Estimating integrated squared density derivatives: Sharp best order of convergence estimates.(English)Zbl 0676.62037

Summary: Estimation of the integral of the square of a derivative of the probability density function is considered. The estimators we propose and their properties are a function of the amount of smoothness assumed. The rate of convergence of the appropriate estimator is shown to be optimal given the amount of smoothness assumed. In particular, the appropriate estimator achieves the information bound when estimation at an $$n^{- 1/2}$$ rate is possible.

MSC:

 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference