On residual CUSUM statistic for PINAR(1) model in statistical design and diagnostic of control chart

References

• Al-Osh, M. A., and A. A. Alzaid. 1987. First-order integer-valued autoregressive (INAR(1)) process. Journal of Time Series Analysis 8 (3):261–75. doi: 10.1111/j.1467-9892.1987.tb00438.x.
   Google Scholar
• Alzaid, A., and M. Al-Osh. 1988. First-order integer-valued autoregressive (INAR(1)) process: Distributional and regression properties. Statistica Neerlandica 42 (1):53–61. doi: 10.1111/j.1467-9574.1988.tb01521.x.
   Google Scholar
• Bourguignon, M., and K. Vasconcellos. 2015. Improved estimation for poisson INAR(1) models. Journal of Statistical Computation and Simulation 85 (12):2425–41. doi: 10.1080/00949655.2014.930862.
   Web of Science ®Google Scholar
• Bourguignon, M., K. L. Vasconcellos., et al. 2015. First order non-negative integer valued autoregressive processes with power series innovations. Brazilian Journal of Probability and Statistics 29 (1):71–93. doi: 10.1214/13-BJPS229.
   Web of Science ®Google Scholar
• Csörgö, M., and L. Horváth. 1997. Limit theorems in change-point analysis. New York: Wiley.
   Google Scholar
• Du, J.-G., and Y. Li. 1991. The integer-valued autoregressive (INAR(p)) model. Journal of Time Series Analysis 12 (2):129–42.
   Google Scholar
• Franke, J., C. Kirch, and J. T. Kamgaing. 2012. Changepoints in times series of counts. Journal of Time Series Analysis 33 (5):757–70. doi: 10.1111/j.1467-9892.2011.00778.x.
   Web of Science ®Google Scholar
• Franke, J., and T. Seligmann. 1993. Conditional maximum likelihood estimates for INAR(1) processes and their application to modelling epileptic seizure counts. In Developments in time series analysis, ed. T. S. Rao, 310–330. London: Chapman & Hall.
   Google Scholar
• Freeland, R. K., and B. McCabe. 2005. Asymptotic properties of CLS estimators in the poisson AR(1) model. Statistics & Probability Letters 73 (2):147–53. doi: 10.1016/j.spl.2005.03.006.
   Web of Science ®Google Scholar
• Huh, J., H. Kim, and S. Lee. 2017. Monitoring parameter shift with Poisson integer-valued GARCH models. Journal of Statistical Computation and Simulation 87 (9):1754–66. doi: 10.1080/00949655.2017.1284848.
   Web of Science ®Google Scholar
• Hušková, M., Z. Prášková, and J. Steinebach. 2007. On the detection of changes in autoregressive time series I. Asymptotics. Journal of Statistical Planning and Inference 137 (4):1243–59. doi: 10.1016/j.jspi.2006.02.010.
   Web of Science ®Google Scholar
• Jazi, M. A., G. Jones, and C.-D. Lai. 2012. First-order integer valued AR processes with zero inflated poisson innovations. Journal of Time Series Analysis 33 (6):954–63. doi: 10.1111/j.1467-9892.2012.00809.x.
   Web of Science ®Google Scholar
• Kang, J., and S. Lee. 2014. Parameter change test for Poisson autoregressive models. Scandinavian Journal of Statistics 41 (4):1136–52. doi: 10.1111/sjos.12088.
   Web of Science ®Google Scholar
• Kim, H., and S. Lee. 2017. On first-order integer-valued autoregressive process with Katz family innovations. Journal of Statistical Computation and Simulation 87 (3):546–62. doi: 10.1080/00949655.2016.1219356.
   Web of Science ®Google Scholar
• Lee, S., J. Ha, O. Na, and S. Na. 2003. The CUSUM test for parameter change in time series models. Scandinavian Journal of Statistics 30 (4):781–96. doi: 10.1111/1467-9469.00364.
   Web of Science ®Google Scholar
• Lee, S., Y. Lee, and C. W. Chen. 2016. Parameter change test for zero-inflated generalized poisson autoregressive models. Statistics 50 (3):540–57. doi: 10.1080/02331888.2015.1083020.
   Web of Science ®Google Scholar
• Lee, S., Y. Tokutsu, and K. Maekawa. 2004. The CUSUM test for parameter change in regression models with ARCH errors. Journal of the Japan Statistical Society 34 (2):173–88. doi: 10.14490/jjss.34.173.
   Google Scholar
• Li, C., D. Wang, and F. Zhu. 2016. Effective control charts for monitoring the NGINAR(1) process. Quality and Reliability Engineering International 32 (3):877–88. doi: 10.1002/qre.1799.
   Web of Science ®Google Scholar
• McKenzie, E. 1986. Autoregressive moving-average processes with negative-binomial and geometric marginal distributions. Advances in Applied Probability 18 (3):679–705. doi: 10.2307/1427183.
   Web of Science ®Google Scholar
• Meyn, S. P., and R. L. Tweedie. 2012. Markov chains and stochastic stability. 2nd ed. London: Springer.
   Google Scholar
• Montgomery, D. C. 2012. Statistical quality control: A modern introduction. 7th ed. Singapore: Wiley.
   Google Scholar
• Page, E. S. 1954. Continuous inspection schemes. Biometrika 41 (1–2):100–15.
   Web of Science ®Google Scholar
• Pap, G., and T. T. Szabó. 2013. Change detection in INAR(p) processes against various alternative hypotheses. Communications in Statistics - Theory and Methods 42 (7):1386–405. doi: 10.1080/03610926.2012.732181.
   Web of Science ®Google Scholar
• R Core Team 2018. R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.
   Google Scholar
• Rakitzis, A. C., C. H. Weiß, and P. Castagliola. 2017. Control charts for monitoring correlated poisson counts with an excessive number of zeros. Quality and Reliability Engineering International 33 (2):413–30. qre.2017. doi: 10.1002/qre.2017.
   Web of Science ®Google Scholar
• Ristić, M. M., H. S. Bakouch, and A. S. Nastić. 2009. A new geometric first-order integer-valued autoregressive (NGINAR(1)) process. Journal of Statistical Planning and Inference 139 (7):2218–26. doi: 10.1016/j.jspi.2008.10.007.
   Web of Science ®Google Scholar
• Silva, E. D. M., and V. L. Oliveira. 2004. Difference equations for the higher-order moments and cumulants of the INAR(1) model. Journal of Time Series Analysis 25 (3):317–33. doi: 10.1111/j.1467-9892.2004.01685.x.
   Web of Science ®Google Scholar
• Steutel, F. W., and K. van Harn. 1979. Discrete analogues of self-decomposability and stability. The Annals of Probability 7 (5):893–9. doi: 10.1214/aop/1176994950.
   Web of Science ®Google Scholar
• Weiß, C. H. 2007. Controlling correlated processes of poisson counts. Quality and Reliability Engineering International 23 (6):741–54. doi: 10.1002/qre.875.
   Web of Science ®Google Scholar
• Weiß, C. H. 2008. Thinning operations for modeling time series of counts—a survey. AStA Advances in Statistical Analysis 92 (3):319–41. doi: 10.1007/s10182-008-0072-3.
   Web of Science ®Google Scholar
• Weiß, C. H. 2012. Process capability analysis for serially dependent processes of Poisson counts. Journal of Statistical Computation and Simulation 82 (3):383–404. doi: 10.1080/00949655.2010.533178.
   Web of Science ®Google Scholar
• Weiß, C. H. 2015. Spc methods for time-dependent processes of counts—a literature review. Cogent Mathematics 2 (1):1111116.
   Web of Science ®Google Scholar
• Weiß, C. H., and M. C. Testik. 2009. CUSUM monitoring of first-order integer-valued autoregressive processes of Poisson counts. Journal of Quality Technology 41 (4):389–400. doi: 10.1080/00224065.2009.11917793.
   Web of Science ®Google Scholar
• Weiß, C. H., and M. C. Testik. 2015. On the phase I analysis for monitoring time-dependent count processes. IIE Transactions 47 (3):294–306. doi: 10.1080/0740817X.2014.952850.
   Web of Science ®Google Scholar
• Yontay, P., C. H. Weiß, M. C. Testik, and Z. Pelin Bayindir. 2013. A two-sided cumulative sum chart for first-order integer-valued autoregressive processes of Poisson counts. Quality and Reliability Engineering International 29 (1):33–42. doi: 10.1002/qre.1289.
   Web of Science ®Google Scholar
• Zheng, H., I. V. Basawa, and S. Datta. 2007. First-order random coefficient integer-valued autoregressive processes. Journal of Statistical Planning and Inference 137 (1):212–29. doi: 10.1016/j.jspi.2005.12.003.
   Web of Science ®Google Scholar