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J. Japan Statist. Soc. Vol. 37 No. 2 2007 157174 MULTIPLE COMPARISONS understanding ON R-ESTIMATORS IN THE ONE-WAY LAYOUT Taka-aki Shiraishi* In a unidirectional analysis of variance prototype, robust versions ground on R-estimators ar proposed for single-step multiple comparisons procedures discussed by Tukey (1953), Dunnett (1955), and Sche?´ (1953). The robust procedures are two methods e ground on joint ranks and pairwise ranks. It is shown that the two methods are asymptotically equivalent. Although we fail to piss simultaneous tests based on elongate joint ranks, we are able to propose simultaneous tests based on the Restimators. Robustness for asymptotic properties is discussed. The accuracy of asymptotic theme is investigated. Key words and phrases : asymptotic property, robust statistics, simultaneous yardbird?dence intervals, simultaneous tests, single-step procedures. 1. Introduction Let µ1 , . . . , µk be the bastardly responses under k t reatments. consider that, under the i-th treatment, a random sample Xi1 , . . . , Xini is taken. consequently we have the one-way model (1.1) Xij = µi + eij (j = 1, . . . , ni , i = 1, . . . , k) where eij is a random diverse with E (eij ) = 0 for all i, j s. It is further sour that eij s are independent and identically distributed with a straight statistical distribution function (d.f.) F (x). Let Var(eij ) = ? 2 > 0. The model (1.1) is rewritten as chronic by Xij = ? + ?i + eij , where k=1 ni ?i = 0.

Then ? and ?i s are referred to as the grand mean and i elongate treatment e?ects, respectively. We put N = k=1 ni . The least squares i i ¯ ¯ ¯ ¯ estima tor of ?i is precondition by ?i = Xi· ? XÂ! ·Â· , where Xi· = n=1 Xij /ni and X·· = Ëœ j ni k i=1 j =1 Xij /N . The relations of µi ? µi = ?i ? ?i and ¯ ¯ Xi· ? Xi · = ?i ? ?i ˜˜ hold. We discuss single-step procedures. Let ?i ? ?i ? (?i ? ?i ) ˜˜ Ëœ Tii = ? 2 · (1/ni + 1/ni ) Ëœ and Ëœ? Tii = ?2 Ëœ ?i ? ?i ˜˜ , · (1/ni + 1/ni )...If you want to get a full essay, dedicate it on our website: OrderEssay.net

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