an:00843015
Zbl 0838.62055
Erickson, Roy V.; Fabian, Václav; Mařík, Jan
An optimum design for estimating the first derivative
EN
Ann. Stat. 23, No. 4, 1234-1247 (1995).
0090-5364 2168-8966
1995
j
62K05 62L20 15A15
determinants; linear independence; orthogonal polynomials; density estimation; Chebyshev polynomials of the second kind
Summary: 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
\[
\Gamma (x)= \det [1, x^3, \dots, x^{2m-1}]/ \det [x, x^3,\dots, x^{2m-1}]
\]
on the set of all \(m\)-dimensional vectors with components satisfying \(0< x_1< -x_2< \cdots< (-1)^{m-1} x_m\) and \(\prod |x_i|=1\). (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.