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Suppose I have a distribution from a known parametric family f(x; θ). I have a sample from that distribution. From the sample, I estimate values for the parameters. Suppose I have estimators that I know to have some desirable property. For instance, they may be minimum variance unbiased estimators, or maximum likelihood estimators. From the parameters I can numerically integrate and derive moments, if they exist.

Are the good properties of the estimators inherited by such computed moments? If the parameter estimates are known to be minimum variance unbiased, is the mean computed from such an integration a minimum variance unbiased estimate of the mean? If the parameters are the maximum likelihood parameters, is the mean so computed the maximum likelihood estimate of the true mean?

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You have data from some distribution family $f(y; \theta)$ and some estimator $\hat{\theta}$ of $\theta$ with "good properties". But you are interested in some function of $\theta$, say $g(\theta)$ (which might be moments).

Then you want to estimate $g(\theta)$ in the natural way with $g(\hat{\theta})$, and ask if that natural estimator "inherits" the good properties of the estimator $\hat{\theta}$. That is a natural question!

The answer is "it depends". For UMVU estimators, the answer is NO. For maximum likelihood estimators, the answer is "yes", see Invariance property of maximum likelihood estimator? . In fact, that property for the maximum likelihood estimators was one reason they was preferred by Fisher.

There are no general results here, for each property you are interested in, analyze.

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