Laplace center MLE with an even sample

Question

Latency deviations are modeled as Laplace centered at $\mu$ with known scale $b$ , density $f(x \mid \mu) = \dfrac{1}{2b} e^{-|x - \mu|/b}$ . Four deviations are 2, 4, 6, 8.

What is the maximum likelihood estimate of $\mu$ , and why is it not unique here?

Accepted Answer

The MLE minimizes \sumi xi - \mu. With an even sample size the two middle order statistics are 4 and 6, and any \mu in the closed interval [4, 6] gives the same minimal total absolute deviation (moving within that interval trades one unit gained against one unit lost). So the MLE is the whole interval [4, 6], not a single point. The usual convention reports the midpoint, \hat\mu = \frac{4 + 6}{2} = 5, but any value in [4, 6] is an equally valid maximizer. Takeaway. For an even-sized Laplace sample the MLE of the center is the entire interval between the two middle values; the reported median …

Laplace center MLE with an even sample

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