Preferential attachment is widely used to model the power-law behavior of degree distributions in social networks. In this thesis, we study three aspects of a directed preferential attachment model. First, we consider fitting this network model under different data scenarios. We propose both parametric and semi-parametric estimation procedures and compare the corresponding estimating results. Second, we see from empirical studies that statistical estimates of the marginal tail exponent of the power-law degree distribution often use the Hill estimator, even though no theoretical justification has been given. Hence, we study the convergence of the joint empirical measure for in- and out-degrees and prove the consistency of the Hill estimator for the preferential attachment model. Finally, we consider a widely adopted threshold selection procedure when estimating the power-law index in practice and examine the asymptotic behavior of the selected threshold as well as the corresponding power-law index given.