Maximum likelihood estimator for skew Brownian motion: the convergence rate

TL;DR

This paper analyzes the asymptotic properties of the maximum likelihood estimator for the skewness parameter in Skew Brownian Motion, establishing its convergence rate and distribution, and exploring its behavior near boundary cases.

Contribution

It proves the conjecture that the MLE has a mixed normal distribution with a convergence rate of 1/4 and provides a series expansion and detailed asymptotic analysis.

Findings

MLE converges at rate 1/4 to a mixed normal distribution.

Series expansion of the MLE reveals behavior near boundary skewness values.

Quantifies the explosion of expansion coefficients when skewness approaches ±1.

Abstract

We give a thorough description of the asymptotic property of the maximum likelihood estimator (MLE) of the skewness parameter of a Skew Brownian Motion (SBM). Thanks to recent results on the Central Limit Theorem of the rate of convergence of estimators for the SBM, we prove a conjecture left open that the MLE has asymptotically a mixed normal distribution involving the local time with a rate of convergence of order $1/4$. We also give a series expansion of the MLE and study the asymptotic behavior of the score and its derivatives, as well as their variation with the skewness parameter. In particular, we exhibit a specific behavior when the SBM is actually a Brownian motion, and quantify the explosion of the coefficients of the expansion when the skewness parameter is close to $-1$ or $1$.