How to prove $\mathcal{I}_{1}(\eta) = \mathcal{I}_{1}(\theta)[h'(\eta)]^{2}$ where $\mathcal{I}_{1}$ is the Fisher information and $\theta = h(\eta)$?

Question

I am trying to apply the following definition of Fisher Information: \begin{align*} \mathcal{I}_{1}(\theta) = \mathbb{E}_{\theta}\left[\left(\frac{\partial}{\partial\theta}\ln f(x_{1}|\theta)\right)^{2}\right] \end{align*}

But I have not succeeded so far. This is my attempt so far: \begin{align*} \mathcal{I}_{1}(\eta) & = \mathbb{E}_{\eta}\left[\left(\frac{\partial}{\partial\eta}\ln f(x_{1}|\eta)\right)^{2}\right]\\\\ & = \mathbb{E}_{\eta}\left[\left(\frac{\partial\theta}{\partial\eta}\frac{\partial}{\partial\theta}\ln f(x_{1}|\eta)\right)^{2}\right] \end{align*}

Can someone help me to finish the demonstration?

Answer 1

It's just a consequence of the chain rule. The density of $X$ is $f(x; \theta) = f(x; h(\eta))$. Therefore, viewing it as a function of $\eta$, we have \begin{align} \frac{d\ln f(x; h(\eta))}{d\eta} = \frac{1}{f(x; h(\eta))}\frac{df(x; \theta)}{d\theta}\frac{d\theta}{d\eta} = \frac{1}{f(x; \theta)}\frac{df(x; \theta)}{d\theta}h'(\eta) = \frac{d\ln f(x; \theta)}{d\theta}h'(\eta). \end{align}

It then follows that \begin{align} I(\eta) &= \int_{\mathbb{R}}\left[\frac{d\ln f(x; h(\eta))}{d\eta}\right]^2f(x;h(\eta))dx \\ &= \int_{\mathbb{R}}\left[\frac{d\ln f(x; \theta)}{d\theta}h'(\eta)\right]^2f(x; \theta)dx \\ &= [h'(\eta)]^2\int_{\mathbb{R}}\left[\frac{d\ln f(x; \theta)}{d\theta}\right]^2f(x; \theta)dx \\ &= [h'(\eta)]^2I(\theta). \end{align}

Answer 2

Start out from likelihood function $L(\theta;Y_1)$ (for a single observation) and make the reparametrization $\theta = h(\eta)$ to get the reparametrized log-likelihood $L(\eta;Y_1) = L(h(\eta);Y_1)$. Then the log-likelihood function of $\eta$ is $\ell(\eta;Y_1) = \ell(h(\eta);Y_1)$ so the Fisher information for a single observation is

\begin{align*} I_1(\eta) & = E_\theta\left[\left(\frac{d \ell(\eta;Y_1)}{d\eta}\right)^2\right]\\ & = E_\theta\left[\left(\frac{d \ell(h(\eta;Y_1))}{d h(\eta)}\frac{dh(\eta)}{d\eta}\right)^2\right]\\ & = E_\theta\left[\left(\frac{d \ell(\theta;Y_1)}{d \theta}\right)^2\right]\left(\frac{dh(\eta)}{d\eta}\right)^2\\ & = I_1(\theta)\left(\frac{dh(\eta)}{d\eta}\right)^2. \end{align*}