Constrained Simultaneous Confidence Intervals for Multiple Comparisons with the Best

For comparing k treatment effects θ₁, θ₂, ⋯ θₖ, often the parameters of primary interest are θᵢ - maxj≠ iθⱼ, i = 1, ⋯, k. In this article, we develop constrained 100P% two-sided simultaneous confidence intervals for θᵢ - maxj ≠ i θⱼ 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% as MCB. We also extend our method to give 100P% simultaneous confidence intervals for θᵢ - θ⁽ⁱ⁾₍ₖ₋ₜ₎, i = 1, ⋯, k, where θ⁽ⁱ⁾₍ₖ₋ₜ₎ is the tth largest among the θ's excluding θᵢ.