A minimal contrast estimator for the linear fractional stable motion

In this paper we present an estimator for the three-dimensional parameter (σ,α,H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\sigma , \alpha , H)$$\end{document} of the linear fractional stable motion, where H represents the self-similarity parameter, and (σ,α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\sigma , \alpha )$$\end{document} are the scaling and stability parameters of the driving symmetric Lévy 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 self-similarity 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 Ljungdahl and Podolskij (A note on parametric estimation of Lévy moving average processes, p 294, 2019) and Mazur et al. (Bernoulli 26(1): 226–252, 2020) in which parameter estimation for the linear fractional stable motion and related Lévy moving average processes has been studied.