On parameter estimation for $N(\mu,\sigma^2 I_3)$ based on projected data into $\mathbb{S}^2$

TL;DR

This paper investigates parameter estimation for a normal distribution projected onto the 2-sphere, demonstrating that intrinsic covariance and expectation properties facilitate effective estimation of the original distribution's parameters.

Contribution

It establishes a correspondence between the covariance of the normal and the intrinsic covariance of the projected distribution, enabling parameter estimation from projected data.

Findings

Expectation commutes with projection on the sphere.

Covariance of the normal relates one-to-one with intrinsic covariance.

Method allows estimation of original normal parameters from projected data.

Abstract

We consider the projected normal distribution, with isotropic variance, on the 2-sphere using intrinsic statistics. We show that in this case, the expectation commutes with the projection and that the covariance of the normal variable has a 1-1 correspondence with the intrinsic covariance of the projected normal distribution. This allows to estimate, after model identification, the parameters of the underlying normal distribution that generates the data.