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?