By Vandenbussche D., Nemhauser G. L.
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Extra info for A branch-and-cut algorithm for nonconvex quadratic programs with box constraints
3) then $-bJ(i)=- 5,ei l-hi' Since n 2 (x=x)-lx;xi(x'x)-l=(X'X)--', i- 1 it follows that [see Mosteller and Tukey (1977)] n var( $) = u2 2 c;. k= I Thus a scaled measure of change can be defined as %ee Appendixes 2A and 2B for details on the computation of the hi. 4) DETECTING INFLUENTIAL OBSERVATIONS AND OUTLIERS 14 where we have replaced s2, the usual estimate of u2,by in order to make the denominator stochastically independent of the numerator in the Gaussian (normal) case. A simple formula for s ( i ) results from ( n - p - l)s2(i)=(n-p)s2- e; 1-hi’ In the special case of location, ei DFBETA,= n- 1 and fi e, DFBETAS, = ( n - l ) s ( i ) ’ (2-9) As we might expect, the chance of getting a large DFBETA is reduced in direct proportion to the increase in sample size.
36) for model size by raising them to the l/pth power. 40) would be 1 - ( 3 / n ) and 1 + ( 3 / n ) respectively. 38). We are also interested in how the variance of jji changes when an observation is deleted. To do this we compute var( j j i ) = s2hi [- var(ji(i))=var(xib(i)) = s2(i) l:hi]7 and form the ratio FVARATIO= s2(i) s2( 1 - hi) ‘ This expression is similar to COVRATIO except that s2(i)/s2is not raised to thepth power. As a diagnostic measure it will exhibit the same patterns of behavior with respect to different configurations of hi and the studentized residual as described above for COVRATIO.
To use A we examine the smaller values for each m = 1, 2, and so on. 76) This is only approximate, since we are interested in extreme values. It would be even better to know the distribution of A conditional on X, but 38 DETECTING INFLUENTIAL OBSERVATIONS AND OUTLIERS this remains an open problem. More than just the smallest value of A should be examined for each m, since there may be several significant groups. Gaps in the values of A are also usually worth noting. Andrews and Pregibon (1978) have proposed another method based on Z.