Laplace scale MLE after the median

Question

Trading errors are modeled as Laplace with unknown center $\mu$ and unknown scale $b$ , density $f(x \mid \mu, b) = \dfrac{1}{2b} e^{-|x - \mu|/b}$ . Five errors are 1, 3, 4, 6, 11.

Find the maximum likelihood estimate of the scale $b$ .

Accepted Answer

The log-likelihood is \ell(\mu, b) = -n\ln(2b) - \frac{1}{b}\sumi xi - \mu. Maximizing over \mu first gives \hat\mu = 	ext{median}. Sorting 1, 3, 4, 6, 11, the median is \hat\mu = 4. Now maximize over b with \mu fixed at \hat\mu. Differentiating -n\ln b - \frac{1}{b}\sumi xi - \hat\mu and setting it to zero, -\frac{n}{b} + \frac{1}{b^2}\sumi xi - \hat\mu = 0 \;\Longrightarrow\; \hat b = \frac{1}{n}\sumi xi - \hat\mu. The absolute deviations from 4 are 3, 1, 0, 2, 7, summing to 13, so \hat b = \frac{13}{5} = 2.6. Takeaway. The Laplace scale MLE is the mean absolute deviation from the median, …

Laplace scale MLE after the median

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