Constrained simultaneous confidence intervals for multiple comparisons with the best (Q761725)

Let \(\pi_ 1,\pi_ 2,...,\pi_ k\) be k treatments and \(\theta_ 1,\theta_ 2,...,\theta_ k\) be their treatment effects. It is assumed that \(\theta_ i\) \((i=1,2,...,k)\) are location parameters in the distributions of the random variables \(Y_{i\alpha}\), \(\alpha =1,2,...,n\). Let \(Y_ i\) denote the statistic for the treatment \(\theta_ i\). The paper obtains the constrained two sided simultaneous confidence intervals for \(\theta_ i-\max_{j\neq i}\theta_ j\). Both the parametric and non-parametric intervals are obtained. In the parametric case, the intervals are of the type \[ [-(Y_ i-\max_{j\neq i}Y_ j-d)^-,\quad (Y_ i-\max_{j\neq i}Y_ j+d)^+] \] where \(x^+=\max (x,0)\) and \(x^-=\max (-x,0)\). The upper confidence bounds of the intervals imply subset selection inference and the lower confidence bounds, indifference zone selection inference. The extension of the method for the differences between \(\theta_ i\) and the jth largest of the remaining parameters is also discussed.