在本篇論文中，我們考慮整數值自我迴歸模型 (INAR) 具有散度偏大的對數常態干擾項。Freeland and McCabe (2004) 推導出 INAR(1) 模型具有卜瓦松干擾項的分數函數 (score function)，資訊矩陣 (information matrix) 以及殘差。在本篇論文中，我們延伸他們的結果在 INAR(1) 模型具有對數常態干擾項上，並提出累計和管制圖來監測 INAR(1) 模型具有對數常態干擾項的均值偏移以及變異數偏移。

In this study we consider the integer-valued autoregressive (INAR) model with lognormal innovations which leads to overdispersion property. Freeland and McCabe (2004) derived the score function, the information matrix and new types of residuals for the INAR(1) model with Poisson innovations. In this study, we extend their results to INAR(1) model with lognormal innovations. Cumulative sum (CUSUM) control chart for monitoring the mean and the variance shift of the INAR(1) model with lognormal innovations are proposed.

論文審定書....................i
誌謝.............................ii
摘要............................iii
Abstract..................... iv
1 Introduction...............1
2 Model...................... 2
2.1 The Poisson INAR(1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 The Poisson INAR(1) Model with overdispersion . . . . . . . . . . . . . . . . . . . 5
3 Parameter Estimation........................................ 7
3.1 MM estimation of the Poisson INAR(1) Model with overdispersion . . . . . . . . 7
3.2 ML estimation of the Poisson INAR(1) Model with overdispersion . . . . . . . . 8
3.3 Gauss-Laguerre quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Residuals of the Poisson INAR(1) Model with overdispersion . . . . . . . . . . . 12
3.5 Estimation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Control Chart.................................................... 15
4.1 Observation-based CUSUM Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Residuals-based CUSUM Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Simulation Study............................................................... 16
5.1 Out-of-Control Shift Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 Chart Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.3 Performance of control chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.4 Special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6 Conclusion............................................................................... 19
References.................................................................................... 21
Appendix ......................................................................................22
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

[1] Al-Osh, M.A. and Alzaid, A.A. (1987). First-order integer valued autoregressive
(INAR(1)) process. Journal of Time Series Analysis 8, 261-275.
[2] Concus, P., Cassatt, D., Jaehnig, G. and Melby, E. (1963). Tables For The Evaluation By Gauss-Laguerre Quadrature, Math. Comp. 17, 245-256.
[3] Freeland, R.K. (1998). Statistical Analysis of Discrete Time Series with Applications to the Analysis of Workers Compensation Claims Data. Ph.D. Thesis, The University of British Columbia, Canada.
[4] Freeland, R.K. and McCabe, B.P.M (2004). Analysis of low count time series data by Poisson autoregression. Journal of time series analysis 25, No. 5 .
[5] Jin-Guan, D. and Yuan, L. (1991). The integer-valued autoregressive (INAR(p)) model. Journal of Time Series Analysis 12 , 129-142.
[6] McKenzie, E. (1988). Some ARMA models for dependent sequences of Poisson counts. Advances in Applied Probability 20, 822-835.
[7] McKenzie, E. (2003). Discrete variate time series. In Stochastic Processes: Modelling and Simulation (eds Shanbhag D.N. and Rao C.R.). North Holland: Elsevier, 573-606.
[8] Reid, D.D. (1981). The Poisson lognormal distribution and its use as a model of plankton aggregation. In: Statistical Distributions in Scientific Work, 6, Taillie, C., Patil G.P. and Baldessari B.(Eds.), Reidel Publishing Company, Dordrecht, Holland, 303-316.
[9] Weib, C.H. and Testik, M.C. (2009). CUSUM monitoring of first-order integer-valued autoregressive processes of Poisson counts. Journal of Quality Technology, 41(4), 389–400.
[10] Weib, C.H. and Testik, M.C. (2011). The Poisson INAR(1) CUSUM chart under overdispersion and estimation error. IIE Transactions, 43(11), 805-818.
[11]Weib, C.H. and Testik, M.C. (2012). Detection of abrupt changes in count data time series: cumulative sum derivations for INARCH(1) models. Journal of quality technology 44 , No.3, 249-264.