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


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 $\gamma= \frac{1}{\theta} = \text{mean of X~exp}$?

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

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.