AN OPTIMUM DESIGN FOR ESTIMATING THE FIRST DERIVATIVE

被引：2

作者：

ERICKSON, RV ^{[1
]}

FABIAN, V ^{[1
]}

MARIK, J ^{[1
]}

机构：

[1] MICHIGAN STATE UNIV,DEPT MATH,E LANSING,MI 48824

来源：

ANNALS OF STATISTICS | 1995年 / 23卷 / 04期

关键词：

STOCHASTIC APPROXIMATION; DETERMINANTS; LINEAR INDEPENDENCE; ORTHOGONAL POLYNOMIALS; CHEBYSHEV POLYNOMIALS OF THE 2ND-KIND;

D O I：

10.1214/aos/1176324707

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

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.

引用

页码：1234 / 1247

页数：14