Performance analysis of the weighted least-squares and maximum likelihood estimators in the joint estimation of source flux and background

Abstract

This thesis studies the possible benefits of simultaneous inference when estimating the brightness of a point source in the sky and the background that the signal is embedded into, and how this joint estimation scheme is empowered by including as much information, in the form of image pixels, as possible. The first part of this analysis resorts to fundamental limits of parametric estimation theory, the classic Cramér-Rao Bound, to show how the incorporation of information allows the estimates to decouple from each other to some extent, leading to precision levels comparable of those of separate inference of one quantity with knowledge of the other. Such behavior emerges for a wide range of observational configurations and objects. For the second part of the thesis, previous work on implicit estimators is extended as a new mathematical framework to allow bounding the momenta of multidimensional estimators defined implicitly through some optimization problem. Different flavors of the Weighted Least-Squares estimator allow us to validate these mathematical tools, which are then employed to show that the Maximum Likelihood estimator approaches the fundamental precision limits tightly and consistently. Moreover, potential use of these mathematical tools as a mean for validation of the implementation of an estimation algorithm is explored.