Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
10.2. Sample Median and the Sign Test 585 10.2.2.Show that the test given by (10.2.6) has asymptotically levelα;thatis, show tha ...
586 Nonparametric and Robust Statistics 10.2.9.LetX 1 ,X 2 ,...,Xnbe a random sample that follows the location model (10.2.1). I ...
10.3. Signed-Rank Wilcoxon 587 other hand, ifH 1 is true, then we expect more than half of theXis to be positive and further, th ...
588 Nonparametric and Robust Statistics Basedonthisobservation,forssuch that−∞<s<∞,themgfofTis E[exp{sT}]=E ⎡ ⎣exp ⎧ ⎨ ⎩ ∑ ...
10.3. Signed-Rank Wilcoxon 589 Note that the support ofTis much denser than that of the sign test, so the normal approximation i ...
590 Nonparametric and Robust Statistics Table 10.3.1:Signed Ranks for Darwin Data, Example 10.3.1 Cross- Self- Signed- Pot Ferti ...
10.3. Signed-Rank Wilcoxon 591 Letcαdenote the critical value of a levelαtest of the hypotheses (10.3.1) based on the signed-ran ...
592 Nonparametric and Robust Statistics This latter probability can be expressed as follows: P 0 (X 1 +X 2 >− 2 θn)=E 0 [P 0 ...
10.3. Signed-Rank Wilcoxon 593 We now derive some AREs between the Wilcoxon and thet-test. As noted above, the parameterτW is a ...
594 Nonparametric and Robust Statistics Table 10.3.2:AREs among the sign, the Signed-Rank Wilcoxon, and thet-Tests for Contamina ...
10.3. Signed-Rank Wilcoxon 595 we then have that 1 −α = Pθ[cW 1 <T+(θ)<n−cW 1 ] = Pθ[WcW 1 +1≤θ<Wm−cW 1 ], (10.3.33) wh ...
596 Nonparametric and Robust Statistics As sketched in Exercise 10.3.2, under the assumptions of symmetry and location estimator ...
10.3. Signed-Rank Wilcoxon 597 10.3.2.Consider the location Model (10.3.35). Assume that the pdf of the random errors,f(x), is s ...
598 Nonparametric and Robust Statistics 1.‖v‖≥0and‖v‖=0ifandonlyifv= 0. 2.‖av‖=|a|‖v‖, for allasuch that−∞<a<∞. 3.‖u+v‖≤‖u ...
10.4. Mann–Whitney–Wilcoxon Procedure 599 In the shift case, the parameter Δ is independent of what location functional is used. ...
600 Nonparametric and Robust Statistics The variance is displayed below (10.4.10) and a derivation of a more general case is giv ...
10.4. Mann–Whitney–Wilcoxon Procedure 601 ofXiandYjare continuous; hence, we treat the observations as distinct. Thus R(Yj)=#i{X ...
602 Nonparametric and Robust Statistics horizontal axis and thenon the vertical axis is replaced byn 1 n 2 ;thatis,U(Δ) is a dec ...
10.4. Mann–Whitney–Wilcoxon Procedure 603 where we have applied the mean value theorem to obtain the last line. Putting together ...
604 Nonparametric and Robust Statistics Note that this is the same ARE as derived in the last section between the signed- rank W ...
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