Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
9.9. The Independence of Certain Quadratic Forms 565 SinceQ 2 ≥0, each of the matricesA,A 1 ,andA 2 is positive semidefinite. Be ...
566 Inferences About Normal Linear Models This section concludes with a proof of a frequently quoted theorem due to Cochran. The ...
9.9. The Independence of Certain Quadratic Forms 567 9.9.4.LetAbe the real symmetric matrix of a quadratic formQin the observati ...
568 Inferences About Normal Linear Models multivariate normal distribution with meanβand variance–covariance matrix σ^2 (X′X)−^1 ...
Chapter 10 Nonparametric and Robust Statistics 10.1LocationModels In this chapter, we present some nonparametric procedures for ...
570 Nonparametric and Robust Statistics Remark 10.1.1(Natural Nonparametric Estimators). Functionals induce non- parametric esti ...
10.1. Location Models 571 Next, supposeY=aX.Ifa>0, thenFY(y)=FX(y/a) and, hence, FY(aT(FX)) =FX(aT(FX)/a)=FX(T(FX)) = 1/ 2. T ...
572 Nonparametric and Robust Statistics Proof:By (10.1.2), we have T(FX−a)=T(FX)−a. (10.1.5) Since the distribution ofXis symmet ...
10.2. Sample Median and the Sign Test 573 whereε 1 ,ε 2 ,...,εnare iid with cdfF(x), pdff(x), and median 0. Note that in terms o ...
574 Nonparametric and Robust Statistics The following symmetric decision rule seems appropriate: RejectH 0 in favor ofH 1 ifS(θ ...
10.2. Sample Median and the Sign Test 575 0.9 0.8 0.7 0.6 Width to length ratios Panel A 0.9 0.8 0.7 0.6 1.5 Width to length r ...
576 Nonparametric and Robust Statistics Y 1 Y 2 Yc 1 Yc 1 + 1 Yn–c 1 Yn n – c 1 n – 1 c 1 0 S( ) ......... ......... Figure 10.2 ...
10.2. Sample Median and the Sign Test 577 Recall from Definition 4.5.4 of Chapter 4 that the size of the test for a composite nu ...
578 Nonparametric and Robust Statistics θn, we can approximate the mean of this test as follows: Eθn [ 1 √ n ( S(0)− n 2 )] = E ...
10.2. Sample Median and the Sign Test 579 mean given in (10.2.14) in terms of another concept calledefficacy.Consider another st ...
580 Nonparametric and Robust Statistics and σ^2 X(0) =V 0 (X)= σ^2 n . (10.2.22) Thus, by (10.2.21) and (10.2.22), the efficacy ...
10.2. Sample Median and the Sign Test 581 Note that this is the same relative efficiency that was discussed in Example 6.2.5 whe ...
582 Nonparametric and Robust Statistics whereφ(z) is the pdf of a standard normal random variable. As shown in Section 3.4, the ...
10.2. Sample Median and the Sign Test 583 whose solution is the sample medianQ 2 , (4.4.4). Because our observations are continu ...
584 Nonparametric and Robust Statistics 10.2.3 Confidence Interval for the Median In Section 4.4, we obtained a confidence inter ...
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