The Existence Question for Maximum-Likelihood Estimators in Time-of-Arrival-Based Localization

Abstract

We investigate the existence question for maximum-likelihood estimation for a time-of-arrival-based localization problem where the signal transmission time is unknown. We have previously shown that for some inputs, the maximum-likelihood estimator (MLE) does not exist. Here we go further: under a genericity condition, we prove that there is nonzero probability that the MLE does not exist. Besides this theoretical result, we draw attention to practical issues with finding the estimator using local optimization techniques.