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Huber M-estimates are obtainable from the function rreg in the statistical package S-PLUS (Becker, Chambers, and Wilks, 1988, p. 571) and from the robust regression packages ROBSYS (Marazzi, 1987) and ROSEPACK (Holland and Welsch, 1977). If you have a computer package that performs least-squares regression, you can apply it iteratively to obtain Huber M-estimates. Let y denote the vector of observed response variables and X the matrix of explanatory variables. To begin, let y(l) = y. Apply the package to y(l) and X to obtain the vector b(l) of least-squares regression estimates and the vector y( I) of predicted y-values. Let e( I) = Y - y( I) be the vector of residuals. Calculate 0-( I) = 1.483 median( Ie~ I) I) and truncate the residuals by defining f ) = max( - 1.50-(1), min(e~ I >, 1.50-(1 )). Let y(2) = y( I) + I( I) and repeat the calculations with y(2) in place of y( I). That is, apply the package to y(2) and X to obtain b(2) and y(2). Let e(2) = y - y(2) and then calculate 0-(2) and 1(2). Let y(3) = y(2) + J<2) and repeat the calculations with y(3). Continue until consecutive estimates b(m - I) and b(m) are sufficiently close to one another; for example, continue until IbJ"'-I) - bt)l/ Ibt-I)I < 0.00001 for all j. If the estimates b(l), b(2), ... do not converge, try a different starting vector b(l). The test statistic FM in (5.6) can be calculated as follows. Let be the vector of M-estimates, y = = y - y, 0- = 1.483 median( leil), !; = max( -1.50-, min(e i, 1.50-)), and gi = max(O, leil - 1.50-). Let SI = '[f and S2 = '[gi Then STR rull = SI + 30-S 2 . Similar calculations using the reduced
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model produce STRreduced' And A = nSI/(m(n - p - 0), where m is the number of g;'s equal to O. Now apply formula (5.6). Iteratively Reweighted Least Squares. The computational procedure used in ROBSYS, ROSEPACK, and S-PLUS is an iteratively reweighted leastsquares procedure. This is a more general method than the one described in Section 5.6 and can be used to compute all types of M-estimates. For the case of Huber M-estimation, the procedure is described below. To find the minimum of (5.4), for a fixed value of U, take its derivatives with respect to b o, b I' ... ,bp and set them equal to O. This gives us p + 1 equations in p + 1 unknowns:
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for j = 0,1, ... , p, where we let x iO = 1 for all i. These are nonlinear equations in the unknowns bo, b l , ... , bp , but they can be approximated by linear equations as follows. Consider an iterative procedure in which bg, b\), ... ,b2 are current estimates and b o, b l , ... , bp represent improved estimates. Let e;) = Yi - (bg + b xiJ + ... +b2xi) and ei = Yi - (b o + blxiJ + ... +bpx ip )' To solve for the improved estimates, write p'(e) = (p'(e)/e)e i : : : (p'(e )/e)))e i. Let Wi = p'(e )/e ; that is, if le)'1 ~ 1.5u if le))1 > 1.5u Then p'(e) : : : wie i and we can approximate (5.7) by the linear equations
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Let W be the diagonal matrix with diagonal entries Wi' In terms of matrices we have X'W(y - Xb) = O. Solving for b, we obtain
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To start, let bO be the vector of least-squares estimates. At each iterative step, use the vector bO of current estimates to calculate the vector eO = y Xbo of residuals. Then use the residuals to obtain u and the weights Wi' The vector b of improved estimates can now be computed as in (5.8). Iterate until convergence. The vector b in (5.8) is called a weighted least-squares estimate. Some least-squares regression packages also perform weighted regression. For
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