# Solved – Proof of invariance property of MLE

I am reading the proof of the invariance property of MLE from Casella and Berger.

In this proof we parametrize :
\$eta = tau(theta)\$

There we define the induced likelihood function:

\$ L_{1}^*(eta|x) = sup_{theta|tau(theta) = eta} L(theta|x) tag{1}\$

I have subscripted L*(\$eta\$|x) by 1 to differentiate between the induced likelihood of \$eta \$ and the Likelihood of \$eta\$ which are both denoted by \$L^*(eta|x)\$

I am not sure why this is being done. (In what follows,L* is the likelihood of \$eta\$ ).
If \$theta_1\$ and \$theta_2\$ are such that \$tau(theta_1) = tau(theta_2)\$ then \$L(theta_1|x)\$ = \$L^*(eta = tau(theta_1)|x)\$= \$L^*(eta = tau(theta_2)|x)\$ = \$L(theta_2|x\$) since \$tau(theta_1)\$ =\$tau(theta_2)\$

Hence there is no need of the supremum in (1).

Where do I misunderstand?

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Perhaps the issues here are best understood in the context of an example. Suppose that we are interested in estimating the mean of a normal model with variance 1 i.e. we are considering models of the form \$N(theta,1)\$. In this case, the likelihood (for a single data point \$x\$) is (ignoring the constant) \$L(theta | x)=exp(-(x-theta)^2/2)\$.

Suppose that we are actually interested in a function of the mean, call it \$eta=tau(theta)\$. How to define the likelihood \$L(eta|x)\$? If \$tau\$ is invertible then we just define \$L(eta|x)\$ to be \$L(theta=tau^{-1}(eta) | x)\$ i.e. we set \$theta\$ equal to the unique value corresponding to the chosen value of \$eta\$. e.g. if \$tau(theta)=2theta\$ then \$L(eta | x):=L(theta=frac{eta}{2} |x)\$.

What if \$tau\$ is not invertible? e.g. \$tau(theta)=theta^2\$. Should \$L(eta|x)\$ be \$L(theta=+sqrt{eta} | x)\$ or should it be defined as \$L(theta=-sqrt{eta} | x)\$? These two values will usually be different, so the likelihood \$L(eta|x)\$ is undefined. Hence Casella and Berger define the induced likelihood. With the chosen definition, it turns out that the invariance property (which is obvious when \$tau\$ is invertible) still holds.

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# Solved – Proof of invariance property of MLE

I am reading the proof of the invariance property of MLE from Casella and Berger.

In this proof we parametrize :
\$eta = tau(theta)\$

There we define the induced likelihood function:

\$ L_{1}^*(eta|x) = sup_{theta|tau(theta) = eta} L(theta|x) tag{1}\$

I have subscripted L*(\$eta\$|x) by 1 to differentiate between the induced likelihood of \$eta \$ and the Likelihood of \$eta\$ which are both denoted by \$L^*(eta|x)\$

I am not sure why this is being done. (In what follows,L* is the likelihood of \$eta\$ ).
If \$theta_1\$ and \$theta_2\$ are such that \$tau(theta_1) = tau(theta_2)\$ then \$L(theta_1|x)\$ = \$L^*(eta = tau(theta_1)|x)\$= \$L^*(eta = tau(theta_2)|x)\$ = \$L(theta_2|x\$) since \$tau(theta_1)\$ =\$tau(theta_2)\$

Hence there is no need of the supremum in (1).

Where do I misunderstand?