Abstract
In this paper, we introduce a new stationary integer-valued autoregressive process of the first order with geometric marginal based on mixing Pegram and generalized binomial thinning operators. The count series of the process consists of dependent Bernoulli count variables. Various properties of the process are obtained, including the distribution of its innovation process. Maximum likelihood estimation by EM algorithm is applied to estimate the parameters of the process and the performance of the estimates is checked by Monte Carlo simulation. We investigate applicability of the process using a real count data set and compare the process to many competitive INAR(1) models via some goodness-of-fit statistics. As a result, forecasting of the data is discussed under the proposed process.
Similar content being viewed by others
References
Al-Osh MA, Aly E-EAA (1992) First order autoregressive time series with negative binomial and geometric marginals. Commun Stat Theory Methods 21:2483–2492
Al-Osh MA, Alzaid AA (1987) First-order integer-valued autoregressive (INAR(1)) process. J Time Ser Anal 8:261–275
Aly E-EAA, Bouzar N (1994a) Explicit stationary distributions for some Galton–Watson processes with immigration. Commun Stat Stoch Models 10:499–517
Aly E-EAA, Bouzar N (1994b) On some integer-valued autoregressive moving average models. J Multivar Anal 50:132–151
Alzaid AA, Al-Osh MA (1988) First-order integer-valued autoregressive (INAR(1)) process: distributional and regression properties. Stat Neerl 42:53–61
Alzaid AA, Al-Osh MA (1993) Some autoregressive moving average processes with generalized Poisson marginal distributions. Ann Inst Stat Math 45:223–232
Bakouch HS, Ristić MM (2010) Zero truncated Poisson integer-valued AR(1) model. Metrika 72:265–280
Biswas A, Song PX-K (2009) Discrete-valued ARMA processes. Stat Probab Lett 79:1884–1889
Chen Y, Gupta MR (2010) EM demystified: an expectation-maximization tutorial. University of Washington, Dept. of EE, Technical Report No. -2010-0002
Khoo WC, Ong SH, Biswas A (2015) Modeling time series of counts with a new class of INAR(1) model. Stat Pap. https://doi.org/10.1007/s00362-015-0704-0
McKenzie E (1986) Autoregressive moving-average processes with negative binomial and geometric distributions. Adv Appl Probab 18:679–705
McKenzie E (1988) Some ARMA models for dependent sequences of Poisson counts. Adv Appl Probab 20:822–835
Pegram GGS (1980) An autoregressive model for multilag markov chains. J Appl Probab 17(350):362
Ristic MM, Bakouch HS, Nastic AS (2009) A new geometric first-order integer-valued autoregressive (NGINAR(1)) process. J Stat Plann Inference 139:2218–2226
Ristić MM, Nastić AS, Bakouch HS (2012) Estimation in an integer-valued autoregressive process with negative binomial marginals (NBINAR(1)). Commun Stat Theory Methods 41:606–618
Ristić MM, Nastić AS, Miletić AV (2013) Geometric time series model with dependent Bernoulli counting series. J Time Ser Anal 34:466–476
Schweer S, Weiß CH (2014) Compound Poisson INAR(1) processes: stochastic properties and testing for overdispersion. Comput Stat Data Anal 77:267–284
Steutel FW, van Harn K (1979) Discrete analogues of self-decomposability and stability. Ann Probab 7:893–899
Weiß CH (2008) Thining operations for modeling time series of count-a survey. AStA Adv Stat Anal 92:319–341
Acknowledgements
The authors would like to thank the two referees for their comments and suggestions that improved this manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shirozhan, M., Mohammadpour, M. & Bakouch, H.S. A New Geometric INAR(1) Model with Mixing Pegram and Generalized Binomial Thinning Operators. Iran J Sci Technol Trans Sci 43, 1011–1020 (2019). https://doi.org/10.1007/s40995-017-0345-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-017-0345-3
Keywords
- Generalized binomial thinning operator
- Mixture distributions
- Pegram operator
- Extreme order statistics
- EM algorithm