Open Access
June 1997 Estimation of integral functionals of a density and its derivatives
Béatrice Laurent
Bernoulli 3(2): 181-211 (June 1997).


We consider the problem of estimating a functional of a density of the type φ(f,f ,...,f ( k),.) . The estimation of φ(f,.) has already been studied by the author: starting from efficient estimators of linear and quadratic functionals of f and its derivatives and using a Taylor expansion of φ , we construct estimators which achieve the n - 1/2 rate whenever f is smooth enough. Moreover, we show that these estimators are efficient. We also obtain the optimal rate of convergence when the n - 1/2 rate is not achievable and when k >0 . Concerning the estimation of quadratic functionals, more precisely of integrated squared density derivatives, Bickel and Ritov have already constructed efficient estimators. Here we propose an alternative construction based on projections, an approach which seems more natural.


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Béatrice Laurent. "Estimation of integral functionals of a density and its derivatives." Bernoulli 3 (2) 181 - 211, June 1997.


Published: June 1997
First available in Project Euclid: 25 April 2007

zbMATH: 0872.62044
MathSciNet: MR1466306

Keywords: estimation of a density and its derivatives , Fourier series , kernel estimators , projection methods , semi-parametric Cramér-Rao bound

Rights: Copyright © 1997 Bernoulli Society for Mathematical Statistics and Probability

Vol.3 • No. 2 • June 1997
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