On Linear Estimators for some Stable Vectors
🤖 AI Summary
This study investigates optimal linear estimation for jointly symmetric α-stable (SαS) random variables under specific dependence structures. Focusing on two canonical models—invertible linear transformations of independent stable variables and sub-Gaussian SαS vectors—the work derives the conditional mean estimator by leveraging stable distribution theory, conditional expectation analysis, and spectral representation techniques, and establishes its linear form. The key contribution lies in demonstrating that, within the SαS framework, the conditional mean estimator coincides with the dispersion-optimal linear estimator. This result systematically extends the classical Gaussian estimation theory to the broader family of stable distributions, thereby unifying and generalizing existing results in the field.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
stable vectors
estimation problem
conditional mean estimator
sub-Gaussian symmetric α-stable
linear estimators
Innovation
Methods, ideas, or system contributions that make the work stand out.
linear estimator
stable distribution
sub-Gaussian
conditional mean
dispersion optimality
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