By Lloyd Fisher, Z. W. Birnbaum, E. Lukacs, John N. McDonald

ISBN-10: 0122573501

ISBN-13: 9780122573507

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C) Plot t(x0). (d) Does t seem robust against outliers? 13 (Robustness continued) One approach to making a two-sample test that is robust against outliers is to turn to a rank test. Let the X and Y observations be arranged in nondecreasing order from the smallest observation to the largest. The rank of an observation is its position in line; that is, the smallest observation has rank 1, the next smallest has rank 2, up to the largest observation which has rank m + n. ) Let T be the sum of the ranks of the Y observations.

Suppose we are interested in testing whether or not all the μ/s are equal. If all the μ^ are equal, then X ~ Ν(μ1,σ2Ι). Note that 1 lies in the subspace generated by {v 1? ,vfc}; in fact, 1 = vt + · · · + vfc. Since the Pythagorean theorem holds for a proper subspace of a vector space, we have, using the theorem, ||PiX + · · · + P*X|| 2 = IIP5XII2 + ||PrX||2, where S is the subspace spanned by the 1 vector and T is the orthogonal complement of S relative to the subspace spanned by {v l 5 ... ,vk} [a (k — 1)dimensional subspace].

This projection operator is P M — P H (Theorem A30). If the reduced model is true, the statistic 1ΚΡ Μ -Ρ Η )Χ11 2 /|1Ρ^Χ1| 2 dim(M nH1) / dim