Geometric Methods for Point Estimation

Hoff, Peter D

McCormack, Andrew R

2023-06-08T18:22:00Z

2023-06-08T18:22:00Z

2023

Statistical Science

The focus of this dissertation is on geometric aspects of point estimation problems. In the first half of this work, we examine the estimation of location parameters for non-Euclidean data that lies in a known manifold or metric space. Ideas from statistical decision theory motivate the construction of new estimators for location parameters. The second half of this work explores information geometric aspects of covariance matrix estimation. In a regular statistical model the Fisher information metric endows the parameter space with a Riemannian manifold structure. Parameter estimation can therefore also be viewed as problem in non-Euclidean data analysis.

Chapter 2 introduces and formalizes the problem of estimating Frechet mean location parameters for metric space valued data. We highlight the importance of the isometry group of distance preserving transformations, and how this group interacts with Frechet means. Pitman's minimum risk equivariant estimator for location models in Euclidean space is generalized to manifold settings, where we discuss aspects of the performance and computation of this minimum risk equivariant estimator.

Turning from equivariant estimation to shrinkage estimation, Chapter 3 introduces a shrinkage estimator for Frechet means that is inspired by Stein's estimator. This estimator utilizes metric space geodesics to shrink an estimate towards a pre-specified, shrinkage point. It is shown that the performance of this geodesic James-Stein estimator depends on the curvature of the underlying metric space, where shrinkage is especially beneficial in non-positively curved spaces.

Chapter 4 discusses shrinkage estimation for covariance matrices that approximately have a Kronecker product structure. The idea of geodesic shrinkage can be applied with respect to alpha-geodesics that arise from the information geometry of a statistical model. These alpha-geodesics also lead to interpretable parameter space decompositions. In a Wishart model we propose an empirical Bayes procedure for estimating approximate Kronecker covariance matrices; a procedure which can be viewed as shrinkage along (-1)-geodesics.

The last chapter of this work further discusses information geometric aspects of covariance estimation, with a view towards asymptotic efficiency. The asymptotic performance of the partial trace estimator for a Kronecker covariance is contrasted with the maximum likelihood estimator. A correction to the partial trace estimator is proposed which is both asymptotically efficient and simple to compute.

https://hdl.handle.net/10161/27672

Statistics

Equivariance

Fisher Information

Frechet Mean

Kronecker Covariance

Metric Space

Shrinkage

Geometric Methods for Point Estimation

Dissertation