Ben Bolker sent me a private email rightfully correcting me that was
factually wrong when I wrote that ML /is/ a numerical method (I had written
sloppily and under time pressure). He is of course right to point out that
not all maximum likelihood estimators require numerical methods to solve.
Further, only numerical optimization will show the behavior discussed in
this post for the given reasons. (I hope this post isn't yet another blooper
of mine at 5 a.m. in the morning).
Best,
Daniel
Daniel Malter wrote:
>
> With respect, your statement that R's optim does not give you a reliable
> estimator is bogus. As pointed out before, this would depend on when optim
> believes it's good enough and stops optimizing. In particular if you
> stretch out x, then it is plausible that the likelihood function will
> become flat enough "earlier," so that the numerical optimization will stop
> earlier (i.e., optim will "think" that the slope of the likelihood
> function is flat enough to be considered zero and stop earlier than it
> will for more condensed data). After all, maximum likelihood is a
> numerical method and thus an approximation. I would venture to say that
> what you describe lies in the nature of this method. You could also follow
> the good advice given earlier, by increasing the number of iterations or
> decreasing the tolerance.
>
> However, check the example below: for all purposes it's really close
> enough and has nothing to do with optim being "unreliable."
>
> n<-1000
> x<-rnorm(n)
> y<-0.5*x+rnorm(n)
> z<-ifelse(y>0,1,0)
>
> X<-cbind(1,x)
> b<-matrix(c(0,0),nrow=2)
>
> #Probit
> reg<-glm(z~x,family=binomial("probit"))
>
> #Optim reproducing probit (with minor deviations due to difference in
> method)
> LL<-function(b){-sum(z*log(pnorm(X%*%b))+(1-z)*log(1-pnorm(X%*%b)))}
> optim(c(0,0),LL)
>
> #Multiply x by 2 and repeat optim
> X[,2]=2*X[,2]
> optim(c(0,0),LL)
>
> HTH,
> Daniel
>
>
>
> yehengxin wrote:
>>
>> What I tried is just a simple binary probit model. Create a random data
>> and use "optim" to maximize the log-likelihood function to estimate the
>> coefficients. (e.g. u = 0.1+0.2*x + e, e is standard normal. And y =
>> (u > 0), y indicating a binary choice variable)
>>
>> If I estimate coefficient of "x", I should be able to get a value close
>> to 0.2 if sample is large enough. Say I got 0.18.
>>
>> If I expand x by twice and reestimate the model, which coefficient should
>> I get? 0.09, right?
>>
>> But with "optim", I got something different. When I do the same thing in
>> both Gauss and Matlab, I can exactly get 0.09, evidencing that the
>> coefficient estimator is reliable. But R's "optim" does not give me a
>> reliable estimator.
>>
>
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