A Proof of Strong Consistency of Maximum Likelihood Estimator for Independent Non-Identically Distributed Data
Ricardo Ferreira, Filipa Valdeira, Marta Guimarães, Cláudia Soares
TL;DR
The paper proves the strong consistency of the maximum likelihood estimator for independent non-identically distributed data under a set of regularity assumptions, building on the foundational work of Wald, Goel, and Ferguson. It then specialized this general result to an orbit-determination problem with a network of radars measuring range, angle, and Doppler, modeling noise with Gaussian and von Mises-Fisher distributions and showing identifiability of the state vector. By carefully verifying compactness, identifiability, measurability, and drift/variance conditions, the authors demonstrate that the orbit-determination MLE satisfies the strong-consistency criteria. This provides a theoretical guarantee for the convergence of the orbital state estimates derived from i.n.i.d. radar data in practical tracking scenarios.
Abstract
We give a general proof of the strong consistency of the Maximum Likelihood Estimator for the case of independent non-identically distributed (i.n.i.d) data, assuming that the density functions of the random variables follow a particular set of assumptions. Our proof is based on the works of Wald~\cite{wald1949note}, Goel~\cite{goel1974note}, and Ferguson~\cite{ferguson2017course}. We use this result to prove the strong consistency of a Maximum Likelihood Estimator for Orbit Determination.