50,278 research outputs found
-optimal designs for second-order response surface models
-optimal experimental designs for a second-order response surface model
with predictors are investigated. If the design space is the
-dimensional unit cube, Galil and Kiefer [J. Statist. Plann. Inference 1
(1977a) 121-132] determined optimal designs in a restricted class of designs
(defined by the multiplicity of the minimal eigenvalue) and stated their
universal optimality as a conjecture. In this paper, we prove this claim and
show that these designs are in fact -optimal in the class of all approximate
designs. Moreover, if the design space is the unit ball, -optimal designs
have not been found so far and we also provide a complete solution to this
optimal design problem. The main difficulty in the construction of -optimal
designs for the second-order response surface model consists in the fact that
for the multiplicity of the minimum eigenvalue of the "optimal information
matrix" is larger than one (in contrast to the case ) and as a consequence
the corresponding optimality criterion is not differentiable at the optimal
solution. These difficulties are solved by considering nonlinear Chebyshev
approximation problems, which arise from a corresponding equivalence theorem.
The extremal polynomials which solve these Chebyshev problems are constructed
explicitly leading to a complete solution of the corresponding -optimal
design problems.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1241 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Implementing Brouwer's database of strongly regular graphs
Andries Brouwer maintains a public database of existence results for strongly
regular graphs on vertices. We implemented most of the infinite
families of graphs listed there in the open-source software Sagemath, as well
as provided constructions of the "sporadic" cases, to obtain a graph for each
set of parameters with known examples. Besides providing a convenient way to
verify these existence results from the actual graphs, it also extends the
database to higher values of .Comment: 18 pages, LaTe
Complex spherical codes with two inner products
A finite set in a complex sphere is called a complex spherical -code
if the number of inner products between two distinct vectors in is equal to
. In this paper, we characterize the tight complex spherical -codes by
doubly regular tournaments, or skew Hadamard matrices. We also give certain
maximal 2-codes relating to skew-symmetric -optimal designs. To prove them,
we show the smallest embedding dimension of a tournament into a complex sphere
by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel
matrix.Comment: 10 pages, to appear in European Journal of Combinatoric
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