State-space models for count time series with excess zeros.

*(English)*Zbl 07258978Summary: Count time series are frequently encountered in biomedical, epidemiological and public health applications. In principle, such series may exhibit three distinctive features: overdispersion, zero-inflation and temporal correlation. Developing a modelling framework that is sufficiently general to accommodate all three of these characteristics poses a challenge. To address this challenge, we propose a flexible class of dynamic models in the state-space framework. Certain models that have been previously introduced in the literature may be viewed as special cases of this model class. For parameter estimation, we devise a Monte Carlo Expectation-Maximization (MCEM) algorithm, where particle filtering and particle smoothing methods are employed to approximate the high-dimensional integrals in the E-step of the algorithm. To illustrate the proposed methodology, we consider an application based on the evaluation of a participatory ergonomics intervention, which is designed to reduce the incidence of workplace injuries among a group of hospital cleaners. The data consists of aggregated monthly counts of work-related injuries that were reported before and after the intervention.

##### MSC:

62-XX | Statistics |

##### Keywords:

autocorrelation; interrupted time series; intervention analysis; overdispersion; particle methods; state-space models; zero-inflation##### Software:

astsa
Full Text:
DOI

**OpenURL**

##### References:

[1] | Akaike, H (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, 716-23. · Zbl 0314.62039 |

[2] | Andrieu, C, Doucet, A, Holenstein, R (2010) Particle Markov chain Monte Carlo methods. Journal of the Royal Statistical Society Series B, 72, 269-342. · Zbl 1411.65020 |

[3] | Cameron, AC, Trivedi, PK (2013) Regression analysis of count data. 2nd edn. Cambridge University Press. · Zbl 1301.62003 |

[4] | Chan, KS, Ledolter, J (1995) Monte Carlo EM estimation for time series models involving counts. Journal of the American Statistical Association, 90, 242-52. · Zbl 0819.62069 |

[5] | Cox, DR (1981) Statistical analysis of time series: Some recent developments. Scandinavian Journal of Statistics, 8, 93-115. · Zbl 0468.62079 |

[6] | Dalrymple, ML, Hudson, IL, Ford, RPK (2003) Finite mixture, zero-inflated Poisson and hurdle models with application to SIDS. Computational Statistics & Data Analysis, 41, 491-504. · Zbl 1429.62513 |

[7] | Davis, RA, Dunsmuir, WTM, Streett, SB (2003) Observation-driven models for Poisson counts. Biometrika, 90, 777-90. · Zbl 1436.62418 |

[8] | Davis, RA, Wu, R (2009) A negative binomial model for time series of counts. Biometrika, 96, 735-49. · Zbl 1170.62062 |

[9] | Dempster, AP, Laird, NM, Rubin, DB (1977) Maximum likelihood estimation from incomplete data via the EM algorithm. Journal of the Royal Statistical Society Series B, 39, 1-39. · Zbl 0364.62022 |

[10] | Doucet, A, Freitas, ND, Gordon, N (2001) Sequential Monte Carlo methods in practice. New York: Springer. · Zbl 0967.00022 |

[11] | Fokianos, K (2011) Some recent progress in count time series. Statistics, 45, 49-58. · Zbl 1291.62164 |

[12] | Fokianos, K, Rahbek, A, Tjøstheim, D (2009) Poisson autoregression. Journal of the American Statistical Association, 104, 1430-9. · Zbl 1205.62130 |

[13] | Freeland, RK, McCabe, BPM (2004) Analysis of low count time series data by Poisson autoregression. Journal of Time Series Analysis, 25, 701-22. · Zbl 1062.62174 |

[14] | Godsill, SJ, Doucet, A, West, M (2004) Monte Carlo smoothing for nonlinear time series. Journal of the American Statistical Association, 99, 156-68. · Zbl 1089.62517 |

[15] | Gordon, NJ, Salmond, DJ, Smith, AFM (1993) Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F, Radar and Signal Processing, 140, 107-13. |

[16] | Jazi, MA, Jones, G, Lai, CD (2012) First-order integer valued AR processes with zero inflated Poisson innovations. Journal of Time Series Analysis, 33, 954-63. · Zbl 1281.62197 |

[17] | Kedem, B, Fokianos, K (2002) Regression models for time series analysis. New Jersey: Wiley. · Zbl 1011.62089 |

[18] | Kim, J, Stoffer, DS (2008) Fitting stochastic volatility models in the presence of irregular sampling via particle methods and the EM algorithm. Journal of Time Series Analysis, 29, 811-33. · Zbl 1199.62042 |

[19] | Lambert, D (1992) Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics, 34, 1-14. · Zbl 0850.62756 |

[20] | Lawless, JF (1987) Negative binomial and mixed Poisson regression. The Canadian Journal of Statistics, 15, 209-25. · Zbl 0632.62060 |

[21] | Lee, AH, Wang, K, Yau, KKW, Carrivick, PJW, Stevenson, MR (2005) Modelling bivariate count series with excess zeros. Mathematical Biosciences, 196, 226-37. · Zbl 1071.62079 |

[22] | Levine, RA, Casella, G (2001) Implementations of the Monte Carlo EM algorithm. Journal of Computational and Graphical Statistics, 10, 422-39. |

[23] | Louis, TA (1982) Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society Series B, 44, 226-33. · Zbl 0488.62018 |

[24] | Nelson, KP, Leroux, BG (2006) Statistical models for autocorrelated count data. Statistics in Medicine, 25, 1413-30. |

[25] | Oh, MS, Lim, YB (2001) Bayesian analysis of time series Poisson data. Journal of Applied Statistics, 28, 259-71. · Zbl 0993.62080 |

[26] | Rue, H, Martino, S, Chopin, N (2009) Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations (with discussion). Journal of the Royal Statistical Society Series B, 71, 319-92. · Zbl 1248.62156 |

[27] | Shumway, RH, Stoffer, DS (1982) An approach to time series smoothing and forecasting using the EM algorithm. Journal of Time Series Analysis, 3, 253-64. · Zbl 0502.62085 |

[28] | Wang, P (2001) Markov zero-inflated Poisson regression models for a time series of counts with excess zeros. Journal of Applied Statistics, 28, 623-32. · Zbl 0991.62064 |

[29] | Yang, M, Zamba, GKD, Cavanaugh, JE (2013) Markov regression models for count time series with excess zeros: A partial likelihood approach. Statistical Methodology, 14, 26-38. · Zbl 07000877 |

[30] | Yau, KKW, Lee, AH, Carrivick, PJW (2004) Modeling zero-inflated count series with application to occupational health. Computer Methods and Programs in Biomedicine, 74, 47-52. |

[31] | Yau, KKW, Wang, K, Lee, AH (2003) Zero-inflated negative binomial mixed regression modeling of over-dispersed count data with extra zeros. Biometrical Journal, 45, 437-52. · Zbl 1441.62543 |

[32] | Zeger, SL (1988) A regression model for time series of counts. Biometrika, 75, 621-9. · Zbl 0653.62064 |

[33] | Zeger, SL, Qaqish, B (1988) Markov regression models for time series: A quasi-likelihood approach. Biometrics, 44, 1019-31. · Zbl 0715.62166 |

[34] | Zhu, F (2010) A negative binomial integer-valued GARCH model. Journal of Time Series Analysis, 32, 54-67. · Zbl 1290.62092 |

[35] | Zhu, F (2012) Zero-inflated Poisson and negative binomial integer-valued GARCH models. Journal of Statistical Planning and Inference, 142, 826-39. · Zbl 1232.62121 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.