An Optimum Design for Estimating the First Derivative

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 Γ 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 Π|xi| = 1. (In the determinants, 1 is the column vector with all components 1, and xihas components of x raised to the i-th power.) The minimum of Γ is shown to be m, and the point at which the minimum is attained is characterized by Chebyshev polynomials of the second kind.