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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.

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