Building Unbiased Estimators from Non-Gaussian Likelihoods with Application to Shear Estimation

TL;DR

This paper introduces a general framework for constructing unbiased estimators from non-Gaussian likelihoods, with applications to shear estimation in lensing, improving accuracy over previous methods.

Contribution

It develops a formalism to generate unbiased estimators to a specified order and applies it to shear estimation, enhancing existing quadratic estimators with higher-order unbiased versions.

Findings

First-order estimator reduces bias similar to previous work

Third-order estimator achieves 0.1% bias for shears up to 0.2

Framework generalizes to non-constant shear and power spectrum calculations

Abstract

We develop a general framework for generating estimators of a given quantity which are unbiased to a given order in the difference between the true value of the underlying quantity and the fiducial position in theory space around which we expand the likelihood. We apply this formalism to rederive the optimal quadratic estimator and show how the replacement of the second derivative matrix with the Fisher matrix is a generic way of creating an unbiased estimator (assuming choice of the fiducial model is independent of data). Next apply the approach to estimation of shear lensing, closely following the work of Bernstein and Armstrong (2014). Our first order estimator reduces to their estimator in the limit of zero shear, but it also naturally allows for the case of non-constant shear and the easy calculation of correlation functions or power spectra using standard methods. Both our first-order estimator and Bernstein and Armstrong's estimator exhibit a bias which is quadratic in true shear. Our third-order estimator is, at least in the realm of the toy problem of Bernstein and Armstrong, unbiased to 0.1% in relative shear errors $\Delta g/|g|$ for shears up to $|g|=0.2$.