Non efficiency and non Gaussianity of a maximum likelihood estimator at high signal-to-noise ratio and finite number of samples

In estimation theory, the asymptotic efficiency of the maximum likelihood (ML) method for independent identically distributed observations and when the number of observations, T, tends to infinity is a well known result. In some scenarios, the number of snapshots may be small, making this result inapplicable. In the array processing framework, for Gaussian emitted signals, we fill this lack at high signal-to-noise ratio (SNR). In this situation, we show that the ML estimation is asymptotically (with respect to SNR) inefficient and non Gaussian.