Statistics of the MLE and Approximate Upper and Lower Bounds - Part 1: Application to TOA Estimation

TL;DR

This paper introduces new methods to approximate the mean-squared-error and bounds of the maximum likelihood estimator in nonlinear parameter estimation, especially at low to medium SNRs, with application to TOA estimation.

Contribution

It proposes novel MSE approximations and bounds using the method of interval estimation and Taylor series expansion, improving evaluation of MLE performance in challenging SNR regimes.

Findings

Proposed MSE approximations closely match empirical results.

Derived tight approximate bounds for MLE performance.

Validated methods through time-of-arrival estimation example.

Abstract

In nonlinear deterministic parameter estimation, the maximum likelihood estimator (MLE) is unable to attain the Cramer-Rao lower bound at low and medium signal-to-noise ratios (SNR) due the threshold and ambiguity phenomena. In order to evaluate the achieved mean-squared-error (MSE) at those SNR levels, we propose new MSE approximations (MSEA) and an approximate upper bound by using the method of interval estimation (MIE). The mean and the distribution of the MLE are approximated as well. The MIE consists in splitting the a priori domain of the unknown parameter into intervals and computing the statistics of the estimator in each interval. Also, we derive an approximate lower bound (ALB) based on the Taylor series expansion of noise and an ALB family by employing the binary detection principle. The accurateness of the proposed MSEAs and the tightness of the derived approximate bounds are validated by considering the example of time-of-arrival estimation.