Analysis and Comparison of Biased Affine Estimators

Affine biased estimation is particularly useful when there is some a-priori knowledge on the parameters that can be exploited in adverse situations (when the number of samples is low, or the noise is high). Three different affine estimation strategies are discussed, namely the Deepest Minimum Criterion (DMC), the Min-Max (MM), and the Linear Matrix Inequality (LMI) strategies, and closed form expressions are obtained for all of them, for the case when the a priori knowledge is given in the form of ellipsoidal constraints on the parameter space, and when the covariance matrix of the unbiased estimator is constant. A relationship between affine estimation and Bayesian estimation of the mean of a multivariate Gaussian distribution with Gaussian prior is established and it is shown how affine estimation theory can help in the choice of the Gaussian prior distribution.