Fisher Information of a function of a parameter

Suppose that $X$ is a random variable for which the p.d.f. or the p.f. is $f(x|\theta)$, where the value of the parameter $\theta$ is unknown but must lie in an open interval $\Omega$. Let $I_0(\theta)$ denote the Fisher information in $X.$ Suppose now that the parameter $\theta$ is replaced by a new parameter $\mu$, where $\theta = \psi(\mu)$ and $\psi$ is a differentiable function. Let $I_1(\mu)$ denote the Fisher information in $X$ when the parameter is regarded as $\mu.$ Show that $$I_1(\mu) = [\psi'(\mu)]^2 I_0[\psi(\mu)].$$

How would I do this? Do I need to use a Taylor expansion? Regardless, I would appreciate a written proof. This isn't for class but the above statement has been mentioned in texts without any detail whatsoever.

Thanks!

Solution 1:

By definition $I_{0}(\theta)=-\mathbb{E}\left[\frac{d^{2}\log f\left(X\vert\theta\right)}{d\theta^{2}}\right].$ So $I_{1}(\mu)=-\mathbb{E}\left[\frac{d^{2}\log f\left(X\vert\mu\right)}{d\mu^{2}}\right]$.By the chain rule we have $$I_{1}\left(\mu\right) = -\mathbb{E}\left[\frac{d^{2}\log f\left(X\vert\theta\right)}{d\theta^{2}}\left(\frac{d\theta}{d\mu}\right)^{2}+\frac{d\log f\left(X\vert\theta\right)}{d\theta}\frac{d^{2}\theta}{d\mu^{2}}\right]$$ $$= -\mathbb{E}\left[\frac{d^{2}\log f\left(X\vert\theta\right)}{d\theta^{2}}\right]\left(\frac{d\theta}{d\mu}\right)^{2}+\mathbb{E}\left[\frac{d\log f\left(X\vert\theta\right)}{d\theta}\right]\frac{d^{2}\theta}{d\mu^{2}}. $$

But $\mathbb{E}\left[\frac{d\log f\left(X\vert\theta\right)}{d\theta}\right]=0.$ So we get
$$ I_{1}\left(\mu\right)=I_{0}\left(\theta\right)\left(\frac{d\theta}{d\mu}\right)^{2}.$$