D-optimal chemical balance weighing designs with autoregressive errors

In this paper, we consider the estimation problem of individual weights of three objects. For the estimation we use the chemical balance weighing design and the criterion of D-optimality. We assume that the error terms <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\varepsilon_{i},\ i=1,2,\dots,n,}$$</EquationSource> </InlineEquation> are a first-order autoregressive process. This assumption implies that the covariance matrix of errors depends on the known parameter ρ. We present the chemical balance weighing design matrix <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\widetilde{\bf X}}$$</EquationSource> </InlineEquation> and we prove that this design is D-optimal in certain classes of designs for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${\rho\in[0,1)}$$</EquationSource> </InlineEquation> and it is also D-optimal in the class of designs with the design matrix <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${{\bf X} \in M_{n\times 3}(\pm 1)}$$</EquationSource> </InlineEquation> for some ρ ≥ 0. We prove also the necessary and sufficient conditions under which the design is D-optimal in the class of designs <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${M_{n\times 3}(\pm 1)}$$</EquationSource> </InlineEquation> , if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$${\rho\in[0,1/(n-2))}$$</EquationSource> </InlineEquation> . We present also the matrix of the D-optimal factorial design with 3 two-level factors. Copyright The Author(s) 2013