R help - Jul 2006 - Problems when computing the 1rst derivative of mixtures of densities

Hi everybody,

I am currently working on mixtures of two densities ( f(xi,teta)(1-teta)*f1(xi)
+ teta*f2(xi) ),
particularly on the behavior of the variance for teta=0 (so sample only
comes from the first distribution).
To determine the maximum likelihood estimator I use the Newton-Rapdon
Iteration. But when
computing the first derivative I get a none linear function (with several
asymptotes) which is completely
absurd.

This is my function to compute the first derivative:

phy=function(teta,vect1,vect2){
    return( sum(( vect2 - vect1) / (( 1 - teta) * vect1 + teta * vect2)))
}

note: vect1 and vect2 contains  values of the  two distributions computed
from sample previously extracted.

Beside, vect2 - vect1 is constant and ( 1 - teta) * vect1 + teta * vect2) is
linear and always defined so I am expecting
a linear function for the first derivative. To my mind, it's likely comes
from the operation of division but I don't understand
why results are skewed.

Have you got any suggestions, please?

-- 
Clément Viel
Student in engineering school of Polytech-Lille
http://www.polytech-lille.fr

	[[alternative HTML version deleted]]

What problem are you trying to solve?  The mixture proportion you are 
trying to estimate violates the assumptions that provide a normal 
approximation to the distribution of maximum likelihood estimators. 
These situations even violate the assumptions for 2*log(likelihood 
ratio) being approximately chi-square.

	  If you want hypothesis tests or confidence intervals, Pinheiro and 
Bates (2000) recommended using 2*log(likelihood ratio) modified as 
suggested by Monte Carlo results.  More recently, Bates has incorporated 
an 'mcmcsamp' function in the 'lme4' and 'Matrix'
libraries.

	  The best references I know on testing a parameter at a boundary are 
the following:

Pinheiro and Bates (2000) Mixed-Effects Models in S and S-PLUS 
(Springer, sec. 2.4)

Crainiceanu, Ruppert and Vogelsang (2003) ?Some properties of Likelihood 
Ratio tests in linear mixed models?

Crainiceanu, Ruppert, Claeskens, and Wand (2005) ?Exact Likelihood Ratio 
Tests for Penalized Splines?, Biometrika, 92(1)

	  Hope this helps.
	  Spencer Graves
p.s.  See also inline.

Cl?ment Viel wrote:
> Hi everybody,
> 
> I am currently working on mixtures of two densities ( f(xi,teta)>
(1-teta)*f1(xi) + teta*f2(xi) ),
> particularly on the behavior of the variance for teta=0 (so sample only
> comes from the first distribution).
> To determine the maximum likelihood estimator I use the Newton-Rapdon
> Iteration. But when
> computing the first derivative I get a none linear function (with several
> asymptotes) which is completely
> absurd.
Why do you think this is absurd?  Nonlinear estimation can be very 
difficult.

	  Also, have you reviewed the capabilities in R for working with 
mixtures of distributions?  I just got 167 hits for 
RSiteSearch("mixtures") and 49 for RSiteSearch("mixture
distributions",
"functions").
> 
> This is my function to compute the first derivative:
> 
> phy=function(teta,vect1,vect2){
>     return( sum(( vect2 - vect1) / (( 1 - teta) * vect1 + teta * vect2)))
> }
> 
> note: vect1 and vect2 contains  values of the  two distributions computed
> from sample previously extracted.
> 
> Beside, vect2 - vect1 is constant and ( 1 - teta) * vect1 + teta * vect2)
is
> linear and always defined so I am expecting
> a linear function for the first derivative. To my mind, it's likely
comes
> from the operation of division but I don't understand
> why results are skewed.
> 
> Have you got any suggestions, please?
> 
> 
> 
> ------------------------------------------------------------------------
> 
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