Unconditional Maximum Likelihood Performance at Finite Number of Samples and High Signal-to-Noise Ratio

This correspondence deals with the problem of estimating signal parameters using an array of sensors. In source localization, two main maximum-likelihood methods have been introduced: the conditional maximum-likelihood method which assumes the source signals nonrandom and the unconditional maximum-likelihood method which assumes the source signals random. Many theoretical investigations have been already conducted for the large samples statistical properties. This correspondence studies the behavior of unconditional maximum likelihood at high signal-to-noise ratio for finite samples. We first establish the equivalence between the unconditional and the conditional maximum-likelihood criterions at high signal-to-noise ratio. Then, thanks to this equivalence we prove the non-Gaussianity and the non-efficiency of the unconditional maximum-likelihood estimator. We also rediscover the closed-form expressions of the probability density function and of the variance of the estimates in the one source scenario and we derive a closed-form expression of this estimator variance in the two sources scenario.