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Table 6.2 Pairwise Slopes for the Forearm Length Data, in Increasing Order, and Their Weights
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Slope bij - 86.000 -48.500 -40.000 -36.500 -36.000 -31.000 2.604 2.667 2.667 2.683 2.684 2.684 37.500 39.000 43.000 49.000 51.000 91.000 Weight
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Cumulative Sum of Weights 0.000125 0.000374 0.000623 0.000873 0.001247 0.001496 0.497632 0.499127 0.499501 0.504612 0.506981 0.511718 0.999252 0.999377 0.999626 0.999751 0.999875 1.000000
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0.000125 0.000249 0.000249 0.000249 0.000374 0.000249 0.006607 0.001496 0.000374 0.005111 0.002368 0.004737 0.000249 0.000125 0.000249 0.000125 0.000125 0.000125
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A Description of ~ in Terms of Ranks. The estimates of a and f3 should be chosen so that the residuals i = Yi - (& + ~x) are "small". A sensible way to measure their smallness is by means of a weighted sum of the absolute values of the residuals, LwileJ The weights Wi should be nonnegative. In least-squares estimation, we choose & and ~ to minimize the weighted sum with weights Wi = leJ In least-absolute-deviations estimation, we minimize the weighted sum with weights Wi = 1. An intermediate procedure, intermediate between weighting the residuals equally and weighting them according to their absolute values, would be to weight them according to the ranks of their absolute values, Wi = rank( lei I). (The ranking is done from smallest to largest, the smallest value of leil being given the smallest rank, 1.) This would limit the influence of large residuals to a greater extent than least-squares estimation, since rank(leil) can be no larger than n whereas leil could be arbitrarily large, and to a lesser extent than least-absolute-deviations estimation. We will not use this procedure exactly but will use a similar procedure. Rather than choose estimates that minimize Lrank(leil)leil, we choose them to minimize
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Both sums yield approximately the same estimates, at least when the population of errors has a symmetric distribution and when the sample size n is large. Sum (6.2) is expected to give better results when the distribution of the population of errors is not symmetric. Below we will see that the nonparametric estimate ~, presented above as a weighted median, can also be characterized as the value of b that minimizes (6.2), where e i = Yi - (a + bxJ But first we verify the close relationship between the two sums.
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Relationship Between the Rank-Weighted Sum of Absolute Residuals and Sum (6.2). If the distribution of the population of errors is approximately symmetric, then we can expect rank(leil) to be approximately equal to 2Irank(e) - t(n + 01. This is because t(n + 1) is the rank of the median residual, which can be expected to be near 0, and because the absolute values of the negative residuals should be approximately uniformly interspersed with the positive residuals. For example, suppose the residuals are -23,-18,-11,2,16,19,29. The values of rank(leil) are, respectively, 6,4,2,1,3,5,7. The values of 2Irank(e) - t(n + 1)1 are, respectively, 6,4,2,0,2,4,6. Note that dividing all the weights Wi by the same constant 2 will not affect the minimization of the weighted sum [wileJ So using the weights Wi = rank(leil) is approximately equivalent to using the weights Wi = Irank(e) - t(n + 01. If the median of the residuals is 0, then the negative residuals have rank less than t(n + 1) and the positive residuals have rank greater than t(n + 1). This implies that rank(e) - t(n + 0 has the same sign as ei' so Irank(e) - t(n + 01 leil = [rank(e) - t(n + 1)]ei For example, suppose the residuals are - 23, -18, -11,0,16,19,29. The values of rank(e) - t(n + 0 for these residuals are - 3, - 2, -1,0, 1,2,3. Note that I - 31 I - 231 = (-3)( -23), ... ,101101 = (0)(0), ... ,1311291 = (3)(29). Therefore, if the distribution of errors is approximately symmetric, in which case the median of the residuals can be expected to be near 0, then the weighted sum [wileil with weights Wi = Irank(e) - t(n + 01 should be approximately equal to sum (6.2). Equivalence of the Two Descriptions of~. Let a and b denote candidates for Ii and ~, and put e i = Yi - (a + bx) in (6.2). First note that the value of (6.2) is unaffected by the value of a, and hence E[rank( Yi - a bx - t(n + 1)](Yi - a - bx) = E[rank(Yi - bx) - t(n + O](Yi - bxJ This is because shifting all the residuals by the same amount a does not change their ranks, so that rank( Yi - a - bx) = rank( Yi - bx), and because the sum of the n ranks must be n(n + 1)/2, so that [[rank( Yi - bx) t(n + O]a = 0. Therefore the non parametric estimate ~ is the value of b
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