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# Re: st: RE: RE: A math question

 From Yuval Arbel To statalist@hsphsun2.harvard.edu Subject Re: st: RE: RE: A math question Date Fri, 21 Dec 2012 08:26:51 -0800

```I address you to my previous response regarding the LAD estimator. It
seems from Johnston that the median is obtained if we minimize the
absolute value of the random disturbance term.

On Fri, Dec 21, 2012 at 8:09 AM, CJ Lan <CJ@jupiter.fl.us> wrote:
> What happens is the following:
>
> Recall the Huber weighting functions in the robust estimator is defined as (k= tuning constant and sigma is the robust estimate of the "scale" used to standardize the errors)
> w=1, if |e|<=k*sigma
> w=k*sigma/|e|, if |e|>k*sigma
>
> Sigma (aka scale) in several software including Stata (I believe) is defaulted as MAE/norminv(0.75), where MAE = median of absolute errors = median{|ei|, i=1,...,n}.  n=no. of data points.  "Norminv" is the inverse of standard normal fun.
> e = estimation error = y - u(b,x).
>
> Now, what I try to do is to compute the derivative as well as Hessian of the weights, w, with respect to the parameters b.  Both Sigma and errors are a function of b.
>
> I am stuck at taking derivative of the Sigma wrt b, because it involves with the "median".
>
> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of David Hoaglin
> Sent: Friday, December 21, 2012 10:52 AM
> To: statalist@hsphsun2.harvard.edu
> Subject: Re: st: RE: RE: A math question
>
> The median is a function of a distribution.  You did not specify the distribution of x.  If you explained the context for your question, we might be able to shed better light.
>
> David Hoaglin
>
> On Fri, Dec 21, 2012 at 10:42 AM, CJ Lan <CJ@jupiter.fl.us> wrote:
>> Al,
>>
>> I think the "median" is a function, just like the "mean" is a function, not just a number.
>>
>> BTW, the derivative of abs(x^3) should be 3*x*abs(x) because the
>> derivative can be derived as
>>  abs(x^3)/(x^3)*(3*x^2) = 3*abs(x)*x^2/x = 3*x*abs(x).
>>
>> I wish others could shed a light too?  Thx.
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--
Dr. Yuval Arbel
4 Shaar Palmer Street,
Haifa 33031, Israel
e-mail1: yuval.arbel@carmel.ac.il
e-mail2: yuval.arbel@gmail.com

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```