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Some questions

Hi,

I have some problems understanding these parts of the course/exercises

1) In series 12, why is the density of y_j, g(y_j-x_j^T beta), where g() is the denstiy of epsilon_j, my guess would be that y_j-x_j^T beta = epsilon_j but then why do they have different density functions ? Or are they actually the same "modulo a shift" ?

2)On the slides about Robust/Resistent Regression: M-estimation as Weighted Regression : when taking the derivative with respect to gamma, why is there still x_i^T  in the sum with psi, since according to the chain rule we take the transpose of x_i^T ?

Thanks in advance.

Posted by Jean-Claude Ton on Thursday 16 January 2014 at 0:13
Comments
1) The density functions are the same modulo a shift by x_j^T beta. This results from a simple change of variables in g(epsilon_j) with epsilon_j = y_j - x_j^T beta.

2) The chain rule gives indeed x_i instead of x_i^T, but you can just take the transpose of the resulting equation to obtain the one on the slides (psi(.) is a scalar here).
Posted by Mikael Kuusela on Thursday 16 January 2014 at 18:30
For 2), does this mean that in the theory that follows (asymptotic distribution of M-Estimators), we need to remove the transpose from X and transpose psi ?
Posted by Jean-Claude Ton on Thursday 16 January 2014 at 18:48
Thanks for your answer by the way.
Posted by Jean-Claude Ton on Thursday 16 January 2014 at 18:49
No, X^T psi(gamma) = 0 on that slide is fine (remember that x_i^T is the ith row of X). This is equivalent to sum_{i=1}^n x_i psi((y_i-x_i^T gamma)/sigma) = 0 which is equivalent to the formula on the slides by transposing both sides. Alternatively, one could of course write psi(gamma)^T X = 0.
Posted by Mikael Kuusela on Thursday 16 January 2014 at 20:42
OK, thanks
Posted by Jean-Claude Ton on Thursday 16 January 2014 at 21:19