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I am trying to show that the median is a M-estimator, and the definition is $t$ is an M-estimator if there exist a function $\psi$ such that $$ t=\arg\min_s\mathbb{E}(\psi(s,X))\qquad(1)$$ so for empirical median it is known that $\psi(s,X)=|X-s|$ and $\mathbb{E}$ is the empirical mean, but I do not understand why $$t=median(X_1,...,X_n)$$ is solution of the equation (1).

Any help, hint, or observation is welcome. Thank you!.

Don P.
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  • Are you sure a median is the solution of that? – Daeyoung May 17 '22 at 04:04
  • No, I was missundestanding somethings, the median has to solve the equality $$\frac{1}{n}\sum_{i=1}^nsign(X_i-t)=0$$. – Don P. May 17 '22 at 04:50
  • I woul recommend Robust Statistics, Theory and Methods by Maronna. You have a derivation in section 2.2.1 Generalizing maximum likelihood. – Hugo B May 17 '22 at 03:01

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