The Non-Uniqueness of Some Designs for Discriminating Between Two Polynomial Models in One Variable Anthony C. Atkinson Abstract T-optimum designs for discriminating between two nested polynomial re- gression models in one variable that differ in the presence or absence of the two highest order terms are studied as a function of the values of the parameters of the true model. For the value of the parameters corresponding to the absence of the next-highest order term, the optimum designs are not unique and can contain an additional support point. A numerical exploration of the non-uniqueness reveals a connection with D s -optimality for models which do contain the next highest term. Brief comments are given on the analysis of data from such designs 1 Introduction T-optimum designs for discriminating between two regression models were intro- duced by Atkinson and Fedorov (1975). More recently, Dette and Titoff (2008) ex- plored the structure of T-optimum designs in some detail. One of their examples was of discrimination between linear and cubic models in one variable. For particular parameter values the T-optimum design was not unique, consisting of convex com- binations of two extreme designs. This example can be thought of as an extension of Example 1 of Atkinson and Fedorov in which designs were found for discrimination between a constant and a general quadratic. The paper illustrates how the designs depend upon the parameters of the true model and gives a geometric interpretation of the occurrence of non-unique designs as a function of the response. The non-unique designs occur when the larger model contains a term of order x k and all lower order terms except that of order x k−1 , the smaller model containing terms up to order x k−2 . The structure of these non-unique designs is explored nu- merically for k in the range two to six. A relationship is indicated with D s -optimum designs for the estimation of the coefficient of x k in a polynomial model which adds a term in x k−1 to those of the larger model. Prof. Anthony C. Atkinson London School of Economics, London WC2A 2AE, UK, e-mail: a.c.atkinson@lse.ac.uk A. Giovagnoli et al. (eds.), C. May (co-editor), mODa 9 – Advances in Model-Oriented 9 Design and Analysis, Contributions to Statistics, DOI 10.1007/978-3-7908-2410-0 2, c  Springer-Verlag Berlin Heidelberg 2010