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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


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.


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Roy V. Erickson. Vaclav Fabian. Jan Marik. "An Optimum Design for Estimating the First Derivative." Ann. Statist. 23 (4) 1234 - 1247, August, 1995.


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

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

Primary: 62K05
Secondary: 15A15 , 62L20

Keywords: Chebyshev polynomials of second kind , determinants , linear independence , orthogonal polynomials , stochastic approximation

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 4 • August, 1995
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