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My provided definition of the Fisher information $\mathcal{I}(\theta)$ is the expected value of the observed information $I(\theta)$, where $I(\theta)$ is the second derivative of the log-likelihood function of some distribution $F(x|\theta)$.

There is an exercise in my notes which mentions that the median of the exponential distribution can be estimated using maximum likelihood, using

$$ Q(p=0.5|\hat{\theta}) $$

where $Q$ is the quartile function. The exercise asks for me to find the Fisher information for this quantity.

I can't really understand what this means. Clearly $$ Q(p=0.5|\hat{\theta}) = \theta^{-1} \ln{2} $$ which I understand to be the median, but now what is the Fisher information? I don't think I have a distribution, do I? $\theta$ is the true value of the parameter, so it seems to me like I have an unknown quantity here... which leaves me with no idea how to derive the Fisher information.

I'm not looking for a drawn out answer if that's what is required here, but rather just an idea of what is meant by the Fisher information in this context.

Thanks!

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Hint:

Notice that the quantile function is a 1-to-1 transformation of $\theta$; therefore, we can invoke the invariance property of maximum likelihood estimators (see p16 here). This property implies that the log-likelihood of a particular median estimate (given $\theta$) is equal to the log-likelihood of $\theta$ (i.e., $\mathcal{L}_{\theta}(\theta)$), which allows us to derive the log-likelihood for the median $\mathcal{L}_m(m)$ by substitution and transformation:

$$\mathcal{L}_m(m) = \mathcal{L}_{\theta}\left(\frac{\ln{2}}{m}\right) \implies \mathcal{I}(\hat{m}) = E\left[\frac{d^2}{dm^2}\mathcal{L}_{\theta}\left(\frac{\ln{2}}{m}\right)_{m=\hat{m}}\right]$$

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