New observations on efficiency of variance estimation of white Gaussian signal with unknown mean

The uniformly minimum variance unbiased estimator (UMVUE) for mean and variance of white Gaussian noise is known to be not efficient. This is due to the fact that according to the Cramér-Rao bound (CRB), no coupling exists between mean and variance of Gaussian observations, while it is clear that knowledge or lack of knowledge of the mean has impact on estimation of the variance. In this work, we consider the problem of variance estimation in the presence of unknown mean of white Gaussian signals, where the unknown mean is considered to be a nuisance parameter. For this purpose, a Cramér-Rao-type bound on the mean-squared-error (MSE) of non-Bayesian estimators, which has been recently introduced, is analyzed. This bound considers no unbiasedness condition on the nuisance parameters. Alternatively, Lehmann's concept of unbiasedness is imposed for a risk that measures the distance between the estimator and the locally best unbiased estimator, which assumes perfect knowledge of the model parameters. It is analytically shown that the MSE of the well-known UMVUE coincides with the proposed risk-unbiased CRB, and therefore it is called risk-efficient estimator.