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I've calculated the $\theta_{est}$ using MLE for the exponential distribution (with parameter $\theta$).

Then I've calculated the median $Md$ using $\theta_{est}$.

How can I use the Delta method to calculate $s.e. Md_{est}$?

The Delta method (for estimating $Var(\theta)$) requires function $g(\gamma)$ and a "parametrization" $\gamma$ of $\theta$. Then

$$Var(\theta)=g'(\gamma)^2Var(\gamma)$$

But what to use as $\gamma$ to get $Var(Md_{est})$?

Could I use $\gamma=\theta_{est}$ since that's the only thing I have here? But isn't its derivative 0? What about using $\gamma= \frac{1}{\theta} = \text{mean of X~exp}$?

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  • $\begingroup$ Are you sure you need the delta method? $\endgroup$ Feb 25, 2017 at 23:53

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You don't need the delta method here! The density function of the exponential distribution https://en.wikipedia.org/wiki/Exponential_distribution is $$ f(x;\theta) = \frac1\theta e^{-x/\theta}, \qquad x> 0 $$ Then you can find the likelihood function, maximize it to find the maximum likelihood estimator $\hat{\theta} = \bar{x}$ which has variance $\theta^2/n$. But we can calculate that the median is $M=\ln 2 \theta$ so the variance of the maximum likelihood estimator of the median is simply $(\ln 2)^2 \theta^2 / n$. No need for delta method here.

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