Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
10.7. Simple Linear Model 625 10.6.3.LetF(x) be a distribution function of a distribution of the continuous type that is symmetr ...
626 Nonparametric and Robust Statistics i=1,...,n 1 ,andZn 1 +i=Yi,fori=n 1 +1,...,n,wheren=n 1 +n 2 .Letcibe 0 or 1 depending o ...
10.7. Simple Linear Model 627 Hence the corresponding estimate ofβis given byβ̂φ, which solves the estimating equations Tφ(β̂φ)≈ ...
628 Nonparametric and Robust Statistics Number of calls 0 5 10 15 20 50 55 60 Year 65 70 LS–Fit Wilcoxon–Fit Figure 10.7.1: Plot ...
10.7. Simple Linear Model 629 Differentiating this last expression, we have μ′T(β)= ∑n i=1 (xi−x)xi ∫∞ −∞ φ′(F(y+xiβ))f(y+xiβ)f( ...
630 Nonparametric and Robust Statistics Example 10.7.3(Distance of Punts).Rasmussen (1992), page 562, presents a data set concer ...
10.8. Measures of Association 631 10.7.6.Consider the sequence of local alternatives given by the hypotheses H 0 :β=0versusH 1 n ...
632 Nonparametric and Robust Statistics bivariate distribution (discrete or continuous). We say these pairs areconcordantif sgn{ ...
10.8. Measures of Association 633 null variance ofKis given by expression (10.8.6); see, for instance, page 205 of Hettmansperge ...
634 Nonparametric and Robust Statistics Table 10.8.1:Data for Example 10.8.1 Year 1500 m Marathon∗ Year 1500 m Marathon∗ 1896 37 ...
10.8. Measures of Association 635 Proof: Under independence,XiandYj are independent for alli andj; hence, in particular,R(Xi) is ...
636 Nonparametric and Robust Statistics whereγ=P[(X 2 −X 1 )(Y 3 −Y 1 )>0]. For largen,E(rS)≈6(γ− 1 /2), which is a harder pa ...
10.8. Measures of Association 637 a= 0. As in expression (10.5.6), lets^2 a= ∑n i=1a (^2) (i). Consider the rank correlation coe ...
638 Nonparametric and Robust Statistics Show essentially that rqc= 1 n {#(Xi,Yi)∈I+#(Xi,Yi)∈III−#(Xi,Yi)∈II−#(Xi,Yi)∈IV}. (10.8. ...
10.9. Robust Concepts 639 whereθis a location parameter (functional) andεihas cdfF(t)andpdff(t). Let FX(t)andfX(t) denote the cd ...
640 Nonparametric and Robust Statistics In contrast, consider the sample median in which the sample sizenis odd. In this case, t ...
10.9. Robust Concepts 641 The solution to the above equation is, of course,T(FX)=E(X). Likewise, we can write the estimating equ ...
642 Nonparametric and Robust Statistics Definition 10.9.1.The estimatorθ̂is said to berobustif|IF(x;θ̂)|is bounded for allx. Ham ...
10.9. Robust Concepts 643 Evaluating this partial derivative at = 0, we arrive at the influence function of the median: IF(x;̂θL ...
644 Nonparametric and Robust Statistics Example 10.9.2(Asymptotic Distribution of the Sample Median).The influence function for ...
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