 Invariance property of maximum likelihood estimation

If $\hat{\theta }$ maximizes a likelihood function $L(\theta )$ and the target is $\eta =g(\theta )$, then $g(\hat{\theta })$ maximizes the induced likelihood $L_{\eta }(\eta )={\sup }_{\theta :g(\theta )=\eta }L(\theta )$. Every fibre supremum is at most $L(\hat{\theta })$, and the fibre containing $\hat{\theta }$ 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.