On shrinking minimax convergence in nonparametric statistics

摘要

'...if we are prepared to assume that the unknown density has k derivatives, then ... the optimal mean integrated squared error is of order n~(-2k/(2k+1)) ...' The citation is from Silverman [(1986), Density Estimation for Statistics and Data Analysis, London: Chapman & Hall] and its assertion is based on a classical minimax lower bound which is the pillar of the modern nonparametric statistics. This paper proposes a new minimax methodology that implies a faster decreasing minimax lower bound that is attainable by a data-driven estimator, and the same estimator is also minimax under the classical approach. The recommendation is to test performance of estimators via the new and classical minimax approaches.