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By A. Dold, B. Eckmann

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Extra info for Seminaire de Probabilites. Universite de Strasbourg, Novembre 1966 - Fevrier 1967

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1. Many of the results that follow in this and the subsequent chapters hinge on these approaches. 3 Optimality of Balanced Uniform Designs In this section, results on the optimality of balanced uniform designs are presented following Hedayat and Afsarinejad (1978), Cheng and Wu (1980), Kunert (1984b) and Hedayat and Yang (2003, 2004). As remarked in the preceding section, for an arbitrary d ∈ Ωt,n,p , g-inverses of Cd11 and Cd22 are intractable and so obtaining Cd and C¯d can pose a problem. Hedayat and Afsarinejad (1978) overcame this difficulty by considering a subclass of uniform designs in Ωt,n=µ1 t,p=t .

Zdss = λ2 , 1 ≤ s, s ≤ t. It is clear from the above definitions that a necessary condition for a design d ∈ Ωt,n,p to be uniform on periods is that t|n and a necessary condition for a design to be uniform on subjects is that t|p, where for positive integers a, b, a|b means that b is divisible by a. Thus, for a uniform design d ∈ Ωt,n,p , n = µ1 t and p = µ2 t for some integers µ1 , µ2 ≥ 1. Furthermore, a necessary condition for a design d ∈ Ωt,n,p to be balanced is that t(t−1)|n(p−1) and a necessary condition for d to be strongly balanced is that t2 |n(p − 1).

1 simply means that d∗ is also strongly balanced. In contrast, the following two results by Kunert (1983) demonstrate that non-uniform designs which are neither balanced nor strongly balanced can be universally optimal over Ωt,n,p . 2. Suppose t divides p but not n and let there be a GYD d∗ ∈ Ωt,n,p such that p zd∗ ss = n−1 md∗ si m ¯ d∗ s i , 1 ≤ s, s ≤ t. i=1 Then d∗ is universally optimal for the estimation of direct effects over Ωt,n,p . Proof. 1, we have Cd ≤ Td pr⊥ (P )Td and equality holds if and only if the following orthogonality condition is satisfied: Td pr⊥ (P )([U Fd ]) = 0.

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