Moment-Based Ellipticity Measurement as a Statistical Parameter Estimation Problem

TL;DR

This paper formulates galaxy ellipticity measurement in weak lensing as a statistical parameter estimation problem using unweighted image moments, deriving properties, bounds, and an unbiased estimator, highlighting practical limitations.

Contribution

It introduces a novel statistical framework for ellipticity estimation via image moments, including an unbiased estimator and analysis of its properties and limitations.

Findings

The unbiased estimator has a poorly behaved distribution.

The framework accounts for effects like PSF and pixellation.

Derived Cramér-Rao bounds for ellipticity estimation.

Abstract

We show that galaxy ellipticity estimation for weak gravitational lensing with unweighted image moments reduces to the problem of measuring a combination of the means of three independent normal random variables. Under very general assumptions, the intrinsic image moments of sources can be recovered from observations including effects such as the point-spread function and pixellation. Gaussian pixel noise turns these into three jointly normal random variables, the means of which are algebraically related to the ellipticity. We show that the random variables are approximately independent with known variances, and provide an algorithm for making them exactly independent. Once the framework is developed, we derive general properties of the ellipticity estimation problem, such as the signal-to-noise ratio, a generic form of an ellipticity estimator, and Cram\'er-Rao lower bounds for an unbiased estimator. We then derive the unbiased ellipticity estimator using unweighted image moments. We find that this unbiased estimator has a poorly behaved distribution and does not converge in practical applications, but demonstrates how to derive and understand the behaviour of new moment-based ellipticity estimators.