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Some INAR models generated by the geometric counting sequence are presented and on the basis on their main features are compared to the INAR models based on the well known binomial thinning operator. Crucial properties of the observed models and their parameter estimators are considered. The main attentionis paid to the application of the models to the real data. Finally, on the observed kind of dynamical crime data, the possible reasons for outperforming binomial thinning based models by the corresponding ones generated by the negative binomial thinning operator, are discussed.

INAR(1) models; Binomial thinning; Negative binomial thinning; Geometric marginal distribution