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In this paper, new models are studied by proposing the family of generalized power series distributions with inflated parameter (IGPSD) for the innovation process of the INAR(1) model. The main properties of the process were established, such as mean, variance, autocorrelation and transition probability. The methods of estimation by Yule–Walker and the conditional maximum likelihood were used to estimate the parameters of the models. Two particular cases of the INAR(  1  )$\left(1\right)$model with IGPSD innovation were studied, named IPoINAR(  1  )$\left(1\right)$and IGeoINAR(  1  )$\left(1\right)$. Finally, in the real data example, a good performance of the proposed new models was observed.

INAR(1) Processes with Inflated-parameter Generalized Power Series Innovations