The Symmetric Location Problem: a Song of Efficiency and Robustness

The aim of this Lecture Note is to introduce the Signal Processing (SP) community to a powerful yet still under-utilised tool: the semiparametric statistics. In short, the semiparametric framework allows us to estimate or perform hypothesis testing on a finite-dimensional parameter in the presence of an infinite-dimensional nuisance parameter (i.e. a function), such as the density of the noise. Clearly, this framework is general enough to include almost every SP application. Remarkably, as the title suggests drawing on George R. R. Martin's famous book series, the greatest advantage of semiparametric statistics over parametric and non-parametric ones lies in the fact that it is able to reconcile two seemingly dichotomous concepts: statistical efficiency and robustness. Here, robustness is understood in the sense of distribution-freeness, that is the estimation performance must be robust with respect to the lack of knowledge of the functional form of the generating data distribution. To explain exactly what this means, in this Lecture Note we will focus our attention on the famous and fundamental symmetric location problem. The symmetric location problem is a fundamental problem that can be found (in various forms) in countless areas of SP: source localization, time synchronization, array signal processing, and distributed sensor networks, just to name a few. Furthermore, it is important to note that the methodology we will develop for this specific problem can be extended to much more general semiparametric estimation problems, such as the estimation of the location vector and covariance matrix in elliptical data.