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> 0 (because h'(X;Xc)-lh = f'f, where f = Xc(X;X)-Ih), and so we can choose d = -(X;X)-Ih = -(X;X)-I Vg(bO).
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6.6d. The direction vector d = -(X;Xc)-1 Vg(bO) was suggested by McKean and Hettmansperger (1978, p. 574). The matrix (X;X)-I is approximately proportional to the variance-covariance matrix of the nonparametric regression estimates of f31' ... ' f3p. See the subsection on the parameter T in Section 6.7 and Note 6.7e. 6.6e. The accuracy of estimates obtained from an iterative algorithm is discussed in Notes 5.3g, 5.3h, and 5.6c. For the education expenditure data, we iterated the non parametric estimation algorithm until two successive vectors of estimates satisfied the condition that the relative differences were less than to- 8 However, the function (6.7), which we are trying to minimize, appears to be rather "flat" and so we are only confident in the accuracy of the estimates to about three significant digits. Using the least-squares estimates (0.07239, 15.52, - 0.04269) as initial estimates, the algorithm converged to (0.05868, 12.54,0.3372). Using the slightly different initial estimates (0.07,16.0, - 0.04), the algorithm converged to (0.05857, 12.56,0.3380). 6.6f. Rather than require convergence of the estimation algorithm, it is sometimes sufficient to do only one or two iterations of the algorithm. McKean and Hettmansperger (1978) found that, provided the sample size is large and the error distribution does not have extremely heavy tails, the estimates obtained from only one iteration are generally quite close to the estimates that would be obtained by iterating until convergence. This can save considerable computation time for large data sets. 6.6g. When ties occur among the differences Yi - b'x i , we need to specify how to assign their ranks. We used midranks in our calculations for the education expenditure data. But if there are not many ties, it should not make much difference how ties are handled. To avoid calculation of mid ranks, one could break ties in some arbitrary manner such as according to the order of their indices. 6.7a. Test statistic FNP in (6.8) was introduced by McKean and Hettmansperger (1976). Also see Hettmansperger (1984, Subsection 5.3.1). To see that our formula for F NP agrees with Hettmansperger's book, note that his sum of rank-weighted residuals in (5.2.5) using a(i) from (5.2.11) is equal to our SRWR multiplied by /(n + 1). If we had used his definition of SRWR, we would have defined c = i in (6.8). In 6 we estimate f3 by minimizing g(b) = '[,a(r)e i with a(r) = r - i(n + 1); whereas Hettmansperger (1984) and others use a(r) = [m/(n + 1)][r - i(n + 1)]. Of course the constant factor m /(n + 1) does not
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affect the minimization. The factor serves simply to standardize the scores
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a( r) so that the average of the squared scores is approximately 1. (The average of the squared scores is exactly equal to (n - 1)/(n + 1); see Note
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6.4d.) 6.7h. Two other tests are available for testing the coefficients in a multiple linear regression model: the aligned rank test, proposed by Koul (1970) and developed by Sen and Puri (1977) and Adichie (1978), and the Wald test, proposed by Hettmansperger and McKean (1983). The aligned rank test can be described in terms of the function g(b) in (6.7). Recall that the estimation procedure in Section 6.6 is based on the fact that the minimum of g(b) should occur for a vector that is near the parameter vector f3. This implies that the partial derivatives of g(b) should be close to 0 at b = f3; that is, (iJg/iJb)(f3) "" O. To test the hypothesis f3 q + I = ... = f3 p = 0, we first estimate f3 under the assumption that the hypothesis is true, by applying the procedure in Section 6.6 to the reduced model to obtain ~I' ... ' ~q and setting ~q+ I = ... = ~p = O. Let Po denote this vector of estimates. If the hypothesis is true, then the vector S2 with entries (iJg/iJb)(Po), j = q + 1, ... , p, should be near O. The aligned rank test statistic is A = S~V-IS2' where V = X;2Xc2 - X;2Xci(X;IXCI)-IX;lXc2 and Xci and Xc2 are, respectively, columns 1 through q and columns q + 1 through p of the matrix Xc of centered explanatory variables. The hypothesis is rejected if A is too large. The p-value is calculated from the chi-squared distribution with p - q degrees of freedom. See Hettmansperger (1984, Section 5.3.2). The phrase "aligned rank" comes from the fact that the test involves the ranks of the residuals Yi - p;)x" which can be regarded as alignments of the observations Yi. An aligned rank test uses ranks of residuals, whereas a "pure" rank test uses the ranks of the observations. The Wald test of the hypothesis f3 q + I = ... = f3 p = 0 is based directly on the vector of estimates ~2 = (~q+I' ... '~p) from the full model. If the hypothesis is true, then P2 should be near O. The Wald test statistic is W = P~VP2/( p - q )1'2, where V is from the preceding paragraph and l' is obtained from (6.9). The hypothesis is rejected if W is too large. The p-value is calculated from the F distribution with p - q and n - p - 1 degrees of freedom. See Hettmansperger (1984, Section 5.3.3). All three tests are equivalent for infinitely large samples. In simulation studies with finite samples, Hettmansperger and McKean (1983) found the test in Section 6.7 to be more stable in its validity and power (see Note 3.5b). The aligned rank test avoids the problem of estimating T. 6.7c. Formula (6.9) for estimating T is recommended by Hettmansperger and McKean 0983, Section 4). The factor f is ad hoc and has no theoretical
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