Geodesic
Binomial ARMA count series from renewal processes
Discussiones Mathematicae. Probability and Statistics, Tome 32 (2012) no. 1-2, pp. 5-16.

Voir la notice de l'article provenant de la source Library of Science

This paper describes a new method for generating stationary integer-valued time series from renewal processes. We prove that if the lifetime distribution of renewal processes is nonlattice and the probability generating function is rational, then the generated time series satisfy causal and invertible ARMA type stochastic difference equations. The result provides an easy method for generating integer-valued time series with ARMA type autocovariance functions. Examples of generating binomial ARMA(p,p-1) series from lifetime distributions with constant hazard rates after lag p are given as an illustration.
Mots-clés : integer-valued time series, stochastic difference equations, autoregressive moving average, renewal process, lifetime distribution, probability generating function, palindromic polynomial, constant hazard rate 
@article{DMPS_2012_32_1-2_a0,
     author = {Koshkin, Sergiy and Cui, Yunwei},
     title = {Binomial {ARMA} count series from renewal processes},
     journal = {Discussiones Mathematicae. Probability and Statistics},
     pages = {5--16},
     publisher = {mathdoc},
     volume = {32},
     number = {1-2},
     year = {2012},
     zbl = {1291.62166},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/DMPS_2012_32_1-2_a0/}
}
TY  - JOUR
AU  - Koshkin, Sergiy
AU  - Cui, Yunwei
TI  - Binomial ARMA count series from renewal processes
JO  - Discussiones Mathematicae. Probability and Statistics
PY  - 2012
SP  - 5
EP  - 16
VL  - 32
IS  - 1-2
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/DMPS_2012_32_1-2_a0/
LA  - en
ID  - DMPS_2012_32_1-2_a0
ER  - 
%0 Journal Article
%A Koshkin, Sergiy
%A Cui, Yunwei
%T Binomial ARMA count series from renewal processes
%J Discussiones Mathematicae. Probability and Statistics
%D 2012
%P 5-16
%V 32
%N 1-2
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/DMPS_2012_32_1-2_a0/
%G en
%F DMPS_2012_32_1-2_a0
Koshkin, Sergiy; Cui, Yunwei. Binomial ARMA count series from renewal processes. Discussiones Mathematicae. Probability and Statistics, Tome 32 (2012) no. 1-2, pp. 5-16. https://geodesic-test.mathdoc.fr/item/DMPS_2012_32_1-2_a0/

[1] Time Series: Theory and Methods (Springer, New York, 1991). doi: 10.1007/978-1-4419-0320-4

[2] Time series formed from the superposition of discrete renewal processes, Journal of Applied Probability 26 (1989) 189-195. doi: 10.2307/3214330

[3] Some properties and a characterization of trivariate and multivariate binomial distributions , Statistics 36 (2002) 211-218. doi: 10.1080/02331880212859

[4] A new look at time series of counts , Biometrika 96 (2009) 781-792. doi: 10.1093/biomet/asp057

[5] Inference in binomial AR(1) models, Statistics and Probability Letters 80 (2010) 1985-1990. doi: 10.1016/j.spl.2010.09.003

[6] On the superposition of renewal processes , Biometrika 41 (1954) 91-99

[7] R.A. Davis and R. Wu, A negative binomial model for time series of counts, Biometrika 96 (2009) 735-749. doi: 10.1093/biomet/asp029

[8] An Introduction to Probability Theory and Its Applications, Volume I (3rd ed., Wiley, New York, 1968).

[9] Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, New York, 1968).

[10] E. McKenzie, Discrete variate time series, in: Stochastic Processes: Modelling and Simulation, Handbook of Statistics, 21, D.N. Shanbhag and C.R. Rao (Ed(s)), (North-Holland, Amsterdam, 1999) 573-606

[11] Spectral Analysis and Time Series (Academic Press, London, 1981).

[12] Stochastic Processes (2nd ed., Wiley, New York, 1995).

[13] First-order random coefficient integer-valued autoregressive processes, Journal of Statistical Planning and Inference 173 (2007) 212-229. doi: 10.1016/j.jspi.2005.12.003