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An optimum design for estimating the first derivative. (English) Zbl 0838.62055
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.

62K05 Optimal statistical designs
62L20 Stochastic approximation
15A15 Determinants, permanents, traces, other special matrix functions
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