  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.
