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Re: st: RE: Inequaltity constraints


From   Stas Kolenikov <skolenik@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: RE: Inequaltity constraints
Date   Thu, 19 Jan 2006 12:15:05 -0600

On 1/19/06, Maarten Buis <M.Buis@fsw.vu.nl> wrote:
> I am not StataCorp, but I do have a question about this trick. If I remember
> correctly from my MLE class, than inference from MLE estimates are based on
> the result that the sampling distributions of MLE estimates are, at least
> in the wonderful Kingdom of Asymptotia, normal distributions. If your
> estimate is at the boundary, than my intuition says that that result would
> break down. Maybe the sampling distribution would become a truncated normal
> or something more exotic. That would at least justify setting the standard
> error to missing. Any thoughts about that?

Your intuition is exactly right. If the true parameter value is on the
boundary, then (think in 1D) with asymptotic normality, you'd have a
positive value half of the time, and the negative value, which you
constrain to be zero, the other half. Hence, the distribution will be
a mixture of point mass at zero with probability 1/2, and half-normal,
with probability 1/2. The distribution of the likelihood ratio will be
a square of that: 1/2*delta(0) + 1/2*chi^2_1. The results are known in
asymptotic theory, but for some reason rarely make it to the books. In
Ferguson, for instance, it is given as an exercise. For basic reading,
find Chernoff (1954) in Annals of Mathematical Statistics, and for
advanced reading, Andrews (1999, 2001) in Econometrica.

The point with the maximizer, however, is that of numeric
optimization. Even if you do know how to correct the asymptotic
distribution and/or s.e.s, there is no way to force it to work... But
generally yes, it is probably better to give a missing instead of a
misleading standard error. Note however that the reported s.e. is
estimated based on the curvature of the likelihood around the maximum,
so the scale parameter of the aforementioned normal distribution
should be estimated correctly (at least if the sandwich/robust
estimator is used; I would not put much trust into OIM/OPG
estimators); it needs to be tweaked around though somewhat to make it
a square root of the variance of the resulting distribution, if that
matters. It is easier to deal with LRT in this situation.

--
Stas Kolenikov
http://stas.kolenikov.name

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