sample mean vs. estimated expected value

Scholars and gentlefolk:

Consider an iid sample {x_1,...x_n} believed to have been generated
from distribution function f(x;\theta,T) with a finite mean
\chi=E[x|\theta,T]. Suppose one estimates the parameters from the
sample and then calculates the conditional mean
\hat{\chi} = E[x|\hat{\theta},\hat{T}].

Assuming f() is the correct model, is there anything that can be said
in general about the relative small-sample bias of the estimated and
sample mean? I imagine they must be asymptotically equal in some sense,
but is there a reason to use one rather than the other? The
distribution in question is a truncated exponential...I suppose that
matters.

Any remarks, advice or pointers to the literature will be most welcome.

Best regards

Steve ***
Boreal Ecosystems Research Ltd.
http://www.berl.ab.ca

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