There has been a growing interest in discrete circular models such as wrapped zero inflated Poisson and wrapped Poisson distributions and the trigonometric moments (see Brobbey et al., 2016 and Girija et al., 2014). Also, characteristic functions of stable processes have been used to study the estimation of the model parameters using estimating function approach (see Thavaneswaran et al., 2013). One difficulty in estimating the circular mean and the resultant mean length parameter of wrapped Poisson (WP) or wrapped zero inflated Poisson (WZIP) is that neither the likelihood of WP/WZIP random variable nor the score function is available in closed form, which leads one to use either trigonometric method of moment estimation (TMME) or an estimating function approach. In this paper, we study the estimation of WZIP distribution and WP distribution using estimating functions and obtain the closed form expression of the information matrix. We also derive the asymptotic distribution of the tangent of the mean direction for both the WZIP and WP distributions.