The integer-valued autoregressive (INAR(p)) model. J Time Ser Anal 12:129–142

    Article  MathSciNet  MATH  Google Scholar 

  • Guo B, Wang BX (2015) The design of the ARL-unbiased \(S^2\) chart when the in-control variance is estimated. Qual Reliab Eng Int 31:501–511

    Article  Google Scholar 

  • Guo B, Wang BX (2016) Control charts for the coefficient of variation. Stat Pap. doi:10.1007/s00362-016-0797-0

  • Guo B, Wang BX, Xie M (2014) ARL-unbiased control charts for the monitoring of exponentially distributed characteristics based on type-II censored samples. J Stat Comput Simul 84:2734–2747

    Article  MathSciNet  Google Scholar 

  • Huang X, Pascual F (2011) ARL-unbiased control charts with alarm and warning lines for monitoring Weibull percentiles using the first-order statistic. J Stat Comput Simul 81:1677–1696

    Article  MathSciNet  MATH  Google Scholar 

  • Huwang L, Huang C-J, Wang Y-HT (2010) New EWMA control charts for monitoring process dispersion. Comput Stat Data Anal 54:2328–2342

    Article  MathSciNet  MATH  Google Scholar 

  • Knoth S (2010) Control charting normal variance—reflections, curiosities, and recommendations. In: Lenz HJ, Wilrich P-Th (eds) Frontiers in statistical quality control. Physica-Verlag, Heidelberg, pp 3–18

    Chapter  Google Scholar 

  • Knoth S, Morais MC (2013) On ARL-unbiased control charts. In: Knoth S, Schmid W, Sparks R (eds) Proceedings of the XIth international workshop on intelligent statistical quality control, 2013)

  • Knoth S, Morais MC (2015) On ARL-unbiased control charts. In: Knoth S, Schmid W (eds) Frontiers in statistical quality control. Springer, Cham, pp 95–117

    Chapter  Google Scholar 

  • Knoth S, Schmid W (2004) Control charts for time series: a review. In: Lenz HJ, Wilrich P-Th (eds) Frontiers in statistical quality control. Physica-Verlag, Heidelberg, pp 210–236

    Chapter  Google Scholar 

  • Knoth S, Morais MC, Pacheco A, Schmid W (2009) Misleading signals in simultaneous residual schemes for the mean and variance of a stationary process. Commun Stat Theory Methods 38:2923–2943

    Article  MathSciNet  MATH  Google Scholar 

  • McKenzie E (1985) Some simple models for discrete variate time series. Water Resour Bull 21:645–650

    Article  Google Scholar 

  • Morais MC (2016a) An ARL-unbiased np-chart. Econ Qual Control 31:11–21

    Article  MathSciNet  MATH  Google Scholar 

  • Morais MC (2016b) An ARL-unbiased geometric chart. Int J Prod Res (Under revision for publication)

  • Morais MC, Pacheco A (2016) On hitting times for Markov time series of counts with applications to quality control. REVSTAT 14:455–479

    MathSciNet  MATH  Google Scholar 

  • Pascual F (2010) EWMA charts for the Weibull shape parameter. J Qual Technol 42:400–416

    Article  Google Scholar 

  • Paulino S, Morais MC, Knoth S (2016) An ARL-unbiased c-chart. Qual Reliab Eng Int 32:2847–2858

    Article  Google Scholar 

  • Pignatiello Jr. JJ, Acosta-Mejía CA, Rao BV (1995) The performance of control charts for monitoring process dispersion. In: 4th industral engineering research conference proceedings, pp 320–328

  • Ramalhoto MF, Morais M (1995) Cartas de controlo para o parâmetro de escala da população Weibull tri-paramétrica. (control charts for the scale parameter of the Weibull distribution.). In: Actas do II Congresso Anual da Sociedade Portuguesa de Estatística, pp 345–371

  • Ramalhoto MF, Morais M (1999) Shewhart control charts for the scale parameter of a Weibull control variable with fixed and variable sampling intervals. J Appl Stat 26:129–160

    Article  MATH  Google Scholar 

  • Silva I (2005) Contributions to the Analysis of Discrete-valued Time Series. Ph.D. thesis, Faculdade de Ciências da Universidade do Porto, Portugal

  • Steutel FW, Harn KV (1979) Discrete analogues of self-decomposability and stability. Ann Probab 7:893–899

    Article  MathSciNet  MATH  Google Scholar 

  • Uhlmann W (1982) Statistische Qualitätskontrolle (2. Aufl.). Teubner

  • Vasilopoulos AV, Stamboulis AP (1978) Modification of control chart limits in the presence of data correlation. J Qual Technol 10:20–30

    Article  Google Scholar 

  • Vazifedan S, Shitan M (2012) Modeling polio data using the first order non-negative integer-valued autoregressive, INAR(1), model. Int J Mod Phys 9:232–239

    Google Scholar 

  • Weiß CH (2007) Controlling correlated processes of Poisson counts. Qual Reliab Eng Int 23:741–754

    Article  Google Scholar 

  • Weiß CH (2009a) Monitoring correlated processes with binomial marginals. J Appl Stat 36:399–414

    Article  MathSciNet  MATH  Google Scholar 

  • Weiß CH (2009b) Controlling jumps in correlated processes of Poisson counts. Appl Stoch Models Bus Ind 25:551–564

    Article  MathSciNet  MATH  Google Scholar 

  • Weiß CH (2009c) EWMA monitoring of correlated processes of Poisson counts. Qual Technol Quant Manag 6:137–153

    Article  MathSciNet  Google Scholar 

  • Weiß CH (2009d) Categorical time series analysis and applications in statistical quality control. Ph.D. thesis, Fakultät für Mathematik und Informatik der Universität Würzburg. dissertation.de—Verlag im Internet GmbH

  • Weiß CH, Kim H (2013) Parameter estimation for binomial AR(1) models with applications in finance and industry. Stat Pap 54:563–590

    Article  MathSciNet  MATH  Google Scholar 

  • Weiß CH, Testik MC (2009) CUSUM monitoring of first-order integer-valued autoregressive processes of Poisson counts. J Qual Technol 41:389–400

    Article  Google Scholar 

  • Weiß CH, Testik MC (2011) The Poisson INAR(1) CUSUM chart under overdispersion and estimation error. IIE Trans 43:805–818

    Article  Google Scholar 

  • Wetherill GB, Brown DW (1991) Statistical process control: theory and practice. Chapman and Hall, London

    Book  MATH  Google Scholar 

  • Yang S, Arnold BC (2016) Monitoring process variance using an ARL-unbiased EWMA-p control chart. Qual Reliab Eng Int 32:1227–1235

    Article  Google Scholar 

  • Yontay P, Weiß CH, Testik MC, Bayindir ZP (2013) A two-sided cumulative sum chart for first-order integer-valued autoregressive processes of Poisson counts. Qual Reliab Eng Int 29:33–42

    Article  Google Scholar 

  • Zeger SL (1988) A regression model for time series of counts. Biometrika 75:621–629

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang CW, Xie M, Goh TN (2006) Design of exponential control charts using a sequential sampling scheme. IIE Trans 38:1105–1116

    Article  Google Scholar 

  • Zhang CW, Xie M, Jin T (2012) An improved self-starting cumulative count of conforming chart for monitoring high-quality processes under group inspection. Int J Prod Res 50:7026–7043

    Article  Google Scholar 

  • Zhang L, Govindaraju K, Bebbington M, Lai CD (2004) On the statistical design of geometric control charts. Qual Technol Quant Manag 2:233–243

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially supported by FCT (Fundação para a Ciência e a Tecnologia) through projects UID/Multi/04621/2013, PEst-OE/MAT/UI0822/2014 and PEst-OE/MAT/UI4080/2014. We are grateful to the Coordinating Editor and the Referees for all the valuable suggestions, which led to an improved version of the original draft of this paper.

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Authors and Affiliations

  1. Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Sofia Paulino

  2. Department of Mathematics and CEMAT (Center for Computational and Stochastic Mathematics), Instituto Superior Técnico, University de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Manuel Cabral Morais

  3. Department of Economics and Social Sciences, Institute of Mathematics and Statistics, Helmut Schmidt University, Postfach 700822, 22008, Hamburg, Germany

    Sven Knoth

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  1. Sofia Paulino
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  2. Manuel Cabral Morais
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  3. Sven Knoth
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Correspondence to Manuel Cabral Morais.

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Paulino, S., Morais, M.C. & Knoth, S. On ARL-unbiased c-charts for INAR(1) Poisson counts. Stat Papers 60, 1021–1038 (2019). https://doi.org/10.1007/s00362-016-0861-9

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  • DOI: https://doi.org/10.1007/s00362-016-0861-9

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