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.

翻译：在约束参数估计中，经典约束克拉美-罗下界（CCRB）与近期提出的莱曼无偏CCRB（LU-CCRB）分别对应均值无偏估计器和莱曼无偏估计器的性能下界。然而，CCRB与LU-CCRB均要求似然函数可微，这一假设具有局限性。此外，这些局部下界无法准确预测约束最大似然（CML）估计器的阈值现象。约束巴兰金型下界（CBTB）作为约束参数估计的非局部均方误差（MSE）下界，无需似然函数可微性，但其在约束集内要求严格的均值无偏性条件。本文针对加权均方误差（WMSE）提出莱曼无偏CBTB（LU-CBTB）。该下界无需似然函数可微，且采用限制性更弱的均匀莱曼无偏性假设（相较于CBTB的均匀均值无偏性）。我们证明：LU-CBTB紧于或等于LU-CCRB，且在线性约束下与CBTB等价；对于非线性约束，LU-CBTB与CBTB存在差异，并且LU-CBTB可作为约束估计器WMSE的下界（在CBTB失效的场景中有效）。仿真实验中，我们考虑方向角估计未知恒定模量离散信号的问题。该情形下似然函数不可微，约束克拉美-罗型下界不存在而CBTB存在。实验表明：由于CML估计器具有莱曼无偏性而非均值无偏性，LU-CBTB对CML估计器性能的预测优于CBTB。