D-Optimal cyclic designs for a covariate model with odd replication number

We consider a linear model in a completely randomized statistical setup (i.e. without blocking) with one treatment at v greater than or equal to 3 levels (or v qualitative factors) and k continuous covariates (or quantitative factors) subject to a first order regression with values on a k-cube, making r(i) observations per treatment level, i = 1,2,...,v. The D-optimality criterion (minimization of the generalized variance) based on the maximization of the determinant of the information matrix M(d) of a design d under consideration, is used for estimating the parameters of the model. When our primary interest is in estimating all the parameters of the model (or only the regression parameters) we restrict our attention to "cyclic designs" d characterized by the property that when k = v and r(i) = r for all i = i, 2,..., v, the allocation matrix of each treatment level is obtained through cyclic permutation of the columns of the allocation matrix of the first treatment level. Necessary and sufficient conditions for the existence of D-optimal cyclic designs with odd replication number (i.e. r = 1 mod 2) are given. The non-existence of a series of D-optimal cyclic designs is also proved. By studying the nonperiodic autocorrelation function of circulant matrices we give a method for constructing such D-optimal cyclic designs. We develop an exhaustive algorithm based on our method and we apply this algorithm for N = rv, N less than or equal to 100 is the total number of observations. Finally, all the corresponding D-optimal cyclic designs are given.