The class of $α$-stable distributions with a wide range of applications in economics, telecommunications, biology, applied, and theoretical physics. This is due to the fact that it possesses both the skewness and heavy tails. Since $α$-stable distribution suffers from a closed-form expression for density function, finding efficient estimators for its parameters has attracted a great deal of attention in the literature. Here, we propose some EM algorithm to estimate the maximum likelihood estimators of the parameters of $α$-stable distribution. The performance of the proposed EM algorithm is demonstrated via comparison study in the presence of other well-known competitors and analyzing three sets of real data.