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 T of observations tends to infinity is a well known result. In some scenarii, the number of snapshots may be small making this result unapplicable. 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) non efficient and non Gaussian.