Dear Dr Granville,
The method you describe is not new. Is in fact the standard multivariate regression.
See for example equation (1) page 2 of the following paper
"Robust Multivariate Regression" by Peter J. Rousseeuw, Stefan Van Aelst, Katrien Van Driessen and Jose Agulló.
Technometrics, Vol. 46, No. 3 (Aug., 2004), pp. 293-305.
Of course this forumal can be found in text books as well, but I find this paper convenient. The standard formula for beta is what you described and as you said, assuming corr(X_i,X_j)=0. Since the formula in that paper allows for some correlation values, even if they are small, the results with the way you describe will naturally be a bit different.
Yor example works for linear regression. You mentioned about logistic regression as well. What happens there?
Also next time you do a simulation study, do it for 50 observations first, maximum 100 and compare it with the standard ones before going to 10,000. When I see 10,000 observations in a simulation study I think something is tricky.
Michail
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