Use of Karush-Kuhn-Tucker conditions for obtaining a closed form expression for the Minimax affine estimator

FERNANDO GAMA; DANIEL CASAGLIA; BRUNO CERNUSCHI FRÍAS

Bariloche

Congreso; XV Reunión de Trabajo en Procesamiento de la Información y Control, RPIC 2013; 2013

Universidad Nacional de Rio Negro

The problem of parameter estimationunder ellipsoidal constraints is addressed in this paper.Affine estimation is analyzed as a method forimproving the mean squared error (MSE) over unbiasedestimators. In particular, the MiniMax AffineEstimation approach is adopted. For the special casewhen the covariance matrix of the unbiased estimatoris constant (i.e. it does not depend on the unknownparameter) a closed form solution for the MiniMaxaffine estimation problem is obtained. The equivalentconvex problem for the MiniMax affine estimationproblem is used to explicitly derive the KKT conditions.Then, the solution proposed is proved tobe optimal through the use of this KKT conditionswhich are necessary and sufficient because the problemis convex. Finally, as an illustrative example, thisclosed form expression for the MiniMax affine estimatoris applied to the problem of parameter estimationof multiple damped complex exponentials in additivenoise. It is observed, through simulations, thatthe affine estimator presents a better performance, interms of the MSE, than the unbiased estimator.