Laplace center MLE for seven readings

Question

Sensor offsets are modeled as Laplace centered at $\mu$ with known scale $b$ , density $f(x \mid \mu) = \dfrac{1}{2b} e^{-|x - \mu|/b}$ . Seven offsets are 10, 2, 7, 15, 4, 1, 9.

Estimate $\mu$ by maximum likelihood.

Accepted Answer

The Laplace center MLE is the sample median. Sorting the seven values gives 1, 2, 4, 7, 9, 10, 15, and the fourth (middle) value is \hat\mu = 7. Note how the median ignores the extreme 15, which is exactly the robustness the heavy-tailed Laplace model builds in. Takeaway. The Laplace MLE is the median, a robust center that is barely moved by outliers.

Laplace center MLE for seven readings

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