The Annals of Statistics

Interpolating Spline Methods for Density Estimation I. Equi-Spaced Knots

Grace Wahba

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Abstract

Statistical properties of a variant of the histospline density estimate introduced by Boneva-Kendall-Stefanov are obtained. The estimate we study is formed for $x$ in a finite interval, $x\in\lbrack a, b\rbrack = \lbrack 0, 1\rbrack$ say, by letting $\hat{F}_n(x), x \epsilon\lbrack 0, 1\rbrack$ be the unique cubic spline of interpolation to the sample cumulative distribution function $F_n(x)$ at equi-spaced points $x = jh, j = 0, 1,\cdots, l + 1, (l + 1)h = 1$, which satisfies specified boundary conditions $\hat{F}_n'(0) = a, \hat{F}_n'(1) = b$. The density estimate $\hat{f}_n(x)$ is then $\hat{f}_n(x) = d/dx\hat{F}_n(x)$. It is shown how to estimate $a$ and $b$. A formula for the optimum $h$ is given. Suppose $f$ has its support on [0, 1] and $f^{(m)}\in\mathscr{L}_p\lbrack 0, 1\rbrack$. Then, for $m = 1,2,3$ and certain values of $p$, it is shown that $E(f_n(x) - f(x))^2 = O(n^{-(2m - 2/p)/(2m+1-2/p)}).$ Bounds for the constant covered by the "$O$" are given. An extension to the $\mathscr{L}_p$ case of known convergence properties of the derivative of an interpolating spline is found, as part of the proofs.

Article information

Source
Ann. Statist., Volume 3, Number 1 (1975), 30-48.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342998

Digital Object Identifier
doi:10.1214/aos/1176342998

Mathematical Reviews number (MathSciNet)
MR370906

Zentralblatt MATH identifier
0305.62022

JSTOR
links.jstor.org

Keywords
Spline histospline density estimate optimal convergence rates

Citation

Wahba, Grace. Interpolating Spline Methods for Density Estimation I. Equi-Spaced Knots. Ann. Statist. 3 (1975), no. 1, 30--48. doi:10.1214/aos/1176342998. https://projecteuclid.org/euclid.aos/1176342998


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