Tail Estimation via Sample Splitting: A Two-Sample Framework for Stable-like Distributions

Abstract

Stable distributions provide a flexible framework for modeling heavy-tailed and skewed data, with the stability index $\alpha$ quantifying tail heaviness. We propose a new semiparametric approach that leverages the two-sum closure property of stable distributions within a location-scale framework, \textcolor{black}{where the induced scale parameter satisfies $\sigma = 2^{1/\alpha}$}. The method constructs two pseudo-independent samples from a single observed sample via repeated random splitting and empirical convolution, and then estimates $\sigma$ using weighted least squares applied to empirical quantiles. We also propose a bootstrap procedure that uses a reduced number of sample splits together with extrapolation to estimate standard errors, and we demonstrate good finite-sample performance of this procedure. This approach avoids intractable likelihood calculations and offers computational advantages over maximum likelihood estimation. We establish consistency and asymptotic properties of the estimator and assess its finite-sample performance through simulation studies. Beyond the benefit of substantial computational savings, results indicate competitive accuracy, particularly in heavy-tailed settings.