A distribution for a pair of unit vectors generated by Brownian motion

TL;DR

This paper introduces a new bivariate distribution for dependent unit vectors generated by Brownian motion, explores its properties, estimation methods, and applications to wind data and planar distributions.

Contribution

It proposes a novel dependent bivariate unit vector model based on Brownian motion with properties, estimation techniques, and generalizations including applications to wind direction data.

Findings

Model has uniform marginals on the sphere

Parameter estimators are studied with simulation results

Generalizations include planar and cylindrical distributions

Abstract

We propose a bivariate model for a pair of dependent unit vectors which is generated by Brownian motion. Both marginals have uniform distributions on the sphere, while the conditionals follow so-called ``exit'' distributions. Some properties of the proposed model, including parameter estimation and a pivotal statistic, are investigated. Further study is undertaken for the bivariate circular case by transforming variables and parameters into the form of complex numbers. Some desirable properties, such as a multiplicative property and infinite divisibility, hold for this submodel. Two estimators for the parameter of the submodel are studied and a simulation study is carried out to investigate the finite sample performance of the estimators. In an attempt to produce more flexible models, some methods to generalize the proposed model are discussed. One of the generalized models is applied to wind direction data. Finally, we show how it is possible to construct distributions in the plane and on the cylinder by applying bilinear fractional transformations to the proposed bivariate circular model.