= Invariance property of maximum likelihood estimation
If $\widehat\theta$ maximizes a <likelihood function> $L(\theta)$ and the target is $\eta=g(\theta)$, then $g(\widehat\theta)$ maximizes the induced <likelihood> $L_\eta(\eta)=\sup_{\theta:g(\theta)=\eta}L(\theta)$. Every fibre supremum is at most $L(\widehat\theta)$, and the fibre containing $\widehat\theta$ attains that value. No one-to-one assumption on $g$ is needed. For example, estimating a normal mean $\mu$ by $\overline X$ gives the <maximum-likelihood estimate> $\overline X^2$ of $\mu^2$; the <likelihood> for a nonzero squared mean compares both possible signs. This rule concerns maximization, not unbiasedness.
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