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SUPERPOSITIONED STATIONARY COUNT TIME SERIES

Published online by Cambridge University Press:  23 December 2019

Yisu Jia Open the ORCID record for Yisu Jia [Opens in a new window] ,
Robert Lund  and
James Livsey
Yisu Jia
Affiliation:
Department of Mathematics and Statistics, University of North Florida, 1 UNF Drive, Jacksonville, FL 32224, USA E-mail: [email protected]
Robert Lund
Affiliation:
Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, S.C. 29634-0975, USA
James Livsey
Affiliation:
U.S. Census Bureau, Center for Statistical Research and Methodology, 4600 Silver Hill Rd, Washington, DC 20233, USA
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Abstract

This paper probabilistically explores a class of stationary count time series models built by superpositioning (or otherwise combining) independent copies of a binary stationary sequence of zeroes and ones. Superpositioning methods have proven useful in devising stationary count time series having prespecified marginal distributions. Here, basic properties of this model class are established and the idea is further developed. Specifically, stationary series with binomial, Poisson, negative binomial, discrete uniform, and multinomial marginal distributions are constructed; other marginal distributions are possible. Our primary goal is to derive the autocovariance function of the resulting series.

Keywords

count-valued processesnegative binomial distributionPoisson distributionstationary processestime series
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 3 , July 2021 , pp. 538 - 556
Copyright
© Cambridge University Press 2019

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References

1.Al-Osh, M.A. & Alzaid, A.A. (1987). First order integer-valued autoregressive (INAR(1)) process. Journal of Time Series Analysis 8: 261–275.CrossRefGoogle Scholar
2.Billingsley, P. (2008). Probability and measure. Hoboken, New Jersey, United States: John Wiley & Sons.Google Scholar
3.Blight, P.A. (1989). Time series formed from the superposition of discrete renewal processes. Journal of Applied Probability 26: 189–195.CrossRefGoogle Scholar
4.Cui, Y. & Lund, R.B. (2009). A new look at time series of counts. Biometrika 96: 781–792.CrossRefGoogle Scholar
5.Cui, Y. & Lund, R.B. (2010). Inference for binomial AR(1) models. Statistics and Probability Letters 80: 1985–1990.CrossRefGoogle Scholar
6.Davis, R.A., Holan, S., Lund, R.B., & Ravishanker, N. (2016). Handbook of discrete-valued time series. Boca Raton, Florida, United States: CRC Press.CrossRefGoogle Scholar
7.Fralix, B., Livsey, J., & Lund, R.B. (2012). Renewal sequences with periodic dynamics. Probability in the Engineering and Informational Sciences 26: 1–15.CrossRefGoogle Scholar
8.Gouveia, S., Möller, T.A., Weiß, C.H., & Scotto, M.G. (2018). A full ARMA model for counts with bounded support and its application to rainy-days time series. Stochastic Environmental Research and Risk Assessment 32: 2495–2514.CrossRefGoogle Scholar
9.Heathcote, C.R. (1967). Complete exponential convergence and related topics. Journal of Applied Probability 4: 217–256.CrossRefGoogle Scholar
10.Hilbe, J.M. (2011). Negative Binomial Regression. Cambridge, England: Cambridge University Press.CrossRefGoogle Scholar
11.Holan, S., Lund, R.B., & Davis, G. (2010). The ARMA alphabet soup: A tour of ARMA model variants. Statistics Surveys 4: 232–274.CrossRefGoogle Scholar
12.Jacobs, P.A. & Lewis, P.A. (1978). Discrete time series generated by mixtures I: Correlational and runs properties. Journal of the Royal Statistical Society 40: 94–105.Google Scholar
13.Jacobs, P.A. & Lewis, P.A. (1978). Discrete time series generated by mixtures II: Asymptotic properties. Journal of the Royal Statistical Society 40: 222–228.Google Scholar
14.Jia, Y., Kechagias, S., Livsey, J., Lund, R.B., & Pipiras, V. (2018). Latent Gaussian count time series modeling, eprint arXiv:1811.00203.Google Scholar
15.Joe, H. (1996). Time series models with univariate margins in the convolution-closed infinitely divisible class. Journal of Applied Probability 33: 664–677.CrossRefGoogle Scholar
16.Kachour, M. & Yao, J.F. (2009). First-order rounded integer-valued autoregressive (RINAR(1)) process. Journal of Time Series Analysis 30: 417–448.CrossRefGoogle Scholar
17.Lin, G.D., Dou, X., Kuriki, S., & Huang, J.S. (2014). Recent developments on the construction of bivariate distributions with fixed marginals. Journal of Statistical Distributions and Applications 1: 14SEP.CrossRefGoogle Scholar
18.Livsey, J., Lund, R.B., Kechagais, S., & Pipiras, V. (2018). Multivariate count time series with flexible autocovariances and their application to hurricane counts. Annals of Applied Probability 12: 408–431.Google Scholar
19.Lund, R.B. & Livsey, J. (2015). Renewal-based count time series. In Davis, R.A., Holan, S., Lund, R.B. & Ravishanker, N. (eds.), Handbook of Discrete-Valued Time Series. New York City: CRC Press, pp. 101–120.Google Scholar
20.Lund, R.B., Holan, S., & Livsey, J. (2015). Long memory discrete-valued time series. In Davis, R.A., Holan, S., Lund, R.B. & Ravishanker, N. (eds.), Handbook of Discrete-Valued Time Series, New York City: CRC Press, pp. 447–458.Google Scholar
21.McKenzie, E. (1985). Some simple models for discrete variate time series. Journal of the American Water Resources Association 21: 645–650.CrossRefGoogle Scholar
22.McKenzie, E. (1986). Autoregressive-moving average processes with negative-binomial and geometric marginal distributions. Advances in Applied Probability 18: 822–835.CrossRefGoogle Scholar
23.McKenzie, E. (1988). Some ARMA models for dependent sequences of Poisson counts. Advances in Applied Probability 20: 645–650.CrossRefGoogle Scholar
24.Ng, C.T., Joe, H., Karlis, D., & Liu, J. (2011). Composite likelihood for time series models with a latent autoregressive process. Statistica Sinica 21: 279–305.Google Scholar
25.Pedeli, X. & Karlis, D. (2013). On composite likelihood estimation of a multivariate INAR(1) model. Journal of Time Series Analysis 34: 206–220.CrossRefGoogle Scholar
26.Rose, C. & Smith, M.D. (1996). The multivariate normal distribution. Mathematica Journal 6: 32–37.Google Scholar
27.Steutel, F.W. & Van Harn, K. (1979). Discrete analogues of self-decomposability and stability. The Annals of Probability 7: 893–899.CrossRefGoogle Scholar
28.Van Vleck, J.H. & Middleton, D. (1966). The spectrum of clipped noise. Proceedings of the IEEE 54: 2–19.CrossRefGoogle Scholar
29.Ver Hoef, J.M. & Boveng, P.L. (2007). Quasi-Poisson vs. negative binomial regression: How should we model overdispersed count data?. Ecological Society of America 88: 2766–2772.Google ScholarPubMed
30.Weiß, C.H. (2007). Serial dependence and regression of Poisson INARMA models. Journal of Statistical Planning and Inference 138: 2975–2990.CrossRefGoogle Scholar
31.Weiß, C.H. (2009). A new class of autoregressive models for time series of binomial counts. Communications in Statistics: Theory and Methods 38: 447–460.CrossRefGoogle Scholar
32.Weiß, C.H. (2009). Controlling correlated processes with binomial marginals. Journal of Applied Statistics 36: 399–414.CrossRefGoogle Scholar
33.White, G.C. & Bennetts, R.E. (1996). Analysis of frequency count data using the negative binomial distribution. Ecological Society of America 77: 2549–2557.Google Scholar
34.Whitt, W. (1976). Bivariate distributions with given marginals. The Annals of Statistics 4: 1280–1289.CrossRefGoogle Scholar
35.Yahav, I. & Shmueli, G. (2011). On generating multivariate Poisson data in management science applications. Applied Stochastic Models in Business and Industry 28: 91–102.CrossRefGoogle Scholar
36.Zhu, F. (2010). A negative binomial integer-valued GARCH model. Journal of Time Series Analysis 32: 54–67.CrossRefGoogle Scholar
37.Zhu, F. (2012). Modeling time series of counts with COM-Poisson INGARCH models. Mathematical and Computer Modelling 56: 191–240.CrossRefGoogle Scholar
38.Zhu, R. & Joe, H. (2003). A new type of discrete self-decomposability and its application to continuous-time Markov processes for modeling count data time series. Stochastic Models 19: 235–254.CrossRefGoogle Scholar