Necessary conditions for a maximum likelihood estimate to become asymptotically unbiased and attain the Cramer-Rao lower bound. Part I. General approach with an application to time-delay and Doppler shift estimation

Analytic expressions for the first order bias and second order covariance o f a general maximum likelihood estimate (MLE) are presented. These expressi ons are used to determine general analytic conditions on sample size, or si gnal-to-noise ratio (SNR), that are necessary for a MLE to become asymptoti cally unbiased and attain minimum. variance as expressed by the Cramer-Rao lower bound (CRLB). The expressions are then evaluated for multivariate Gau ssian data. The results can be used to determine asymptotic biases, varianc es, and conditions for estimator optimality in a wide range of inverse prob lems encountered in ocean acoustics and many other disciplines. The results are then applied to rigorously determine conditions on SNR necessary for t he MLE to become unbiased and attain minimum variance in the classical acti ve sonar and radar time-delay and Doppler-shift estimation problems. The ti me-delay MLE is the time lag at the peak value of a matched filter output. It is shown that the matched filter estimate attains the CRLB for the signa l's position when the SNR is much larger than the kurtosis of the expected signal's energy spectrum. The Doppler-shift MLE exhibits dual behavior for narrow band analytic signals. In a companion paper, the general theory pres ented here is applied to the problem of estimating the range and depth of a n acoustic source submerged in an ocean waveguide. (C) 2001 Acoustical Soci ety of America.