Trigonometrically approximated maximum likelihood estimation for stable law

TL;DR

This paper introduces a trigonometrically approximated maximum likelihood estimator for alpha-stable laws, demonstrating its consistency, asymptotic normality, and superior performance over existing methods through simulations and applications.

Contribution

It proposes a novel estimator based on trigonometric projection of the score function, providing explicit formulas and asymptotic properties, and shows convergence to the true MLE as the number of functions increases.

Findings

Estimator outperforms moment-type estimators in simulations

Standard deviation nearly reaches the Cramér-Rao lower bound

Demonstrates asymptotic mixed normality in high-frequency Ornstein-Uhlenbeck processes

Abstract

A trigonometrically approximated maximum likelihood estimation for $\alpha$-stable laws is proposed. The estimator solves the approximated likelihood equation, which is obtained by projecting a true score function on the space spanned by trigonometric functions. The projected score is expressed only by real and imaginary parts of the characteristic function and their derivatives, so that we can explicitly construct the targeting estimating equation. We study the asymptotic properties of the proposed estimator and show consistency and asymptotic normality. Furthermore, as the number of trigonometric functions increases, the estimator converges to the exact maximum likelihood estimator, in the sense that they have the same asymptotic law. Simulation studies show that our estimator outperforms other moment-type estimators, and its standard deviation almost achieves the Cram\'er--Rao lower bound. We apply our method to the estimation problem for $\alpha$-stable Ornstein--Uhlenbeck processes in a high-frequency setting. The obtained result demonstrates the theory of asymptotic mixed normality.