Dear Allstater,
Let theta be the parameter indexing a regression function y=f(x;theta) + eps
where x is the regressor and eps is the error term. Let theta_hat is an
estimate for theta, obtained from a sample (X,Y) ={x_i, y_i, i=1..N}.
For reasons I explain below in detail, I would like to estimate the
UNCONDITIONAL variance of theta_hat (denoted by Var(theta_hat)) which can be
factored as E(Var(theta_hat|x)) + Var(E(theta_hat|x)), where E() denotes the
expected value and Var(theta_hat|x) and E(theta_hat|x) are the usual
variance and expected value for theta_hat CONDITIONALLY to the regressor x.
I guess that E(Var(theta_hat|x)) and Var(E(theta_hat|x)) can be estimated
using boostrapping methods, but i was wondering if there was no other
technics, and I would be very interested if you could point me to a paper
explaining how to estimate E(Var(theta_hat|x)) and Var(E(theta_hat|x)).
Many Txs
Rgds
Tom
Complete question to Allstaters
Let theta be the parameter indexing a regression function y=f(x;theta) + eps
where x is the regressor and eps is the error term. Let theta_hat is an
estimate for theta, obtained from a sample (X,Y) ={x_i, y_i, i=1..N}.
Let delta be the true mean of x, and let delta_hat be an estimate for delta,
obtained from the sample X.
I am interested in estimating f(x;theta) at x=delta, and I am proposing to
use f(delta_hat; theta_hat) with this purpose. I am particularly interested
in the variance of f(delta_hat; theta_hat).
With this purpose, I am using the delta method which approximate
f(delta_hat; theta_hat) by f(delta; theta) + Df/Ddelta * (delta_hat-delta) +
Df/Dtheta * (theta_hat-theta), where Df/Dz denotes the first derivative of
the function f by one of the parameters.
Obviously, the variance of that approximation depends on Var(theta_hat),
Var(delta_hat), and Cov(theta_hat,delta_hat).
Although theta_hat is estimated conditionally to the sample X, I suppose
that Var(theta_hat) to be used in the approximation above is not the usual
variance obtained CONDITIONALLY to the regressor, but is the UNCONDITIONAL
variance Var(theta_hat).
Var(theta_hat) can be easilly factored into E(Var(theta_hat|x)) +
Var(E(theta_hat|x)), where E() denotes the expected value.
As mentioned above, I guess that E(Var(theta_hat|x)) and Var(E(theta_hat|x))
can be estimated using boostrapping methods, but i was wondering if there
was no other technics, and I would be very interested if you could point me
to a paper explaining how to estimate E(Var(theta_hat|x)) and
Var(E(theta_hat|x)). Besides, I would be even more interested if this paper
was in the context of a setting similar to mine, i.e., the use of a delta
method in a regression setting.
PS: It should be noted that in case theta_hat is unbiased estimate for theta
(e.g., when using OLS for a linear regression), E(theta_hat|x) = theta and
thus Var(theta_hat|x) restricts to E(Var(theta_hat|x)), but in most cases,
e.g., non-linear regression, theta_hat is only asymptotically unbiased.
re-PS, the problem of estimation of Cov(theta_hat,delta_hat) will be similar
and is not mentioned here.
Best regards
Tom
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