*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

$(p,q)$ 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

$(p,q)$ 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

$(p,q)$ 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.