Minimization of a sum of absolute values

Asked Nov 09 '16 at 06:23

Active Nov 09 '16 at 06:32

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Let $X_{1},...X_{n}$ be a random sample with some known pdf.

I found the log likelihood. Now I would like to know how to minimise the following expression in order to minimise this likelihood and thus find the maximum likelihood estimate:

$$ \sum_{i=1}^n | x_{i} - \theta | $$

I know without the absolute sign this expression would be 0 if $\theta$ = mean of the distribution.

Thoughts?

edited Nov 09 '16 at 06:32

Jean Marie

81,803

asked Nov 09 '16 at 06:23

Math R'tard

Have a look at the answer I gave here – Jean Marie Nov 09 '16 at 06:39
$\theta$ is a median of $x_i$ – LinAlg Nov 09 '16 at 17:16
http://math.stackexchange.com/questions/113270/the-median-minimizes-the-sum-of-absolute-deviations – Michael Shi Nov 09 '16 at 21:04

Minimization of a sum of absolute values

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