Translator Disclaimer
August, 1995 An Optimum Design for Estimating the First Derivative
Roy V. Erickson, Vaclav Fabian, Jan Marik
Ann. Statist. 23(4): 1234-1247 (August, 1995). DOI: 10.1214/aos/1176324707

Abstract

An optimum design of experiment for a class of estimates of the first derivative at 0 (used in stochastic approximation and density estimation) is shown to be equivalent to the problem of finding a point of minimum of the function $\Gamma$ defined by $\Gamma (x) = \det\lbrack 1, x^3,\ldots, x^{2m-1} \rbrack/\det\lbrack x, x^3,\ldots, x^{2m-1} \rbrack$ on the set of all $m$-dimensional vectors with components satisfying $0 < x_1 < -x_2 < \cdots < (-1)^{m-1} x_m$ and $\Pi|x_i| = 1$. (In the determinants, 1 is the column vector with all components 1, and $x^i$ has components of $x$ raised to the $i$-th power.) The minimum of $\Gamma$ is shown to be $m$, and the point at which the minimum is attained is characterized by Chebyshev polynomials of the second kind.

Citation

Download Citation

Roy V. Erickson. Vaclav Fabian. Jan Marik. "An Optimum Design for Estimating the First Derivative." Ann. Statist. 23 (4) 1234 - 1247, August, 1995. https://doi.org/10.1214/aos/1176324707

Information

Published: August, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0838.62055
MathSciNet: MR1353504
Digital Object Identifier: 10.1214/aos/1176324707

Subjects:
Primary: 62K05
Secondary: 15A15, 62L20

Rights: Copyright © 1995 Institute of Mathematical Statistics

JOURNAL ARTICLE
14 PAGES


SHARE
Vol.23 • No. 4 • August, 1995
Back to Top