Minimum variance estimation for the sparse signal in noise model

We consider estimation of a sparse parameter vector from measurements corrupted by white Gaussian noise. Using the framework of reproducing kernel Hilbert spaces, we derive closed-form expressions of the Barankin bound, i.e., of the minimum locally achievable variance of any estimator with a prescribed bias function, including the unbiased case. We also derive the locally minimum variance (LMV) estimator that achieves the minimum variance, and a necessary and sufficient condition on the prescribed bias function for the existence of finite-variance estimators and, simultaneously, of the LMV estimator. Finally, we present a numerical comparison of the variance of the hard-thresholding estimator with the corresponding minimum achievable variance.