Research on Consistency of Least Squares Estimators for the Tail Index in Power-Type Distribution Families

This paper systematically investigates the large-sample properties of least squares estimation for the parameter α in the power-law distribution family. We establish the weak consistency of the least squares estimator of α under the Pareto distribution family, scale-invariant distribution family, and asymptotically scale-invariant distribution family. Furthermore, we verify the \sqrt n-consistency and asymptotic normality of the least squares estimator for the Pareto distribution family parameters, thereby providing theoretical guarantees for statistical inference in related models.