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barycenter of depth bound is attained bound on efficiency chapter choose X(g coefficients components compute consider constant convex coordinates corollary covariance matrix D-efficiency D-optimal defined denote design for estimating design g design is supported design is uniform det2 g(xM dg(x diagonal matrix eigenvalue equality holds estimating 9 Euclidean norm fixed following Theorem g is optimal gives implies Jl Jn-1 k-point design Karlin and Studden Kiefer 196I lemma Let g lim max lower bound lower right entries matrices equal matrix with entries max d x max lim maximizes maxx maxx D(x minimizes max monomials of degree nonsingular matrix number of points obtain optimal g p=l p permutations points are needed points of support polynomial positive definite pseudo-inverse q-simplex quadratic rc]l regression functions result Setting the upper simplex Theorem 3.1 Theorem of Corresponding unique optimal design variable vector write X s matrix Ya+f Ya+h)2T