Simultaneous selection of treatments better and worse than the best and worst controls

In this paper, we consider $p(p\ge 2)$ and $q(q\ge 2)$ independent treatment and control populations respectively, such that an appropriate probability model for the data from $i\mathrm{th}\left(j\mathrm{th}\right)$ treatment (control) population is a member of absolutely continuous location and scale family of distributions which have common scale parameter and possibly differ in location parameters. For example, there may be $p$ newly invented drugs/varieties of seeds/components which have to compete with their existing $q$ standard competitors in terms of their average responses. A newly invented drug/variety of seed/component is said to be good (bad) if the distance of its average response from the largest (smallest) average response of $q$ control populations is more (less) than ${\delta }_1\left({\delta }_2\right)$ units, where ${\delta }_1$ and ${\delta }_2$ are positive constants to be specified by the experimenter. In this setting a selection procedure is proposed to select simultaneously two subsets $S_U$ and $S_L$ of the $p$ treatment populations such that the subset $S_U$ contains all the good treatments and the subset $S_L$ contains all the bad treatments with probability at least $P^1$, where $P^1$ is a pre-assigned value. The proposed procedure was applied to normal and two parameters exponential probability models separately and the relevant selection constants have been tabulated. The implementation of the proposed methodology is demonstrated through a numerical example based on real life data. The authenticity of numerically computed critical constants have been verified through simulation. Further, if we define the $i\mathrm{th}$ treatment population as bad (good) if the distance of its average response from the largest (smallest) average response of $q$ control populations is less (more) than ${\delta }_3\left({\delta }_4\right)$ units, where ${\delta }_3$ and ${\delta }_4$ are to be specified by the experimenter such that ${\delta }_4>{\delta }_3>0$, then we have proposed a simultaneous selection procedure to select $S_U$ and $S_L$ and a sample size is determined so that the probability of omitting a good (bad) treatment population from $S_U\left(S_L\right)$ or selecting a bad (good) treatment population in $S_U\left(S_L\right)$ is at most $1-P^1$.