#### Vol. 12, No. 1, 2021

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The semialgebraic geometry of saturated optimal designs for the Bradley–Terry model

### Thomas Kahle, Frank Röttger and Rainer Schwabe

Vol. 12 (2021), No. 1, 97–114
##### Abstract

Optimal design theory for nonlinear regression studies local optimality on a given design space. We identify designs for the Bradley–Terry paired comparison model with small undirected graphs and prove that every saturated, locally $D$-optimal design is represented by a path. We discuss the case of four alternatives in detail and derive explicit polynomial inequality descriptions for optimality regions in parameter space. Using these regions, for each point in parameter space we can prescribe a locally $D$-optimal design.

##### Keywords
nonlinear regression, optimal design, polynomial inequalities
##### Mathematical Subject Classification
Primary: 62K05, 62R01
Secondary: 13P25, 14P10, 62J02