A Two-Sample Multiple Decision Procedure for Ranking Means of Normal Populations with a Common Unknown Variance

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.A multiple decision approach to the problem of ranking populations according to their population means has been formulated by Bechhofer (1954). A single-sample solution to this problem was presented by him for the case of normal populations with known variances. In the present paper the case of normal populations with unknown equal variances is considered. A two-sample procedure is proposed as a solution to the latter problem; this procedure is of the type used by Stein (1945) for obtaining a test of Student's hypothesis with power independent of the variance. Tables are available which enable the experimenter to apply this procedure with little computational effort to the ranking of two or three populations with unknown but equal variances. The same tables can be used when the variance ratios are arbitrary and known. Graphic comparisons of the expected sample size are made for the single-sample and two-sample procedures.

An integrated selection formulation for the best normal mean: the unequal and unknown variance case

“…The first inequality follows from the fact that T 1 and T 2 both have students' t distributions and δ 21 = µ [2] − µ [1] From Theorem 2, it is clear that as h * 1 , h * 2 → ∞, the left hand sides of (26) and (27) approach 1.…”

“…, π k with unknown means and unequal and unknown variances σ k . We denote the ordered means as µ [1] ≤ µ [2] ≤ · · · ≤ µ [k] and denote π (i) as the population which corresponds to µ [i] . We also define the best population to be π (k) , the population corresponding to the largest population mean µ [k] .…”

“…Example: Suppose that we are given three normal populations with unequal and unknown variances. Suppose that we wish to use the integrated formulation to select the population having the largest population mean if µ [3] − µ [2] ≥ 1, and to select a subset that contains the longest mean if µ [3] − µ [2] < 1.…”

Proceedings of the Twentieth Conference on the Design of Experiments in Army Research Development and Testing. Part 2. Ranking and Selection Procedures

“…These procedures are generalizations of and Bechhofer, Dunnett, and Sobel [1954). Gupta and Sobel [1958) proposed a single-stage procedure for this problem using the subset approach.…”

A two-sample procedure for selecting the population with the largest mean fromk normal populations

“…Bechhofer, Dunnet and Sobel [5] considered a two-sample procedure for the case in which the populations have a common unknown variance.…”