Discussion
If T is MLE of \(\theta\) and \(\Psi(\theta)\) is one-to-one function of \(\theta\), then \(\Psi(T)\) is MLE of \(\Psi(\theta)\). This follows from :
Answer: A
It follows from the invariance property of MLE.
If T maximises the likelihood \(L(\theta)\) and \(\Psi\) is one-to-one, reparametrise by \(\eta = \Psi(\theta)\). The likelihood in \(\eta\) is \(L(\Psi^{-1}(\eta))\), which is largest when \(\Psi^{-1}(\eta) = T\), that is \(\eta = \Psi(T)\). So the MLE of \(\Psi(\theta)\) is \(\Psi(T)\).
This is an exact property for every sample size. Asymptotic properties, consistency and regularity conditions concern the behaviour of the MLE as n grows and are not what this result rests on.
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