Nonparametric Estimation of the Fisher Information and Its Applications
This work addresses estimation challenges in statistical inference, offering incremental improvements for applications like minimum mean squared error estimation in Gaussian noise.
The paper tackles the problem of estimating Fisher information for location from a random sample, proposing a new clipped estimator that improves convergence rates and reduces sample size requirements compared to an existing method, with simulations showing significant reductions in sample size for specific confidence intervals.
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.