The Annals of Statistics

Remarks on Some Recursive Estimators of a Probability Density

Edward J. Wegman and H. I. Davies

Full-text: Open access

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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62L12: Sequential estimation 60F20: Zero-one laws 60G50: Sums of independent random variables; random walks

Keywords
Recursive estimators asymptotic bias asymptotic variance weak consistency almost sure invariance principle law of the iterated logarithm strong consistency asymptotic distribution sequential procedure

Citation

Wegman, Edward J.; Davies, H. I. Remarks on Some Recursive Estimators of a Probability Density. Ann. Statist. 7 (1979), no. 2, 316--327. doi:10.1214/aos/1176344616. https://projecteuclid.org/euclid.aos/1176344616


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