A new lower bound on the mean-square error of biased estimators

In this paper, the class of lower bounds on the MSE of unbiased estimators, derived in our previous work, is extended to the case of biased estimation. The proposed class is derived by projecting the estimation error on a Hilbert subspace of L2, which contains linear transformations of elements in the domain of an integral transform of the likelihood-ratio function. It is shown that some well known bounds can be derived from the proposed class by modifying the kernel of the integral transform. By decomposing the projection of the estimation error into bias-independent and bias-dependent components, the proposed class is minimized with respect to the bias function subject to a bounded L2-norm of the bias-dependent component. A new computationally manageable bound is derived from the proposed class using the kernel of the weighted Fourier transform. The bound is applied for exploring the bias-variance tradeoff in the problem of direction-of-arrival estimation.