This paper presents a family of models for a stationary non-negative first order integer-valued random variables based on the Pegram and thinning operators. Some fundamental and regression properties, k-step-ahead conditional probability have been obtained. Maximum likelihood estimation by the EM algorithm is applied to estimate the parameters. Comparative study of the proposed model with the thinning and Pegram models has also been conducted. The Fisher information matrix has been derived to estimate the asymptotic distributions of the parameters. Comparison with existing models by AIC showed that the proposed model is much better and illustrates its potential usefulness in empirical modelling. Real count data sets have been used to illustrate its application and the model with Poisson marginal is considered for forecasting.