Optimality of partial geometric designs

The Annals of Statistics · 2001 · Volume 29 · Issue 2

Abstract

We find a sufficient condition on the spectrum of a partial geometric design d* such that, when d* satisfies this condition, it is better (with respect to all convex decreasing optimality criteria) than all unequally replicated designs (binary or not) with the same parameters b, v, k as d*. Combining this with existing results, we obtain the following results: (i) For any q \ge 3, a linked block design with parameters b = q2, v = q2 + q, k = q2 -1 is optimal with respect to all convex decreasing optimality criteria in the unrestricted class of all connected designs with the same parameters. (ii) A large class of strongly regular graph designs are optimal w.r.t. all type 1 optimality criteria in the class of all binary designs (with the given parameters). For instance, all connected singular group divisible (GD) designs with \lambda_1 = \lambda_2 + 1 (with one possible exception) and many semiregular GD designs satisfy this optimality property. Specializing these general ideas to the Acriterion, we find a large class of linked block designs which are Aoptimal in the unrestricted class. We find an even larger class of regular partial geometric designs (including, for instance, the complements of a large number of partial geometries) which are Aoptimal among all binary designs.

Keywords

15A42, 62K05, majorization, Optimal block designs

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