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A procedure for the selection of populations with means falling within certain normal limits. (English) Zbl 0654.62029

Let \(\bar x_ 1,...,\bar x_ K\) be sample means, based on n observations each, from K normal populations with unknown means \(m_ 1,...,m_ K\), respectively, and a common variance \(\sigma^ 2\). Let \(m_{(1)}\leq....\leq m_{(K)}\) and \(\bar x_{[1]}\leq...\leq \bar x_{[K]}\) denote the ordered values of the population means and sample means, respectively. Let \[ G=\{i:\quad m_{(K)}-e_ 1\leq m_ i\leq m_{(1)}+e_ 2\} \] denote the set of “good” populations, where \(e_ 1,e_ 2>0\) are numbers specified by the investigator. It is required to select a subset of the K populations, S say, containing the good populations, such that the probability of correct selection (G\(\subset S)\) is at least as large as a pre-assigned number \(P^*\). In the case \(\sigma\) is known it is shown that \(P\{G\subset S\}\geq P^*\) for S is given by \[ S=\{i:\quad \bar x_{[K]}-\epsilon_ 1-u \sigma n^{- 1/2}\leq \bar x_ i\leq \bar x_{[1]}+\epsilon_ 2-u \sigma n^{-1/2}\} \] where \(u=u(K,P^*)\) denotes the \(P^*\)-quantile of the range distribution of K standard normal random variables. In the case \(\sigma\) is unknown the subset S given above, is modified by substituting \(\hat q\) for q and \({\hat \sigma}\) for \(\sigma\), where \(\hat q\) denotes the \(P^*\)-quantile of the studentized range distribution of K standard normal random variables and \({\hat \sigma}\) denotes the pooled sample standard deviation. Monte Carlo results are given in support of the theory.
Reviewer: K.Alam

MSC:

62F07 Statistical ranking and selection procedures
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