  This paper considers the 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 a clipped estimator, is proposed. Superior upper bounds
on the rates of convergence can be shown for the new estimator 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 and the minimum mean squared
error (MMSE) in Gaussian noise, two corresponding consistent estimators for the
MMSE are proposed. Simulation examples for the Bhattacharya estimator and the
clipped estimator as well as the MMSE estimators are presented. The examples
demonstrate that the clipped estimator can significantly reduce the required
sample size to guarantee a specific confidence interval compared to the
Bhattacharya estimator.
