  The classical unbiasedness condition utilized e.g. by the best linear
unbiased estimator (BLUE) is very stringent. By softening the "global"
unbiasedness condition and introducing component-wise conditional unbiasedness
conditions instead, the number of constraints limiting the estimator's
performance can in many cases significantly be reduced. In this work we
investigate the component-wise conditionally unbiased linear minimum mean
square error (CWCU LMMSE) estimator for different model assumptions. The
prerequisites in general differ from the ones of the LMMSE estimator. We first
derive the CWCU LMMSE estimator under the jointly Gaussian assumption of the
measurements and the parameters. Then we focus on the linear model and discuss
the CWCU LMMSE estimator for jointly Gaussian parameters, and for mutually
independent (and otherwise arbitrarily distributed) parameters, respectively.
In all these cases the CWCU LMMSE estimator incorporates the prior mean and the
prior covariance matrix of the parameter vector. For the remaining cases
optimum linear CWCU estimators exist, but they may correspond to globally
unbiased estimators that do not make use of prior statistical knowledge about
the parameters. Finally, the beneficial properties of the CWCU LMMSE estimator
are demonstrated with the help of a well-known channel estimation application.
