Abstract
The analysis of time series of counts is an emerging field of science. To obtain an ARMA-like autocorrelation structure, many models make use of thinning operations to adapt the ARMA recursion to the integer-valued case. Most popular among these probabilistic operations is the concept of binomial thinning, leading to the class of INARMA models. These models are proved to be useful, especially for processes of Poisson counts, but may lead to difficulties in the case of different count distributions. Therefore, several alternative thinning concepts have been developed. This article reviews such thinning operations and shows how they are successfully applied to define integer-valued ARMA models.
Similar content being viewed by others
References
Al-Osh, M.A., Aly, E.-E.A.A.: First order autoregressive time series with negative binomial and geometric marginals. Commun. Stat. Theory Methods 21(9), 2483–2492 (1992)
Al-Osh, M.A., Alzaid, A.A.: First-order integer-valued autoregressive (INAR(1)) process. J. Time Ser. Anal. 8(3), 261–275 (1987)
Al-Osh, M.A., Alzaid, A.A.: Integer-valued moving average (INMA) process. Stat. Pap. 29, 281–300 (1988)
Al-Osh, M.A., Alzaid, A.A.: Binomial autoregressive moving average models. Commun. Stat. Stoch. Models 7(2), 261–282 (1991)
Alzaid, A.A., Al-Osh, M.A.: First-order integer-valued autoregressive process: distributional and regression properties. Stat. Neerl. 42, 53–61 (1988)
Alzaid, A.A., Al-Osh, M.A.: An integer-valued pth-order autoregressive structure (INAR(p)) process. J. Appl. Probab. 27, 314–324 (1990)
Alzaid, A.A., Al-Osh, M.A.: Generalized Poisson ARMA processes. Ann. Inst. Stat. Math. 45(2), 223–232 (1993)
Ambagaspitiya, R.S., Balakrishnan, N.: On the compound generalized Poisson distributions. Astin Bull. 24(2), 255–263 (1994)
Brännäs, K., Hall, A.: Estimation in integer-valued moving average models. Appl. Stoch. Models Bus. Ind. 17, 277–291 (2001)
Brännäs, K., Hellström, J.: Generalized integer-valued autoregression. Econom. Rev. 20(4), 425–443 (2001)
Brännäs, K., Hellström, J., Nordström, J.: A new approach to modelling and forecasting monthly guest nights in hotels. Int. J. Forecasting 18, 19–30 (2002)
Brockwell, P.J., Davis, R.A.: Introduction to Time Series and Forecasting, 2nd edn. Springer, New York (2002)
Bronstein, I.N., Semendjajew, K.A., Musiol, G., Mühlig, H.: Taschenbuch der Mathematik. Verlag Harri Deutsch (1993) (in German)
Consul, P.C.: Generalized Poisson Distributions—Properties and Applications. Dekker, New York (1989)
Consul, P.C., Mittal, S.P.: A new urn model with predetermined strategy. Biom. Z. 17, 67–75 (1975)
da Silva, I.M.M.: Contributions to the analysis of discrete-valued time series. PhD thesis, University of Porto (2005)
da Silva, I.M.M., da Silva, M.E.: Asymptotic distribution of the Yule-Walker estimator for INAR(p) processes. Stat. Probab. Lett. 76, 1655–1663 (2006)
da Silva, M.E., Oliveira, V.L.: Difference equations for the higher order moments and cumulants of the INAR(1) model. J. Time Ser. Anal. 25(3), 317–333 (2004)
da Silva, M.E., Oliveira, V.L.: Difference equations for the higher order moments and cumulants of the INAR(p) model. J. Time Ser. Anal. 26(1), 17–36 (2005)
Dion, J.-P., Gauthier, G., Latour, A.: Branching processes with immigration and integer-valued time series. Serdica Math. J. 21, 123–136 (1995)
Du, J.-G., Li, Y.: The integer-valued autoregressive (INAR(p)) model. J. Time Ser. Anal. 12(2), 129–142 (1991)
Feller, W.: An Introduction to Probability Theory and its Applications, vol. I, 3rd edn. Wiley, New York (1968)
Freeland, R.K.: Statistical analysis of discrete time series with applications to the analysis of workers compensation claims data. PhD thesis, University of British Columbia, Canada (1998)
Freeland, R.K., McCabe, B.P.M.: Analysis of low count time series data by Poisson autoregression. J. Time Ser. Anal. 25(5), 701–722 (2004)
Freeland, R.K., McCabe, B.P.M.: Asymptotic properties of CLS estimators in the Poisson AR(1) model. Stat. Probab. Lett. 73, 147–153 (2005)
Grunwald, G.K., Hyndman, R.J., Tedesco, L., Tweedie, R.L.: Non-gaussian conditional linear AR(1) models. Aust. N. Z. J. Stat. 42, 479–495 (2000)
Hellström, J.: Unit root testing in integer-valued AR(1) models. Econom. Lett. 70, 9–14 (2001)
Heyde, C.C., Seneta, E.: Estimation theory for growth and immigration rates in a multiplicative process. J. Appl. Probab. 9, 235–256 (1972)
Joe, H.: Time series models with univariate margins in the convolution-closed infinitely divisible class. J. Appl. Prob. 33, 664–677 (1996)
Johansson, P.: Speed limitation and motorway casualties: a time series count data regression approach. Accident Anal. Prev. 28, 73–87 (1996)
Jung, R.C., Tremayne, A.R.: Testing for serial dependence in time series models of counts. J. Time Ser. Anal. 24(1), 65–84 (2003)
Jung, R.C., Tremayne, A.R.: Binomial thinning models for integer time series. Stat. Model. 6, 81–96 (2006a)
Jung, R.C., Tremayne, A.R.: Coherent forecasting in integer time series models. Int. J. Forecast. 22, 223–238 (2006b)
Jung, R.C., Ronning, G., Tremayne, A.R.: Estimation in conditional first order autoregression with discrete support. Stat. Pap. 46, 195–224 (2005)
Jung, R.C., Kukuk, M., Liesenfeld, R.: Time series of count data: Modelling, estimation and diagnostics. Comput. Stat. Data Anal. 51, 2350–2364 (2006)
Kedem, B., Fokianos, K.: Regression Models for Time Series Analysis. Wiley, Hoboken (2002)
Kim, H.Y., Park, Y.S.: A non-stationary integer-valued autoregressive model. In: Proc. of the Spring Conference, Korean Statistical Society, pp. 193–199 (2004)
Künsch, H.R.: The jackknife and the bootstrap for general stationary observations. Ann. Stat. 17(3), 1217–1241 (1989)
Lambert, D., Liu, C.: Adaptive thresholds: Monitoring streams of network counts. J. Am. Stat. Assoc. 101, 78–88 (2006)
Latour, A.: Existence and stochastic structure of a non-negative integer-valued autoregressive process. J. Time Ser. Anal. 19(4), 439–455 (1998)
MacDonald, I.L., Zucchini, W.: Hidden Markov and Other Models for Discrete-Valued Time Series. Chapman & Hall, London (1997)
McKenzie, E.: Some simple models for discrete variate time series. Water Resour. Bull. 21(4), 645–650 (1985)
McKenzie, E.: Autoregressive-moving average processes with negative-binomial and geometric marginal distributions. Adv. Appl. Probab. 18, 679–705 (1986)
McKenzie, E.: Innovation distributions for gamma and negative binomial autoregressions. Scand. J. Stat. 14, 79–85 (1987)
McKenzie, E.: Some ARMA models for dependent sequences of Poisson counts. Adv. Appl. Probab. 20, 822–835 (1988)
McKenzie, E.: Discrete variate time series. In: Rao, C.R., Shanbhag, D.N. (eds.) Handbook of Statistics, pp. 573–606. Elsevier, Amsterdam (2003)
Pokropp, F., Seidel, W., Begun, A., Heidenreich, M., Sever, K.: Control charts for the number of children injured in traffic accidents. In: Lenz, X., Wilrich, X. (eds.) Frontiers in Stat. Qual. Control, vol. 8, pp. 151–171. Physica-Verlag, Heidelberg (2006)
Shenton, L.R.: Quasibinomial distributions. In: Kotz/Johnson: Encyclopedia of Statistical Sciences, vol. 7, pp. 458–460 (1986)
Steutel, F.W., van Harn, K.: Discrete analogues of self-decomposability and stability. Ann. Prob. 7(5), 893–899 (1979)
Weiß, C.H.: Controlling correlated processes of Poisson counts. Qual. Reliab. Engng. Int. 23(6), 741–754 (2007a)
Weiß, C.H.: Serial dependence and regression of Poisson INARMA models. J. Stat. Plan. Inference (2007b, accepted for publication)
Weiß, C.H.: Controlling correlated processes with binomial marginals. Preprint 277, Mathematische Institute der Julius-Maximilians-Universität Würzburg (2007c)
Weiß, C.H.: A new class of autoregressive models for time series of binomial counts. Preprint 279, Mathematische Institute der Julius-Maximilians-Universität Würzburg (2007d)
Weiß, C.H.: The combined INAR(p) models for time series of count. Stat. Probab. Lett. (2008, accepted for publication)
Ye, N., Giordano, J., Feldman, J.: A process control approach to cyber attack detection. Commun. ACM 44(8), 76–82 (2001)
Zheng, H., Basawa, I.V., Datta, S.: Inference for pth-order random coefficient integer-valued autoregressive processes. J. Time Ser. Anal. 27(3), 411–440 (2006)
Zheng, H., Basawa, I.V., Datta, S.: First-order random coefficient integer-valued autoregressive processes. J. Stat. Plan. Inference 173(1), 212–229 (2007)
Zhu, R., Joe, H.: A new type of discrete self-decomposability and its application to continuous-time Markov processes for modeling count data time series. Stoch. Models 19(2), 235–254 (2003)
Zhu, R., Joe, H.: Modelling count data time series with Markov processes based on binomial thinning. J. Time Ser. Anal. 27(5), 725–738 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Weiß, C.H. Thinning operations for modeling time series of counts—a survey. Adv Stat Anal 92, 319–341 (2008). https://doi.org/10.1007/s10182-008-0072-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10182-008-0072-3