Log-normal median by invariance

Question

Server response times are log-normal, so $\ln X_i$ is normal with mean $\mu$ and variance $\sigma^2$ . The logs of three response times are $y_i = 0.5, 1.0, 1.5$ . The median of a log-normal equals $e^{\mu}$ .

What is the maximum likelihood estimate of the median response time?

Accepted Answer

The log-scale mean MLE is the average of the logs: \hat\mu = \frac{0.5 + 1.0 + 1.5}{3} = \frac{3}{3} = 1. The median is e^{\mu}, a one-to-one function of \mu, so by invariance the median's MLE is \widehat{	ext{median}} = e^{\hat\mu} = e^{1} = e \approx 2.718. Takeaway. The MLE of the log-normal median is e^{\hat\mu} with \hat\mu the mean of the logs, thanks to invariance.

Log-normal median by invariance

Your answer

More Statistics questions