A Minimal Contrast Estimator for the Linear Fractional Stable Motion

TL;DR

This paper introduces a minimal contrast estimator for the three parameters of linear fractional stable motion, demonstrating its consistency and limit theorems, and proposing methods for confidence regions.

Contribution

It develops a novel minimal contrast estimation method for the parameters of linear fractional stable motion, including a ratio estimator for the self-similarity parameter H.

Findings

Proves strong consistency of the estimator

Establishes weak limit theorems for the estimator

Suggests methods for feasible confidence regions

Abstract

In this paper we present an estimator for the three-dimensional parameter $(\sigma, \alpha, H)$ of the linear fractional stable motion, where $H$ represents the self-similarity parameter, and $(\sigma, \alpha)$ are the scaling and stability parameters of the driving symmetric L\'evy process $L$. Our approach is based upon a minimal contrast method associated with the empirical characteristic function combined with a ratio type estimator for the selfsimilarity parameter $H$. The main result investigates the strong consistency and weak limit theorems for the resulting estimator. Furthermore, we propose several ideas to obtain feasible confidence regions in various parameter settings. Our work is mainly related to [16, 18], in which parameter estimation for the linear fractional stable motion and related L\'evy moving average processes has been studied.