Constrained Simultaneous Confidence Intervals for Multiple Comparisons with the Best

Abstract

For comparing $k$ treatment effects $\theta_1, \theta_2, \cdots \theta_k$, often the parameters of primary interest are $\theta_i - \max_{j\neq i}\theta_j, i = 1, \cdots, k$. In this article, we develop constrained $100P^\ast{\tt\%}$ two-sided simultaneous confidence intervals for $\theta_i - \max_{j \neq i} \theta_j$ which we refer to as (constrained) MCB intervals. It turns out that the lower bounds of the intervals imply Indifference Zone selection inference, and the upper bounds of these intervals imply Subset Selection inference, each given at the same confidence level $100P^\ast{\tt\%}$ as MCB. We also extend our method to give $100P^\ast{\tt\%}$ simultaneous confidence intervals for $\theta_i - \theta^{(i)}_{(k-t)}, i = 1, \cdots, k$, where $\theta^{(i)}_{(k-t)}$ is the $t$th largest among the $\theta$'s excluding $\theta_i$.