论文标题
最大可能性和其他隐式定义的非随机参数的估计值（扩展版本）的近似MSE表达式
An Approximate MSE Expression for Maximum Likelihood and Other Implicitly Defined Estimators of Non-Random Parameters (extended version)
论文作者
论文摘要
提出了一个近似均方根误差（MSE）表达，用于隐式定义的非随机参数的估计量的性能分析。隐式定义的估计量（IDE）将选定的成本/奖励功能的最小化器/最大化器称为参数估计。最大可能性（ML）和最小二乘估计量是该类别的众所周知的例子。在本文中，给出了具有对称和单峰目标函数的隐式定义估计量的精确MSE表达式。结果表明，在ML和错误指定的ML估计的较大样本量状态下，表达式减少到CRAMER-RAO下限（CRLB）和错误指定的CRLB。该表达式显示出在贝叶斯设置中使用时，可以为最大a后验（MAP）估计器产生Ziv-Zakai结合（无山谷填充函数），即当将A-Priori分布分配给未知参数时。此外，研究了建议的表达式扩展到滋扰参数的情况，并给出一些近似值以简化此情况的计算。数值结果表明，建议的MSE表达不仅可以预测渐近区域的估计量性能。但是，它也适用于阈值区域分析，即使对于目标函数无法满足对称性和单形式假设的IDE。建议的MSE表达的优势是其概念简单性及其相对直接的数值计算，这是由于将估计问题减少到二元假设测试问题上，类似于随机参数估计问题中ZIV-Zakai界限的使用。
An approximate mean square error (MSE) expression for the performance analysis of implicitly defined estimators of non-random parameters is proposed. An implicitly defined estimator (IDE) declares the minimizer/maximizer of a selected cost/reward function as the parameter estimate. The maximum likelihood (ML) and the least squares estimators are among the well known examples of this class. In this paper, an exact MSE expression for implicitly defined estimators with a symmetric and unimodal objective function is given. It is shown that the expression reduces to the Cramer-Rao lower bound (CRLB) and misspecified CRLB in the large sample size regime for ML and misspecified ML estimation, respectively. The expression is shown to yield the Ziv-Zakai bound (without the valley filling function) for the maximum a posteriori (MAP) estimator when it is used in a Bayesian setting, that is, when an a-priori distribution is assigned to the unknown parameter. In addition, extension of the suggested expression to the case of nuisance parameters is studied and some approximations are given to ease the computations for this case. Numerical results indicate that the suggested MSE expression not only predicts the estimator performance in the asymptotic region; but it is also applicable for the threshold region analysis, even for IDEs whose objective functions do not satisfy the symmetry and unimodality assumptions. Advantages of the suggested MSE expression are its conceptual simplicity and its relatively straightforward numerical calculation due to the reduction of the estimation problem to a binary hypothesis testing problem, similar to the usage of Ziv-Zakai bounds in random parameter estimation problems.
