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Abstract
This article aims at providing a comprehensive survey of recent developments in the field of integer-valued time series modelling, paying particular attention to models obtained as discrete counterparts of conventional autoregressive moving average and bilinear models, and based on the concept of thinning. Such models have proven to be useful in the analysis of many real-world applications ranging from economy and finance to medicine. We review the literature of the most relevant thinning operators proposed in the analysis of univariate and multivariate integer-valued time series with either finite or infinite support. Finally, we also outline and discuss possible directions of future research.
References
| Al-Osh, MA, Alzaid, AA (1987) First order integer-valued autoregressive INAR process. Journal of Time Series Analysis, 8, 261–75. Google Scholar | Crossref | |
| Al-Osh, MA, Alzaid, AA (1988) Integer-valued moving average (INMA) process. Statistical Papers, 29, 281–300. Google Scholar | Crossref | |
| Al-Osh, MA, Alzaid, AA (1991) Binomial autoregressive moving average models. Communications in Statistics –Stochastic Models, 7, 261–82. Google Scholar | Crossref | |
| Aly, EEAA, Bouzar, N (1994a) Explicit stationary distributions for some Galton-Watson processes with immigration. Communications in Statistics –Stochastic Models, 10, 499–517. Google Scholar | Crossref | |
| Aly, EEAA, Bouzar,, N (1994b) On some integer-valued autoregressive moving average models. Journal of Multivariate Analysis, 50, 132–51. Google Scholar | Crossref | ISI | |
| Alzaid, AA, Al-Osh, MA (1990) An integer-valued th-order autoregressive structure (INAR()) process. Journal of Applied Probability, 27, 314–24. Google Scholar | Crossref | ISI | |
| Alzaid, AA, Al-Osh, MA (1993) Generalized Poisson ARMA processes. Annals of the Institute of Statistical Mathematics, 45, 223–32. Google Scholar | Crossref | ISI | |
| Alzaid, AA, Omair, MA (2012) An extended binomial distribution with applications. Communications in Statistics –Theory and Methods, 41, 3511–27. Google Scholar | Crossref | ISI | |
| Alzaid, AA, Omair, MA (2014) Poisson difference integer valued autoregressive model of order one. Bulletin of the Malaysian Mathematical Sciences Society, 37, 465–85. Google Scholar | ISI | |
| Anderson, CW (1970) Extreme value theory for a class of discrete distributions with applications to some stochastic processes. Journal of Applied Probability, 7, 99–113. Google Scholar | Crossref | ISI | |
| Andersson, J, Karlis,, D (2014). A parametric time series model with covariates for integers in . Statistical Modelling, 14, 135–56. Google Scholar | SAGE Journals | ISI | |
| Bakouch, HS, Ristić, MM (2010) Zero truncated Poisson integer-valued AR(1) model. Metrika, 72, 265–80. Google Scholar | Crossref | ISI | |
| Barczy, M, Ispány, M, Pap, G (2011) Asymptotic behaviour of unstable INAR processes. Stochastic Processes and their Applications, 121, 583–608. Google Scholar | Crossref | ISI | |
| Barczy, M, Ispány, M, Pap, G, Scotto, MG, Silva, ME (2010) Innovational outliers in INAR models. Communications in Statistics –Theory and Methods, 39, 3343–62. Google Scholar | Crossref | ISI | |
| Barczy, M, Ispány, M, Pap, G, Scotto, MG, Silva, ME (2012). Additive outliers in INAR models. Statistical Papers, 53, 935–49. Google Scholar | Crossref | ISI | |
| Barczy, M, Ispány, M, Pap, G (2014) Asymptotic behaviour of conditional least squares estimators for unstable integer-valued autoregressive models of order 2. Scandinavian Journal of Statistics, 41, 866–92. Google Scholar | Crossref | ISI | |
| Barreto-Souza, W, Bourguignon, M (2015) A skew INAR process on . AStA –Advances in Statistical Analysis, 99, 189–208. Google Scholar | Crossref | ISI | |
| Biswas, A, Song, PXK (2009) Discrete-valued ARMA processes. Statistics and Probability Letters, 79, 1884–89. Google Scholar | Crossref | ISI | |
| Boudreault, M, Charpentier, A (2011) Multivariate integer-valued autoregressive models applied to earthquake counts. arXiv:1112.0929v1 [stat.AP]. Google Scholar | |
| Brännäs, K (1995) Explanatory variables in the AR(1) count data model. Umeå Economic Studies 381. Google Scholar | |
| Brännäs, K, Hall, A (2001) Estimation in integer-valued moving average models. Applied Stochastic Models in Business and Industry, 17, 277–91. Google Scholar | Crossref | ISI | |
| Brännäs, K, Hellström, J (2001) Generalized integer-valued autoregression. Econometric Reviews, 20, 425–43. Google Scholar | Crossref | |
| Brännäs, K, Nordström, J (2006) Tourist accommodation effects of festivals. Tourism Economics, 12, 291–302. Google Scholar | SAGE Journals | |
| Brännäs, K, Quoreshi, AMMS (2010) Integer-valued moving average modelling of the number of transactions in stocks. Applied Financial Economics, 20, 1429–40. Google Scholar | Crossref | |
| Brännäs, K, Hellström, J, Nordström, J (2002) A new approach to modelling and forecasting monthly guest nights in hotels. International Journal of Forecasting, 18, 19–30. Google Scholar | Crossref | ISI | |
| Bu, R, McCabe, BPM, Hadri, K (2008). Maximum likelihood estimation of higher-order integer-valued autoregressive processes. Journal of Time Series Analysis, 29, 973–94. Google Scholar | Crossref | ISI | |
| Bulla, J, Chesneau, C, Kachour, M (2012) A bivariate first-order signed integer-valued autoregressive process. Submitted. Google Scholar | |
| Bulla, J, Chesneau, C, Kachour, M (2014) On the bivariate Skellam distribution. Communications in Statistics –Theory and Methods. To app ear. Google Scholar | |
| Chesneau, C, Kachour, M (2012) A parametric study for the first-order signed integer-valued autoregressive process. Journal of Statistical Theory and Practice, 6, 760–82. Google Scholar | Crossref | |
| Cui, Y, Lund, R (2010) Inference in binomial AR models. Statistics and Probability Letters, 80, 1985–90. Google Scholar | Crossref | ISI | |
| Davis, RA, Lee, T, Rodriguez-Yam, G (2008) Break detection for a class of nonlinear time series models. Journal of Time Series Analysis, 29, 834–67. Google Scholar | Crossref | ISI | |
| Doukhan, P, Latour, A, Oraichi, D (2006) A simple integer-valued bilinear time series model. Advances in Applied Probability, 38, 559–78. Google Scholar | Crossref | ISI | |
| Drost, FC van den, Akker, R, Werker, BJM (2008) Note on integer-valued bilinear time series models. Statistics and Probability Letters, 78, 992–96. Google Scholar | Crossref | ISI | |
| Drost, FC van den, Akker, R, Werker, BJM (2009) Efficient estimation of auto-regression parameters and innovation distributions for semiparametric integer-valued AR() models. Journal of the Royal Statistical Society: Series B, 71, 467–85. Google Scholar | Crossref | |
| Du, JG, Li, Y (1991) The integer valued autoregressive (INAR()) model. Journal of Time Series Analysis, 12, 129–42. Google Scholar | Crossref | |
| Ferland, R, Latour, A, Oraichi, D (2006) Integer-valued GARCH processes. Journal of Time Series Analysis, 27, 923–42. Google Scholar | Crossref | ISI | |
| Fokianos, K (2012) Count time series models. In T Subba Rao S Subba Rao and CR Rao(eds). Handbook of statistics 30: Time series –methods and applications, pp. 315–47. Amsterdam: Elsevier B.V. Google Scholar | Crossref | |
| Fokianos, K, Tjøstheim, D (2012) Nonlinear Poisson autoregression. Annals of the Institute of Statistical Mathematics, 64, 1205–25. Google Scholar | Crossref | ISI | |
| Fokianos, K, Rahbek, A, Tjøstheim, D (2009) Poisson autoregression. Journal of the American Statistical Association, 104, 1430–39. Google Scholar | Crossref | ISI | |
| Franke, J, Subba, Rao T (1993) Multivariate first-order integer-valued autoregression. Technical report, Universität Kaiserslautern. Google Scholar | |
| Freeland, RK (2010) True integer value time series. AStA –Advances in Statistical Analysis, 94, 217–29. Google Scholar | Crossref | ISI | |
| Freeland, RK, McCabe, BPM (2004) Forecasting discrete valued low count time series. International Journal of Forecasting, 20, 427–34. Google Scholar | Crossref | ISI | |
| Ghodsi, A, Shitan, M, Bakouch, HS (2012) A first-order spatial integer-valued autoregressive SINAR model. Communications in Statistics –Theory and Methods, 41, 2773–87. Google Scholar | Crossref | ISI | |
| Gomes, D, Cantoe, Castro L (2009) Generalized integer-valued random coefficient for a first order structure autoregressive (RCINAR) process. Journal of Statistical Planning and Inference, 139, 4088–97. Google Scholar | Crossref | ISI | |
| Grunwald, GK, Hyndman, RJ, Tedesco, L (1997) A unified view of linear AR models. Research report, Department of Statistics, University of Melbourne. Google Scholar | |
| Hall, A (1996) Maximum term of a particular autoregressive sequence with discrete margins. Communications in Statistics –Theory and Methods, 25, 721–36. Google Scholar | Crossref | ISI | |
| Hall, A (2001) Extremes of integer-valued moving averages models with regularly varying tails. Extremes, 4, 219–39. Google Scholar | Crossref | |
| Hall, A, Scotto, MG, Cruz, JP (2010) Extremes of integer-valued moving average sequences. Test, 19, 359–74. Google Scholar | Crossref | ISI | |
| Heinen, A (2003) Modelling time series count data: an autoregressive conditional Poisson model. CORE Discussion Paper 2003/62. Google Scholar | |
| Hudecová, S, Hušková, M, Meintanis, SG (2015) Tests for time series of counts based on the probability-generating function. Statistics, 49, 316–37. Google Scholar | Crossref | ISI | |
| Jacobs, PA, Lewis, PAW (1983) Stationary discrete autoregressive-moving average time series generated by mixtures. Journal of Time Series Analysis, 4, 19–36. Google Scholar | Crossref | |
| 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. Google Scholar | Crossref | ISI | |
| Joe, H (1996) Time series models with univariate margins in the convolution-closed infinitely divisible class. Journal of Applied Probability, 33, 664–77. Google Scholar | Crossref | ISI | |
| Jørgensen, B, Song, PXK (1998) Stationary time-series models with exponential dispersion model margins. Journal of Applied Probability, 35, 78–92. Google Scholar | Crossref | ISI | |
| Jung, RC, Tremayne, AR (2003) Testing for serial dependence in time series models of counts. Journal of Time Series Analysis, 24, 65–84. Google Scholar | Crossref | ISI | |
| Jung, RC, Tremayne, AR (2011a) Useful models for time series of counts or simply wrong ones? AStA –Advances in Statistical Analysis, 95, 59–91. Google Scholar | Crossref | ISI | |
| Jung, RC, Tremayne, AR (2011b) Convolution-closed models for count time series with applications. Journal of Time Series Analysis, 32, 268–80. Google Scholar | Crossref | ISI | |
| Kachour, M, Truquet, L (2011) A -order signed integer-valued autoregressive (SINAR()) model. Journal of Time Series Analysis, 32, 223–36. Google Scholar | Crossref | ISI | |
| Karlis, D, Pedeli, X (2013) Flexible bivariate INAR processes using copulas. Communications in Statistics –Theory and Methods, 42, 723–40. Google Scholar | Crossref | ISI | |
| Kashikar, AS, Rohan, N, Ramanathan, TV (2013) Integer autoregressive models with structural breaks. Journal of Applied Statistics, 40, 2653–69. Google Scholar | Crossref | ISI | |
| Kim, HY, Park, Y (2008) A non-stationary integer-valued autoregressive model. Statistical Papers, 49, 485–502. Google Scholar | Crossref | ISI | |
| Kim, HY, Weiß,, CH (2015) Goodness-of-fit tests for binomial AR(1) processes. Statistics, 49, 291–315. Google Scholar | Crossref | ISI | |
| Kocherlakota, S, Kocherlakota, K (1992) Bivariate discrete distributions. Marcel Dekker, Inc. Google Scholar | |
| Latour, A (1997) The multivariate GINAR process. Advances in Applied Probability, 29, 228–48. Google Scholar | Crossref | ISI | |
| Latour, A (1998) Existence and stochastic structure of a non-negative integer-valued autoregressive processes. Journal of Time Series Analysis, 4, 439–55. Google Scholar | Crossref | |
| Leonenko, NN, Savani, V, Zhigljavsky, AA (2007) Autoregressive negative binomial processes. Annales de l'I.S.U.P LI, 25–47. Google Scholar | |
| Liu, T, Yuan, X (2013) Random rounded integer-valued autoregressive conditional heteroskedastic process. Statistical Papers, 54, 645–83. Google Scholar | Crossref | ISI | |
| McCabe, BPM, Martin, GM, Harris, D (2011) Efficient probabilistic forecasts for counts. Journal of the Royal Statistical Society: Series B, 73, 253–72. Google Scholar | Crossref | |
| McCormick, WP, Park, YS (1992) Asymptotic analysis of extremes from autoregressive negative binomial processes. Journal of Applied Probability, 29, 904–20. Google Scholar | Crossref | ISI | |
| McKenzie, E (1985) Some simple models for discrete variate time series. Water Resources Bulletin, 21, 645–50. Google Scholar | Crossref | |
| McKenzie, E (1988) Some ARMA models for dependent sequences of Poisson counts. Advances in Applied Probability, 20, 822–35. Google Scholar | Crossref | ISI | |
| Monteiro, M, Pereira, I, Scotto, MG (2008) Optimal alarm systems for count processes. Communications in Statistics –Theory and Methods, 37, 3054–76. Google Scholar | Crossref | ISI | |
| Monteiro, M, Scotto, MG, Pereira, I (2010) Integer-valued autoregressive processes with periodic structure. Journal of Statistical Planning and Inference, 140, 1529–41. Google Scholar | Crossref | ISI | |
| Monteiro, M, Scotto, MG, Pereira, I (2012) Integer-valued self-exciting threshold autoregressive processes. Communications in Statistics –Theory and Methods, 41, 2717–37. Google Scholar | Crossref | ISI | |
| Nastić, AS, Ristić, MM, Bakouch, HS (2012) A combined geometric INAR() model based on negative binomial thinning. Mathematical and Computer Modelling, 55, 1665–72. Google Scholar | Crossref | |
| Nastić, AS, Ristić, MM, Djordjević, MS (2014) An INAR model with discrete Laplace marginal distributions. Brazilian Journal of Probability and Statistics. Forthcoming. Google Scholar | |
| Pedeli, X, Karlis, D (2011) A bivariate INAR process with application. Statistical Modelling, 11, 325–49. Google Scholar | SAGE Journals | ISI | |
| Pedeli, X, Karlis, D (2013a) On composite likelihood estimation of a multivariate INAR model. Journal of Time Series Analysis, 34, 206–20. Google Scholar | Crossref | ISI | |
| Pedeli, X, Karlis, D (2013b) On estimation of the bivariate Poisson INAR process. Communications in Statistics –Simulation and Computation, 42, 514–33. Google Scholar | Crossref | ISI | |
| Pedeli, X, Karlis, D (2013c) Some properties of multivariate INAR processes. Computational Statistics and Data Analysis, 67, 213–25. Google Scholar | Crossref | ISI | |
| Pedeli, X, Davison, AC, Fokianos, K (2014) Likelihood estimation for the INAR(1) model by saddlepoint approximation. Journal of the American Statistical Association. Forthcoming. Google Scholar | |
| Pegram, GGS (1980) An autoregressive model for multilag Markov chains. Journal of Applied Probability, 17, 350–62. Google Scholar | Crossref | ISI | |
| Puig, P, Valero, J (2007) Characterization of count data distributions involving additivity and binomial subsampling. Bernoulli, 13, 544–55. Google Scholar | Crossref | ISI | |
| Quoreshi, AMMS (2006) Bivariate time series modelling of financial count data. Communications in Statistics –Theory and Methods, 35, 1343–58. Google Scholar | Crossref | ISI | |
| Quoreshi, AMMS (2014) A long-memory integer-valued time series model INARFIMA, for financial application. Quantitative Finance, 14, 2225–35. Google Scholar | Crossref | ISI | |
| Ristić, MM, Bakouch, HS, Nastić, AS (2009) A new geometric first-order integer-valued autoregressive (NGINAR) process. Journal of Statistical Planning and Inference 139, 2218–26. Google Scholar | Crossref | ISI | |
| Ristić, MM, Nastić, AS, Bakouch, HS (2012a) Estimation in an integer-valued autoregressive process with negative binomial marginals (NBINAR(1)). Communications in Statistics –Theory and Methods, 41, 606–18. Google Scholar | Crossref | ISI | |
| Ristić, MM, Nastić, AS, Jayakumar, K, Bakouch, HS (2012b) A bivariate INAR time series model with geometric marginals. Applied Mathematics Letters, 25, 481–85. Google Scholar | Crossref | ISI | |
| Ristić, MM, Nastić, AS, Miletić, Ilić AV (2013) A geometric time series model with dependent Bernoulli counting series. Journal of Time Series Analysis, 34, 466–76. Google Scholar | Crossref | ISI | |
| Roitershtein, A, Zhong, Z (2013) On random coefficient INAR processes. Science China Mathematics, 56, 177–200. Google Scholar | Crossref | ISI | |
| Scotto, MG, Weiß, CH, Silva, ME, Pereira, I (2014) Bivariate binomial autoregressive models. Journal of Multivariate Analysis, 125, 233–51. Google Scholar | Crossref | ISI | |
| Schweer, S, Weiß, CH (2014) Compound Poisson INAR(1) processes: stochastic properties and testing for overdispersion. Computational Statistics and Data Analysis, 77, 267–84. Google Scholar | Crossref | ISI | |
| Silva, I, Silva, ME, Pereira, I, Silva, N (2005) Replicated INAR processes. Methodology and Computing in Applied Probability, 7, 517–42. Google Scholar | Crossref | ISI | |
| Silva, ME, Oliveira, VL (2004) Difference equations for the higher order moments and cumulants of the INAR model. Journal of Time Series Analysis, 25, 317–33. Google Scholar | Crossref | ISI | |
| Steutel, FW, van, Harn K (1979) Discrete analogues of self-decomposability and stability. Annals of Probability, 7, 893–99. Google Scholar | Crossref | ISI | |
| Sun, J, McCabe, BPM (2013) Score statistics for testing serial dependence in count data. Journal of Time Series Analysis, 34, 315–29. Google Scholar | Crossref | ISI | |
| Tjøstheim, D (2012). Some recent theory for autoregressive count time series. Test, 21, 413–38. Google Scholar | Crossref | ISI | |
| Tong, H (1990) Non-linear time series. Oxford: Science Publications, Oxford. Google Scholar | |
| Triebsch, LK (2008) New integer-valued autoregressive and regression models with state-dependent parameters. Doctoral dissertation, TU Kaiserslautern, Verlag Dr. Hut, Munich. Google Scholar | |
| Turkman, KF, Scotto, MG, de, Zea, Bermudez, P (2014) Non-linear time series: extreme events and integer value problems. Switzerland: Springer-Verlag. Google Scholar | |
| Weiß, CH (2008a) Thinning operations for modelling time series of counts –a survey. AStA –Advances in Statistical Analysis, 92, 319–41. Google Scholar | Crossref | ISI | |
| Weiß, CH (2008b) Serial dependence and regression of Poisson INARMA models. Journal of Statistical Planning and Inference, 138, 2975–90. Google Scholar | Crossref | ISI | |
| Weiß, CH (2008c) The combined INAR() models for time series of counts. Statistics and Probability Letters, 78, 1817–22. Google Scholar | Crossref | ISI | |
| Weiß, CH (2009) A new class of autoregressive models for time series of binomial counts. Communications in Statistics –Theory and Methods, 38, 447–60. Google Scholar | Crossref | ISI | |
| Weiß, CH (2013) Integer-valued autoregressive models for counts showing underdispersion. Journal of Applied Statistics, 40, 1931–48. Google Scholar | Crossref | ISI | |
| Weiß, CH (2015). A Poisson INAR(1) model with serially dependent innovations. Metrika. Forthcoming. Google Scholar | |
| Weiß, CH, Kim, HY (2013a) Binomial AR processes: moments, cumulants, and estimation. Statistics, 47, 494–10. Google Scholar | Crossref | ISI | |
| Weiß, CH, Kim, HY (2013b) Parameter estimation for binomial AR models with applications in finance and industry. Statistical Papers, 54, 563–90. Google Scholar | Crossref | ISI | |
| Weiß, CH, Kim, HY (2014) Diagnosing and modelling extra-binomial variation for time-dependent counts. Applied Stochastic Models in Business and Industry, 30, 588–608. Google Scholar | Crossref | ISI | |
| Weiß, CH, Pollett, PK (2012) Chain binomial models and binomial autoregressive processes. Biometrics, 68, 815–24. Google Scholar | Crossref | Medline | ISI | |
| Weiß, CH, Pollett, PK (2014) Binomial autoregressive processes with density dependent thinning. Journal of Time Series Analysis, 35, 115–32. Google Scholar | Crossref | ISI | |
| Zhang, H, Wang, D, Zhu, F (2010) Inference for INAR processes with signed generalized power series thinning operator. Journal of Statistical Planning and Inference, 140, 667–83. Google Scholar | Crossref | ISI | |
| Zhang, H, Wang, D (2015) Inference for random coefficient INAR process based on frequency domain analysis. Communications in Statistics –Simulation and Computation, 44, 1078–100. Google Scholar | Crossref | ISI | |
| Zheng, H, Basawa, IV, Datta, S (2006) Inference for th-order random coefficient integer-valued autoregressive processes. Journal of Time Series Analysis, 27, 411–40. Google Scholar | Crossref | ISI | |
| Zheng, H, Basawa, IV, Datta, S (2007) First-order random coefficient integer-valued autoregressive processes. Journal of Statistical Planning and Inference, 173, 212–29. Google Scholar | Crossref | ISI | |
| Zhu, R, Joe, H (2003) A new type of discrete self-decomposability and its applications to continuous-time Markov processes for modelling count data time series. Stochastic Models, 19, 235–54. Google Scholar | Crossref | ISI | |
| Zhu, R, Joe, H (2006) Modelling count data time series with Markov processes based on binomial thinning. Journal of Time Series Analysis, 27, 725–38. Google Scholar | Crossref | ISI | |
| Zhu, R, Joe, H (2010) Negative binomial time series models based on expectation thinning operators. Journal of Statistical Planning and Inference, 140, 1874–88. Google Scholar | Crossref | ISI |
