A simple estimator for the $\mathcal{M}$-index of functions in $\mathcal{M}$

TL;DR

This paper introduces a new estimator for the $oldsymbol{ ext{M}}$-index applicable to a broad class of functions, demonstrating strong consistency and asymptotic normality, with improved performance over existing estimators through simulations and real data.

Contribution

It proposes a novel estimator for the $oldsymbol{ ext{M}}$-index applicable to the class $oldsymbol{ ext{M}}$, extending beyond regularly varying functions, with proven consistency and asymptotic normality.

Findings

Estimator is strongly consistent under certain conditions.

Asymptotic normality is established for a wide class of functions.

Simulation and real data show improved performance over existing estimators.

Abstract

An estimator for the $\M$-index of functions of $\mathcal{M}$, a larger class than the class of regularly varying (RV) functions, is proposed. This index is the tail index of RV functions and this estimator is thus a new one on the class of RV functions. This estimator satisfies, assuming suitable conditions, strong consistency. Asymptotic normality of this estimator is proved for a large class of RV functions, showing a better performance than some well-known estimators. Illustrations with simulated and real life datasets are provided.