“Replication-free” Optimal Designs in Regression Analysis

Let $$\matrix{ {{y_i} = f\left( {{x_i},\theta } \right) + {e_i},} & {i = 1,...,n,} & {{x_i} \in B \subset {{\rm{R}}^1}} \cr } $$ be a regression model with a regression function f and i.i.d. error terms e _i . The unknown parameter 8 may possess p ≤n components i.e. θT = (θ₁,…θ _p ) ∈ Ω ⊂Rp. We assume that the usual condition for the asymptotic least squares theory are fulfilled (see Rasch, 1995, chapter 16). By ^θ we denote the least squares estimator of θ and by V the (asymptotic) covariance matrix of ^θ which may or may not be dependent on θ. Let Φ be any functional of V monotonically decreasing with n which is used as an optimality criterion for an optimal choice of the x _i ∈B (i=l,…,n); the x _i chosen are called an exact design. We call a design (locally or globally) Φ-optimal in B of size n if the set of the x _i defining the design is minimizing the functional Φ amongst all possible designs in B of size n. The design is locally optimal, if it depends on θ, otherwise the design is globally optimal. A design of size n with r support points is called an exact r-point design of size n and can be written as 1$$\left( {\matrix{ {{x_1}} \hfill & {{x_2}} \hfill & \ldots \hfill & {{x_r}} \hfill \cr {{n_1}} \hfill & {{n_2}} \hfill & \ldots \hfill & {{n_r}} \hfill \cr } } \right),\,\sum\limits_{i = 1}^r {{n_i} = n,\,{n_i}\,{\rm{integer}}{\rm{.}}} $$ See for more information Pukelsheim (1994).