Parameter estimation and diagnostic tests for INMA(1) processes

The INMA(1) model for count time series, an integer-valued counterpart to the usual moving-average model of order~1, was introduced by Al-Osh & Alzaid (1988) and McKenzie (1988). During the last years, it gained increasing interest for applications. For instance, it was used by Cossette et al. (2011) to model the number of claims in the area of insurance, and Zhang et al. (2015) applied the model in the area of reinsurance. Furthermore, Hu et al. (2017) point out application areas where the claim numbers may exhibit overdispersion. While stochastic properties of this model, in particular for the special case of the Poisson INMA(1) model, have been comprehensively studied in the literature, only little is known about statistical inference concerning this model.

We start with a central limit theorem for Poisson INMA(1) processes, which allows to explicitly derive the asymptotic distribution of moment and frequency related statistics. In particular, we consider the asymptotic distribution (including bias correction) for diverse moment estimators, for the index of dispersion, and for the autocorrelation function. We apply these results for constructing confidence intervals for model parameters, and for deriving hypothesis tests to check the marginal distribution (e.g., with respect to the Poisson’s equidispersion property) as well as the autocorrelation function (to diagnose the moving average structure). We also show simulation results for INMA(1) time series with different parameters to demonstrate the finite-sample performance of the asymptotic approximations for the above mentioned statistics.