  In constrained parameter estimation, the classical constrained Cramer-Rao
bound (CCRB) and the recent Lehmann-unbiased CCRB (LU-CCRB) are lower bounds on
the performance of mean-unbiased and Lehmann-unbiased estimators, respectively.
Both the CCRB and the LU-CCRB require differentiability of the likelihood
function, which can be a restrictive assumption. Additionally, these bounds are
local bounds that are inappropriate for predicting the threshold phenomena of
the constrained maximum likelihood (CML) estimator. The constrained
Barankin-type bound (CBTB) is a nonlocal mean-squared-error (MSE) lower bound
for constrained parameter estimation that does not require differentiability of
the likelihood function. However, this bound requires a restrictive
mean-unbiasedness condition in the constrained set. In this work, we propose
the Lehmann-unbiased CBTB (LU-CBTB) on the weighted MSE (WMSE). This bound does
not require differentiability of the likelihood function and assumes uniform
Lehmann-unbiasedness, which is less restrictive than the CBTB uniform
mean-unbiasedness. We show that the LU-CBTB is tighter than or equal to the
LU-CCRB and coincides with the CBTB for linear constraints. For nonlinear
constraints the LU-CBTB and the CBTB are different and the LU-CBTB can be a
lower bound on the WMSE of constrained estimators in cases, where the CBTB is
not. In the simulations, we consider direction-of-arrival estimation of an
unknown constant modulus discrete signal. In this case, the likelihood function
is not differentiable and constrained Cramer-Rao-type bounds do not exist,
while CBTBs exist. It is shown that the LU-CBTB better predicts the CML
estimator performance than the CBTB, since the CML estimator is
Lehmann-unbiased but not mean-unbiased.
