Is a Brownian skew?

TL;DR

This paper investigates the asymptotic properties of the maximum likelihood estimator for Skew Brownian motion, enabling a statistical test for skewness with applications in biology.

Contribution

It characterizes the convergence rate and limiting distribution of the estimator, facilitating hypothesis testing on skewness in Skew Brownian motion.

Findings

Derived the estimator's convergence speed and distribution as step size approaches zero.

Developed a practical test for skewness based on the estimator.

Validated the approach with numerical simulations and biological data.

Abstract

We study the asymptotic behavior of the maximum likelihood estimator corresponding to the observation of a trajectory of a Skew Brownian motion, through a uniform time discretization. We characterize the speed of convergence and the limiting distribution when the step size goes to zero, which in this case are non-classical, under the null hypothesis of the Skew Brownian motion being an usual Brownian motion. This allows to design a test on the skewness parameter. We show that numerical simulations that can be easily performed to estimate the skewness parameter, and provide an application in Biology.