## The Annals of Statistics

### Efficient estimation of integral functionals of a density

Béatrice Laurent

#### Abstract

We consider the problem of estimating a functional of a density of the type $\int \phi (f, \cdot)$. Starting from efficient estimators of linear and quadratic functionals of f and using a Taylor expansion of $\phi$, we build estimators that achieve the $n^{-1/2}$ rate whenever f is smooth enough. Moreover, we show that these estimators are efficient. Concerning the estimation of quadratic functionals (more precisely, of integrated squared density) Bickel and Ritov have already built efficient estimators. We propose here an alternative construction based on projections, which seems more natural.

#### Article information

Source
Ann. Statist., Volume 24, Number 2 (1996), 659-681.

Dates
First available in Project Euclid: 24 September 2002

https://projecteuclid.org/euclid.aos/1032894458

Digital Object Identifier
doi:10.1214/aos/1032894458

Mathematical Reviews number (MathSciNet)
MR1394981

Zentralblatt MATH identifier
0859.62038

Subjects
Primary: 62G06 62G07: Density estimation 62G20: Asymptotic properties

#### Citation

Laurent, Béatrice. Efficient estimation of integral functionals of a density. Ann. Statist. 24 (1996), no. 2, 659--681. doi:10.1214/aos/1032894458. https://projecteuclid.org/euclid.aos/1032894458

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