  The constrained Cramer-Rao bound (CCRB) is a lower bound on the
mean-squared-error (MSE) of estimators that satisfy some unbiasedness
conditions. Although the CCRB unbiasedness conditions are satisfied
asymptotically by the constrained maximum likelihood (CML) estimator, in the
non-asymptotic region these conditions are usually too strict and the
commonly-used estimators, such as the CML estimator, do not satisfy them.
Therefore, the CCRB may not be a lower bound on the MSE matrix of such
estimators. In this paper, we propose a new definition for unbiasedness under
constraints, denoted by C-unbiasedness, which is based on using
Lehmann-unbiasedness with a weighted MSE (WMSE) risk and taking into account
the parametric constraints. In addition to C-unbiasedness, a Cramer-Rao-type
bound on the WMSE of C-unbiased estimators, denoted as Lehmann-unbiased CCRB
(LU-CCRB), is derived. This bound is a scalar bound that depends on the chosen
weighted combination of estimation errors. It is shown that C-unbiasedness is
less restrictive than the CCRB unbiasedness conditions. Thus, the set of
estimators that satisfy the CCRB unbiasedness conditions is a subset of the set
of C-unbiased estimators and the proposed LU-CCRB may be an informative lower
bound in cases where the corresponding CCRB is not. In the simulations, we
examine linear and nonlinear estimation problems under nonlinear constraints in
which the CML estimator is shown to be C-unbiased and the LU-CCRB is an
informative lower bound on the WMSE, while the corresponding CCRB on the WMSE
is not a lower bound and is not informative in the non-asymptotic region.
