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I have a question about a proof (about MLE) but let me firstly give you appropriate context.

An estimator T is called maximum likelihood estimator of $\theta$, if: \begin{equation}L(T(\boldsymbol{x});\boldsymbol{x}) = max\space{} L(\theta ;\boldsymbol{x}) \end{equation}

Theorem

if $\gamma = g(\theta)$ and $g$ is bijective, i.e $\theta = g^{-1}(\gamma)$, then $\hat{\theta}$ is a MLE for $\theta$ iff $g(\hat{\theta}) = \hat{\gamma}$ is a MLE for $\gamma$

The proof: The likelihood function is $L(\theta;\boldsymbol{x}) = p(\boldsymbol{x};\theta)$. since $g$ is bijective the likelihood function with respect to $\gamma$ is given by $\tilde{L}(\gamma;\boldsymbol{x})=p(\boldsymbol{x};g^{-1}(\gamma))$. Furthermore,we have

$\tilde{L}( \hat{\gamma};\boldsymbol{x}) \geq \tilde{L}( \gamma;\boldsymbol{x})$ for all $\gamma$

$\Leftrightarrow p(\boldsymbol{x};g^{-1}(\hat{\gamma})) \geq p(\boldsymbol{x};g^{-1}(\gamma))$ for all $\gamma$

$\Leftrightarrow p(\boldsymbol{x}:\hat{\theta}) \geq p(\boldsymbol{x};\theta)$ for all $\theta$

$\Leftrightarrow L(\hat{\theta};\boldsymbol{x}) \geq L(\theta;\boldsymbol{x})$ for all $\theta$

The thing that is bothering me about the proof is this statement:

since $g$ is bijective the likelihood function with respect to $\gamma$ is given by $\tilde{L}(\gamma;\boldsymbol{x})=p(\boldsymbol{x};g^{-1}(\gamma))$

But in this case $\tilde{L}(\gamma ;\boldsymbol{x}) = p(\boldsymbol{x}:\theta )$ which is a function of $\theta$ but how can that be? I am missing something important...

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Because $p$ is taken to be a function of $\theta$ and not $\gamma$. So if we want to write the likelihood in terms of $p$ and the parameter $\gamma$, then we need to plug in the value for $\theta$ corresponding to $\gamma$, which is $g^{-1}(\gamma)$.

For instance, take a single observation $x$ from an exponential$(\lambda)$ distribution, where $\lambda$ is the rate parameter. The likelihood function is then $L(\lambda) = \lambda e^{- \lambda x}$. But the likelihood expressed in terms of the mean parameter $\mu = 1 / \lambda$ (so $g(t) = 1 / t$) is $\tilde{L}(\mu) = e^{- x / \mu} / \mu$, which is simply the original likelihood evaluated at $g^{-1}(\mu) = 1 / \mu$.

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