## The Annals of Statistics

### Remarks on Some Recursive Estimators of a Probability Density

#### Abstract

The density estimator, $f^\ast_n(x) = n^{-1}\sum^n_{j = 1}h^{-1}_jK((x - X_j)/h_j)$, as well as the closely related one $f^\dagger_n(x) = n^{-1}h_n^{-\frac{1}{2}}\sum^n_{j = 1}h_j^{-\frac{1}{2}}K((x - X_j)/h_j)$ are considered. Expressions for asymptotic bias and variance are developed. Using the almost sure invariance principle, laws of the iterated logarithm are developed. Finally, illustration of these results with sequential estimation procedures are made.

#### Article information

Source
Ann. Statist., Volume 7, Number 2 (1979), 316-327.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176344616

Digital Object Identifier
doi:10.1214/aos/1176344616

Mathematical Reviews number (MathSciNet)
MR520242

Zentralblatt MATH identifier
0405.62031

JSTOR