One-misrecorded Poisson INAR(1) Model Via Two Random Operators with Application to Crime and Economics Data

In this paper, we propose a first-order non-negative integer-valued autoregressive (INAR(1)) model with one-misrecorded Poisson (OMP) marginal via a combination of the generalized binomial thinning and mixture Pegram operators for zero-inflated and one deflated count time series. The suggested model is suitable for multimodal, equi- and over-dispersed data modeling. It contains two particular cases: the mixture of Pegram and thinning of a first-order integer-valued autoregressive (MPT(1)) with Poisson and one-misrecorded Poisson. The distribution of the innovation term is derived as a mixture of degenerate distributions at 0 and 1, and two Poisson distributions with certain parameters. Regression and several statistical properties of the proposed model are discussed. We investigate the distribution of the runs (the lengths of zeros and ones). The parameters of the model are estimated using the maximum-likelihood, modified Yule–Walker, and modified conditional least squares methods. The estimation of the parameters, their behavior, and their performance are presented through a simulation study. Two practical data sets on the monthly cases of criminal records and weekly sales are applied to check the proposed process’s performance against other relevant INAR(1) models, showing its capabilities in the challenging case of over-dispersed count data. Furthermore, the proposed model discusses data forecasting.