On Linear Estimators for some Stable Vectors

TL;DR

This paper investigates linear estimators for jointly stable random variables under specific dependency models, demonstrating their optimality and generalizing Gaussian results to stable distributions.

Contribution

It shows the conditional mean estimator is linear for certain stable vector models and identifies dispersion optimal linear estimators, extending Gaussian theory.

Findings

Conditional mean estimator is linear for the models considered

Identifies dispersion optimal linear estimators

Generalizes Gaussian conditional mean results to stable vectors

Abstract

We consider the estimation problem for jointly stable random variables. Under two specific dependency models: a linear transformation of two independent stable variables and a sub-Gaussian symmetric $\alpha$-stable (S$\alpha$S) vector, we show that the conditional mean estimator is linear in both cases. Moreover, we find dispersion optimal linear estimators. Interestingly, for the sub-Gaussian (S$\alpha$S) vector, both estimators are identical generalizing the well-known Gaussian result of the conditional mean being the best linear minimum-mean square estimator.