An Asymptotically Distribution-free Multiple Comparison Procedure- Treatments vs. Control

Abstract

Let $X_{i0}$ and $X_{ij} (i = 1, \cdots, n; j = 1, \cdots, k)$ be the independent measurements on the control and $j$th treatment in the $i$th block, with $P(X_{ij} \leqq x) = F_j(x - b_i)$ Here $b_i$ is the block $i$ nuisance parameter and the $F_j; j = 0, \cdots, k$, are assumed continuous. Nemenyi [5] suggests treatment-control comparisons based on the statistic $T = \max_j T_{0j}$ where $T_{0j}$ is defined by (2.1). It is shown here that, under the null hypothesis \begin{equation*}\tag{1.1}H_0:F_j = F \text{(unknown)},\quad j = 0, \cdots, k,\end{equation*} $T$ is neither distribution-free for finite $n$, nor asymptotically distribution-free. We also modify Nemenyi's procedure so that it is asymptotically distribution-free.