Efficiency. of infinite dimensional M‐ estimators

It is well‐known that maximum likelihood estimators are asymptotically normal with covariance equal to the inverse Fisher information in smooth, finite dimensional parametric models. Thus they are asymptotically efficient. A similar phenomenon has been observed for certain infinite dimensional parameter spaces. We give a simple proof of efficiency, starting from a theorem on asymptotic normality of infinite dimensional M ‐estimators. The proof avoids the explicit calculation of the Fisher information. We also address Hadamard differentiability of the corresponding M ‐functionals.