## The Annals of Statistics

### On Improving Convergence Rates for Nonnegative Kernel Density Estimators

#### Abstract

To improve the rate of decrease of integrated mean square error for nonparametric kernel density estimators beyond $0(n^{-\frac{4}{5}}),$ we must relax the constraint that the density estimate be a bonafide density function, that is, be nonnegative and integrate to one. All current methods for kernel (and orthogonal series) estimators relax the nonnegativity constraint. In this paper we show how to achieve similar improvement by relaxing the integral constraint only. This is important in applications involving hazard function and likelihood ratios where negative density estimates are awkward to handle.

#### Article information

Source
Ann. Statist., Volume 8, Number 5 (1980), 1160-1163.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345153

Digital Object Identifier
doi:10.1214/aos/1176345153

Mathematical Reviews number (MathSciNet)
MR585714

Zentralblatt MATH identifier
0459.62031

JSTOR