Dear all,
excuse my insistence, but it is to interesting a problem to be left
ignored ...

On Wednesday 01 February 2006 01:05, Douglas Rivers wrote:
> The quantity to be estimated is \hat p = F(x'\hat b) where x is fixed and
> \hat b is approximately N(b, V). If F were linear, then E(\hat p) =
> F(x'E(b)). If F were convex, then E(\hat p) > F(x'E(b)), by Jensen's
> inequality, etc.

That is true of course (we all read our books ;-), but this is *exactly* why
Inlow is right in rejecting
the computation of confidence levels based on the assumption that
the distribution of \hat p is normal. In that case, it is critical how you
determine the mean of this distribution and thus Jensen's inequality
becomes relevant.
But Inlow's suggestion does *not* rely on a normality
assumption regarding the distribution of the \hat p,
but the delta method does,
which is based either on p ~ N(E(p(x'b)),AVAR(p(x'b)))
or p ~ N(p(x'E(b)),(p(1-p)x'AVAR(b)xp(1-p))).

But that was not my point. The point I wanted to make is the following:
The direct way to find confidence limits would be
to solve the equations
\int_0^q f_{LN}(t,x'E(b),x'AVAR(b)x) dt = \alpha
and
\int_0^q f_{LN}(t,x'E(b),x'AVAR(b)x) dt = 1-\alpha
for q (that is, for the upper integral limit!).
Here, f_{LN}(x,\mu,\sigma^2) is the density of the (one-dimensional)
logistic-normal distribution [Aitchison + Shen 1980
http://links.jstor.org/sici?sici=0006-3444%28198008%2967%3A2%3C261%3ALDSPAU%3E2.0.CO%3B2-I].
That is,
f_{LN}(p,\mu,\sigma^2) =
(2\pi\sigma^2)^{-1/2}(p*(1-p))^{-1}\exp\left(-(1/(2sigma^2)\left(\frac{p}
{1-p}-\mu\right)^2\right).
This logistic-normal distribution arises if a normally distributed random
variable T (in the present case x'b) is transformed to
p(T) = exp(T)/(1+exp(T)).

[OK- there was a mistake in my last email:
That's not exp(t)/(1+exp(t))f_N(x,\mu,\sigma^2)]

Inlow suggests solving instead
\int_{-\infty}^{q^*} f_N(t,x'E(b),x'AVAR(b)x) dt = \alpha
and
\int_{-\infty}^{q^*} f_N(t,x'E(b),x'AVAR(b)x) dt = 1-\alpha
for q^*. This of course is *way* easier than to solve
the previous equations, since there are well known
approximations to the quantile function (inverse CDF) of the
normal distribution.

Hence it remains to be seen whether
\int_{-\infty}^{q^*} f_N(t,x'E(b),x'AVAR(b)x) dt
is equal to
\int_0^{p(q^*)} f_{LN}(t,x'E(b),x'AVAR(b)x) dt
(note the different integrands and the different integral limits).
It might be possible to spell this out via integration by substitution but
that might become tedious. (Nevertheless, I suspect that the equality holds.)

Jensen's inequality in the present context however concerns the comparison of
\int_{-\infty}^{\infty} p(t) f_N(t,x'E(b),x'AVAR(b)x) dt = E[p(x'(b))]
and
p[\int_{-\infty}^{\infty} t f_N(t,x'E(b),x'AVAR(b)x)dt ] = p[x'E(b)]
(note again the integral limits and the different integrands).
Therefore, it does not concern Inlows suggestion, but it
may be relevant if the delta method is used.

Best,

Martin

-------------------------------------------------
Martin Elff
Department of Social Sciences
University of Mannheim
D7, 27, Room 407
68131 Mannheim
Germany

Phone: +49-621-181-2093
Fax: +49-621-181-2099
E-Mail: [log in to unmask]
Homepage:
http://webrum.uni-mannheim.de/sowi/elff
http://www.sowi.uni-mannheim.de/lspwivs/
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