The Annals of Statistics

Optimal Convergence Properties of Variable Knot, Kernel, and Orthogonal Series Methods for Density Estimation

Grace Wahba

Full-text: Open access

Abstract

Let $W_p^{(m)}(M) = \{f: f^{(\nu)} \operatorname{abs}. \operatorname{cont}., \nu = 0, 1,\cdots, m - 1, f^{(m)} \in \mathscr{L}_p, \|f^{(m)}\|_p \leqq M\}$, where $\|\cdot\|_p$ is the norm in $\mathscr{L}_p, m$ is a positive integer and $p$ is a real number, $p \geqq 1$. Let $\{\hat{f}_n(x)\}, n = 1, 2,\cdots$ be any sequence of estimates of a density at the point $x$ where $\hat{f}_n(x)$ depends on $n$ independent observations from some density $f \in W_p^{(m)}(M)$. It is shown that if $\sup_{f\in W_p^{(m)} (M)} E_f(f(x) - \hat{f}_n(x))^2 = b_nn^{-\phi(m, p+\varepsilon)}$, where $\phi(m, p) = (2m - 2/p)/(2m + 1 - 2/p)$, and $\varepsilon > 0$, then there exists a $D_0 > 0$ such that $b_n \geqq D_0$ for infinitely many $n$. Thus the best possible mean square convergence rate for a density estimate, which is uniform over $W_p^{(m)}(M)$, is not better than $n^{-\phi(m,p+\varepsilon)}$ for arbitrarily small $\varepsilon$. The following types of density estimates are shown to have mean square error at a point bounded above by $Dn^{-\phi(m,p)}$, provided that a certain parameter, usually depending on $m, p$ and $M$, is chosen optimally: the polynomial algorithm, kernel-type estimates, certain orthogonal series estimates, and the ordinary histogram. $D$'s for each method are given.

Article information

Source
Ann. Statist. Volume 3, Number 1 (1975), 15-29.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aos/1176342997

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aos/1176342997

Mathematical Reviews number (MathSciNet)
MR362682

Zentralblatt MATH identifier
0305.62021

Citation

Wahba, Grace. Optimal Convergence Properties of Variable Knot, Kernel, and Orthogonal Series Methods for Density Estimation. The Annals of Statistics 3 (1975), no. 1, 15--29. doi:10.1214/aos/1176342997. http://projecteuclid.org/euclid.aos/1176342997.


Export citation