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Weighted Least squares for Robust/Resistant estimators

 

i have a question about the theory. In the WLS estimator (Robust/Resistant), M.Panaretos define 
 
\beta=(X^TV^{-1}X)^{-1}X^TV^{-1}Y
 
but in the course, when we first define the WLS, we say
 
\beta=(X^TV^{1}X)^{-1}X^TV^{1}y
 
hence, why do we have two different definitions? If we define V=diag{w_1, .., w_n} i think the definition \beta=(X^TV^{-1}X)^{-1}X^TV^{-1}Y is not correct?!
 
Thank's
Posted by Mahé Fellay on Thursday 16 January 2014 at 13:11
Comments
Yes, you are right. The slide "Robust/Resistant Alternatives" is indeed inconsistent with the previous definition. If you define the weights as in Lecture 2, then you should define V = diag(1/w_1,...,1/w_n) on the "Robust/Resistant Alternatives" slide. The slides at the end of Lecture 2 are correct.
Posted by Mikael Kuusela on Thursday 16 January 2014 at 18:49
Tank's a lot!
Posted by Mahé Fellay on Friday 17 January 2014 at 8:12