Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
10.4. Mann–Whitney–Wilcoxon Procedure 605 whereD 1 <D 2 <···<Dn 1 n 2 denote the order differencesYj−Xi. Therefore, the ...
606 Nonparametric and Robust Statistics indwil <-0; indt <- 0 for(i in 1:nsims){ x <- rcn(n1,eps,vc); y <- rcn(n2,ep ...
10.5.∗General Rank Scores 607 10.4.5.For the power study of Section 10.4.4, modify the R functionwil2powsim.R to obtain the empi ...
608 Nonparametric and Robust Statistics We first define a general class of rank scores. Letφ(u) be a nondecreasing function defi ...
10.5.∗General Rank Scores 609 As Exercise 10.5.4 shows,s^2 a/n≈1. SinceEH 0 (Wφ)=0,wehave VarH 0 (Wφ)=EH 0 (Wφ^2 )= ∑n^2 j=1 ∑n^ ...
610 Nonparametric and Robust Statistics interval [Δ 1 ,Δ 2 ]. Therefore, since the scores are nondecreasing, Wφ(Δ 1 )−Wφ(Δ 2 )= ...
10.5.∗General Rank Scores 611 Suppose thatF̂n 1 andF̂n 2 are the empirical cdfs of the random samplesX 1 ,...,Xn 1 andY 1 ,...,Y ...
612 Nonparametric and Robust Statistics 10.5.2 Estimating Equations Based on General Scores Suppose we are using the scoresaφ(i) ...
10.5.∗General Rank Scores 613 can be ignored, so we considerc∗φ=( √ λ 1 λ 2 )−^1 cφ. If we make the change of vari- ablesu=F(y) ...
614 Nonparametric and Robust Statistics section. First, though, note an invariance that simplifies matters. SupposeZis a scale a ...
10.5.∗General Rank Scores 615 Table 10.5.1:Data for Example 10.5.3 Sample 1 (X) Sample 2 (Y) Data Ranks Normal Scores Data Ranks ...
616 Nonparametric and Robust Statistics Table 10.5.2:Summary of analyses for Example 10.5.3 Method Test Statistic Standardized p ...
10.5.∗General Rank Scores 617 Other examples are given in the exercises. EXERCISES 10.5.1.In this section, as discussed above ex ...
618 Nonparametric and Robust Statistics (b)Show that part (a) implies that the efficacy, (10.5.20), is invariant to the location ...
10.6.∗Adaptive Procedures 619 (a)Show that under symmetry the optimal two-sample score function (10.5.26) satisfies φf(1−u)=−φf( ...
620 Nonparametric and Robust Statistics error distributions. Or if the distribution of the errors is thought to be quite close t ...
10.6.∗Adaptive Procedures 621 μ 1 ,μ 2 ,andσ^2 , such as the statistics ∑n^1 1 (Xi−X)^2 ∑n^2 1 (Yi−Y)^2 , ∑n^1 1 |Xi−median(Xi)| ...
622 Nonparametric and Robust Statistics are Wi= ∑n^2 j=1 ai[R(Yj)],i=1, 2 , 3 , 4 , (10.6.1) where ai(j)=φi[j/(n+1)], and the fo ...
10.6.∗Adaptive Procedures 623 left. The second selector statistic is Q 2 = U. 05 −L. 05 U. 5 −L. 5 . (10.6.3) Large values ofQ 2 ...
624 Nonparametric and Robust Statistics X 1 ,...,Xn 1 ,Y 1 ,...,Yn 2 are not identically distributed. There are adaptive proce- ...
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