We show that the balanced crossover designs given by Patterson [Biometrika 39 (1952) 32–48] are (a) universally optimal (UO) for the joint estimation of direct and residual effects when the competing class is the class of connected binary designs and (b) UO for the estimation of direct (residual) effects when the competing class of designs is the class of connected designs (which includes the connected binary designs) in which no treatment is given to the same subject in consecutive periods. In both results, the formulation of UO is as given by Shah and Sinha [Unpublished manuscript (2002)].
Further, we introduce a functional of practical interest, involving both direct and residual effects, and establish (c) optimality of Patterson’s designs with respect to this functional when the class of competing designs is as in (b) above.
"Universal optimality of Patterson’s crossover designs." Ann. Statist. 33 (6) 2854 - 2872, December 2005. https://doi.org/10.1214/009053605000000723