1) Since F is neither concave nor convex, Jensen's inequality doesn't apply.

2) Exact distributions aren't available for this problem, though 2nd order
asymptotics have been worked out (search under "curved exponential family")
that are quite good.

3) The argument behind what the Stata guy says is, presumably, that the
parameter space for b is unbounded, so the usual asymptotic approximation
for its distribution, should generally be ok.  Since F is monotone,
applying these limits to F(x'b) is also (asymptotically) valid. Of course,
since p is bounded, there is nothing that keeps the first interval bounded
between zero and one. It's known that these approximations can be very bad,
as for example with some Wald-type tests.

Doug Rivers


Quoting "Franzese, Robert" <[log in to unmask]>:

> Hi all,
>
>
>
> In preparing some pedagogy, my coauthor and I came across the following
> in Stata's stats FAQ:
>
>
>
> "Generating the confidence intervals for the index [i.e., for XB] and
> then converting them to probabilities to get confidence intervals for
> the predicted probabilities is better than estimating the standard error
> of the predicted probabilities and then generating the confidence
> intervals directly from that standard error. The distribution of the
> predicted index is closer to normality than the predicted probability."
>
> Title                Prediction confidence intervals after logistic
> regression
>
> Author            Mark Inlow, StataCorp
>
> Date                April 1999; minor revisions May 2005
>
> http://www.stata.com/support/faqs/stat/prep.html
>
>
>
> This struck us as problematic; given Jensen's inequality, it would seem,
> if we are correct, that this would be accurate, if at all, only in near
> p=0,.5,1, where the function p=F(XB) is/becomes
> approximately/increasingly linear. An alternative, the delta method:
>
> V(p) approx= [dF/dB]'V(B)[dF/dB] for levels,
>
> V(p) approx= [dF(X1,*)/dB- dF(X0,*)/dB]'V(B)[ dF(X1,*)/dB- dF(X0,*)/dB]
> for differences,
>
> V(dp/dx)= [d(dF/dx)/dB]'V(B)[ d(dF/dx)/dB] for marginals,
>
> seems more appropriate to us (and apparently to Greene also: pp. 674-76
> of 5th ed.). But, then again, it, too, relies on a (Taylor series)
> linear approximation, so maybe it's no more or less appropriate than the
> easier tack suggested by Inlow of calculating V(XB) and c.i.'s for that,
> and transforming those results to p, delta(p), or d(p). (The
> Taylor-series linear approximation aspect of the delta method, for its
> part, would seem to us least problematic, if it's problematic at all,
> for the last and first of these.)
>
>
>
> Anyone have some thoughts or definitive references on the matter? Are we
> right to doubt the quality of answers derived from the simpler
> V(XB)-first-then-plug-and-chug method? (Yes, we know you can just
> simulate any of these instead; we want some valid analytics also
> though.)
>
>
>
> Thanks!
>
> Rob
>
>
>
> ************************************************************************
> **
>
> Robert (Rob) J. Franzese, Jr.                  US Mail:   (ISR Room
> 4256)
>
> Assoc. Prof. Political Science                              P.O. Box
> 1248
>
> The University of Michigan                       Ann Arbor, MI
> 48106-1248
>
> Research Assoc. Prof.                        TeleComm:
> [log in to unmask]
>
> Center for Political Studies                        734-936-1850
> (office)
>
> Institute for Social Research,                         734-764-3341
> (fax)
>
> 426 Thompson St., Room 4256
> http://www-personal.umich.edu/~franzese
>
> ************************************************************************
> **
>
>
>
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