Simultaneous selection of treatments better and worse than the best and worst controls |
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Institution: | 1. Department of Statistics, Panjab University, Chandigarh, 160 014, India;2. Department of Mathematics, Guru Nanak Dev University, Amritsar, India;1. Faculty of Medicine, Toho University, 5-21-16 Omori-nishi, Ota-ku, Tokyo 143-8540, Japan;2. Faculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan |
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Abstract: | In this paper, we consider and independent treatment and control populations respectively, such that an appropriate probability model for the data from 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 newly invented drugs/varieties of seeds/components which have to compete with their existing 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 control populations is more (less) than units, where and are positive constants to be specified by the experimenter. In this setting a selection procedure is proposed to select simultaneously two subsets and of the treatment populations such that the subset contains all the good treatments and the subset contains all the bad treatments with probability at least , where 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 treatment population as bad (good) if the distance of its average response from the largest (smallest) average response of control populations is less (more) than units, where and are to be specified by the experimenter such that , then we have proposed a simultaneous selection procedure to select and and a sample size is determined so that the probability of omitting a good (bad) treatment population from or selecting a bad (good) treatment population in is at most . |
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