On Detecting the Best Treatment

We observe independent random variable $Y_i \sim N(\theta_i,1), i=1,2,\cdots,k$, and we are interested in detecting the treatment with the largest $\theta_i$. We consider a procedure which infers $\theta_{(k)} \geq max\theta_i (i\neq(k))$ whenever $Y_{(k)} \geq maxY_i+C (i\neq(k))$. The maximum probability of a false inference is found, and it is shown that the inference can be made with the two-sample one-sided critical value for the usual error levels. The result also holds in the case of common unknown variance.