On Nonparametric Estimation of the Fisher Information

This paper considers a problem of estimation of the Fisher information for location from a random sample of size n. First, an estimator proposed by Bhattacharya is revisited and improved convergence rates are derived. Second, a new estimator, termed clipped estimator, is proposed. The new estimator is shown to have superior rates of convergence as compared to the Bhattacharya estimator, albeit with different regularity conditions. Third, both of the estimators are evaluated for the practically relevant case of a random variable contaminated by Gaussian noise. Moreover, using Brown's identity, which relates the Fisher information to the minimum mean squared error (MMSE) in Gaussian noise, a consistent estimator for the MMSE is proposed.