Minimum-variance Pseudo-unbiased Reduced-rank Estimator (mv-pure) and Its Applications to Ill-conditioned Inverse Problems

This paper presents mathematically novel estimator for the linear regression model named Minimum-Variance Pseudo-Unbiased Reduced-Rank Estimator (MV-PURE), designed specially for applications where the model matrix under consideration is ill-conditioned, and auxiliary knowledge on the unknown deterministic parameter vector is available in the form of linear constraints. We demonstrate closed algebraic form of MV-PURE estimator and provide a numerical example of its application, where we employ our estimator to the ill-conditioned problem of reconstructing a 2-D image subjected to linear constraints from blurred, noisy observation. It is shown that MVPURE estimator achieves much smaller MSE for all values of SNR not only than the minimum-variance unbiased Gauss-Markov (BLUE) estimator, but also than the minimum-variance conditionally unbiased affine estimator subject to linear restrictions and the recently introduced generalized Marquardt’s reduced-rank estimator. In particular, it will be shown that all of the aforementioned estimators are particular cases of MV-PURE estimator, if the rank constraint on estimator and/or the linear constraints on the unknown deterministic vector of parameters are not imposed.