In fractional replication, the problem is to estimate a subvector, $\alpha$, of $S = p^s$ pre-assigned parameters in the presence of an additional subvector of nuisance parameters. Estimation procedures for randomized fractional replicates in $2^m$ factorial systems have been treated by Ehrenfeld and Zacks ,  and Zacks ,  under an orthogonalized form of the statistical model. Extension to systems with general prime $p$ has recently been studied under a "fully orthogonalized" model by Lentner ; this is discussed in Section 3. The generalized inverse operator is then applied to the study of estimation procedures in general $N = p^m$ fractional replication. The present work differs from that of Zacks  where the problem of estimating the entire vector of $N$ parameters was considered. The class of all type-$g$ (generalized inverse) solutions for $\alpha$ is given and the class is investigated with respect to unbiasedness and optimality. Specifically, it is shown that there is a unique unbiased type-$g$ estimator which coincides with the classical estimator under the assumption of zero nuisance parameters. Using the trace of the mean square error matrix as the risk function, it is shown that there is a coincidence of Bayes, minimax, admissible, and classical procedures under certain conditions.
"Generalized Least-Squares Estimation of a Subvector of Parameters in Randomized Fractional Factorial Experiments." Ann. Math. Statist. 40 (4) 1344 - 1352, August, 1969. https://doi.org/10.1214/aoms/1177697506