On Robust Estimation in Incomplete Block Designs

Abstract

The object of the present investigation is to generalize the results of Greenberg (1966) to a wider class of robust estimators which includes her estimator as a special case. As in Greenberg (1966), we consider an incomplete block design $D$ consisting of $J$ blocks of (constant) size $b$ to which $c(> b)$ treatments are applied, there being $n_j$ replications of the $j$th block, for $j = 1, \cdots, J$. Let $n = \sum^J_{j=1} n_j$ and let $S_j$ consist of the numbers of the $b$ treatments applied in the $j$th block, for $j = 1, \cdots , J$. The observable random variables are then \begin{align}\tag{1.1}X_{ij\alpha} = v + \xi_i + \mu_j + \beta_{j\alpha} + U_{ij\alpha}, \\ \alpha = 1, \cdots, n_j; \quad i \varepsilon S_j; \quad j = 1, \cdots, J,\end{align} where $\xi_i$ is the $i$th treatment-effect, $\mu_j$ the $j$th replication effect, $\beta_{j\alpha}$ the effect of the $\alpha$th block in the $j$th replication set and $U_{ij\alpha}$'s are independent and identically distributed residual error components with a common distribution $F(u)$. We may set (without any loss of generality) that \begin{equation*}\tag{1.2}\sum^c_{i=1} \xi_i = 0, \quad \sum^J_{j=1} \mu_j = 0; \quad \sum^{n_j}_{\alpha=1} \beta_{j\alpha} = 0 \quad\text{for all} j = 1, \cdots, J.\end{equation*} Our intention is to provide some robust estimators of contrasts among $\xi_i$'s and to study their various properties.