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RE: st: Likelihood function of uniform distribution


From   "Mostafa Beshkar" <mostafa.beshkar@vanderbilt.edu>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st: Likelihood function of uniform distribution
Date   Wed, 2 Apr 2008 11:02:50 -0500

Thanks to Stas, Jay and Rodrigo for their comments.

I think I should explain my question in more detail, since I think there has
been some misunderstanding.

I want to estimate the following probability model (this comes from my
game-theoretic model):

Pr(s=1|X)=Pr(p>BX)
Pr(s=0|X)=1-Pr(p>BX)

where s is a binary variable, X is the vector of observable variables, B is
the vector of parameters to be estimated, and p is an unobservable random
varibale that is distributed according to F on the interval [0,1].

I want to use MLE to estimate parameters in B. To do this, I should first
make an assumption on the distribution of p. For example, I can assume that
p is uniformly distributed over [0,1], that is,

f(p)=1 if 0<p<1
    =0 otherwise.

Here is the stata commands that I hoped to do the job:
-------------------------
  program myunif
	    args lnf theta
	    quietly replace `lnf' =ln(`BX') if $ML_y1==1
	    quietly replace `lnf' =ln(1-`BX') if $ML_y1==0
   end
   ml model lf myunif (s = x1 x2 x3 x4)
   ml maximize	
-------------------------

Note that X=(x1 x2 x3 x4).

The problem that I have with this formulation is that the "program myunif"
defines a likelihood function based on the assumption that f(p)=1, while the
correct specification is {f(p)=1 if 0<p<1 and f(p)=0 otherwise}. I think
this problem can be resolved if I can first define the uniform distribution
function, U, in stata and then run the following:
-------------------------
  program myunif
	    args lnf theta
	    quietly replace `lnf' =ln(U(`BX')) if $ML_y1==1
	    quietly replace `lnf' =ln(1-U(`BX')) if $ML_y1==0
   end
   ml model lf myunif (s = x1 x2 x3 x4)
   ml maximize	
-------------------------

I have a similar problem when I assume that p follows a triangular
distribution.

Any help would be greatly appreciated.
Mostafa



-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Stas Kolenikov
Sent: Tuesday, April 01, 2008 3:54 PM
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: Likelihood function of uniform distribution

On 4/1/08, Bob Hammond <robert.g.hammond@vanderbilt.edu> wrote:
>     program myunif
>     args lnf theta
>     quietly replace `lnf' =ln(`theta') if $ML_y1==1
>     quietly replace `lnf' =ln(1-`theta') if $ML_y1==0
>     end

what this does is

f(x|theta) = theta^y (1-theta)^{1-y)

that is, the binomial/Bernoulli likelihood.

What you need is

program myunif
    args lnf theta
    quietly replace `lnf' = 0 if $ML_y1>0 & $ML_y1 <1
    quietly replace `lnf' = . if $ML_y1<=0 & $ML_y1 >=1
end

Note that theta is not used. Which is proper since there are no
parameters to estimate, you know everything in perfection already.

As for the triangular distribution (again, nothing to estimate):

>  Also, how do you define the following triangular probability
>  distribution function?
>
>  f(x)= 4x if 0<x<0.5
>  =4-4x if 0.5<x<1
>  =0 otherwise.

prog def mytriang
  args lnf theta
  qui replace `lnf' = ln(4*$ML_y1) if $ML_y1 >0 & $ML_y1 < 0.5
  qui replace `lnf' = ln(4*(1-$ML_y1)) if $ML_y1 >=0.5 & $ML_y1 <=1
  qui replace `lnf' = . if $ML_y1 < 0 & $ML_y1 >1
end

Since the likelihood does not change with theta, either function will
fail -ml check-. You really need to estimate something. Stata's
optimizer does not handle non-smooth densities like those of uniform
or triangular very graciously; you'll see a lot of error messages
(derivatives cannot be computed) when it has to work near the boundary
of the support.

-- 
Stas Kolenikov, also found at http://stas.kolenikov.name
Small print: Please do not reply to my Gmail address as I don't check
it regularly.
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