I'm reading this book and on page 301 he states this theorem:
I found it very similar with the plug-in estimator, the application of this theorem is the plug-in estimator?
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This is called Functional Invariance property of ML. But, the estimator obtained is the plug-in estimator. So, the plug-in estimator for the function of the parameter is the MLE of the function of the parameter when the plugged in value is the MLE of the original parameter.
There's more than one thing called the plug-in estimator/principle.
Among things you might use that name for, this theorem is distinctive in how weak the assumptions are. $g$ is an arbitrary (measurable) function -- there are no assumptions that $g$ is (piecewise) smooth or continuous or is one:one at $\hat\theta$ or anything like that.