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Efficient order selection algorithms for integer-valued ARMA processes. (English) Zbl 1224.62053
P. Neal and T. Subba Rao [ibid. 28, No. 1, 92–110 (2007; Zbl 1164.62049)] presented a Markov chain Monte Carlo (MCMC) algorithm for obtaining samples from the posterior distributions of parameters of a general INARMA$\left(p,q\right)$ process with known orders $p$ and $q$. For an excellent review of integer-valued time-series, and in particular integer-valued ARMA (INARMA) models, see E. McKenzie [D.N. Shanbhag et al. (eds), Stochastic processes: Modelling and simulation. Amsterdam: North-Holland, Handb. Stat. 21, 573–606 (2003; Zbl 1064.62560)]. One of the main limitations of the algorithm of Neal and Subba Rao is that there is assumed that $p$ and $q$ are known/fixed. The authors of this article propose two approaches for treating $p$ and $q$ as parameters in the model. The first is an extension of the MCMC algorithm of Neal and Subba Rao. They present an efficient reversible jump (RJ)MCMC algorithm (introduced by P.J. Green [Biometrika 82, No. 4, 711–732 (1995; Zbl 0861.62023)]) for inference on the orders $p$ and $q$ of the INARMA$\left(p,q\right)$ model. The algorithm is shown to be very successful at detecting the correct order for simulated data sets and to produce good results for real-life data sets. Furthermore, the algorithm freely moves between the INARMA$\left(p,q\right)$ and the sub-models INAR$\left(p\right)$ and INMA$\left(q\right)$. An alternative to MCMC is given in the form of the EM algorithm for determining the order of an integer-valued autoregressive INAR$\left(p\right)$ process. This algorithm performs very similar to the MCMC algorithm.
##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 65C05 Monte Carlo methods 65C60 Computational problems in statistics