Methods for improving estimators of truncated circular parameters

TL;DR

This paper extends decision theoretic estimation methods to circular parameters, developing conditions for estimator inadmissibility, invariance principles, and improved estimators for directional distributions on manifolds.

Contribution

It introduces a framework for estimating circular parameters using convexity and invariance, extending Euclidean results to curved manifolds like circles.

Findings

Derived conditions for estimator inadmissibility on circular manifolds

Established a complete class theorem for equivariant estimators

Provided improved estimators for various directional distributions

Abstract

In decision theoretic estimation of parameters in Euclidean space $\mathbb{R}^p$, the action space is chosen to be the convex closure of the estimand space. In this paper, the concept has been extended to the estimation of circular parameters of distributions having support as a circle, torus or cylinder. As directional distributions are of curved nature, existing methods for distributions with parameters taking values in $\mathbb{R}^p$ are not immediately applicable here. A circle is the simplest one-dimensional Riemannian manifold. We employ concepts of convexity, projection, etc., on manifolds to develop sufficient conditions for inadmissibility of estimators for circular parameters. Further invariance under a compact group of transformations is introduced in the estimation problem and a complete class theorem for equivariant estimators is derived. This extends the results of Moors [J. Amer. Statist. Assoc. 76 (1981) 910-915] on $\mathbb{R}^p$ to circles. The findings are of special interest to the case when a circular parameter is truncated. The results are implemented to a wide range of directional distributions to obtain improved estimators of circular parameters.