Abstract: Yule-Simon distribution has a wide range of practical applications, such as in network science, biology and humanities. A lot of work focuses on the study of how well the empirical data fits Yule-Simon distribution or how to estimate the parameter. There are still some open problems, such as the error analysis of parameter estimation, the theoretical proof of the convergence of the iterative algorithm for maximum likelihood estimation of parameters. The Yule-Simon distribution is a heavy-tailed distribution and the parameter is usually less than 2, so the variance does not exist. This makes it diffcult to give an interval estimation of the parameter. Using the compression transformation, this paper proposes a method of interval estimation based on the central limit theorem. This method can be applied to many heavy-tailed distributions. The other two asymptotic confidence intervals of the parameter are obtained based on the maximum likelihood and the mode method. These estimation methods are compared in simulations and applications to empirical data.