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 Gamma defined by Gamma(x) = det[1, x(3), ..., x(2m-1)]/det[x, x(3), ..., x(2m-1)] on the set of all m-dimensional vectors with components satisfying 0 < x(1) < -x(2) < ... < (-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.