Shifted exponential MLE for sensor floors

Question

A sensor reads values above an unknown baseline $a$ with an exponential tail, density $f(x \mid a, \lambda) = \lambda e^{-\lambda(x-a)}$ for $x \ge a$ , both parameters unknown. Five readings are 10, 14, 11, 20, 15.

Find the maximum likelihood estimates of $a$ and $\lambda$ .

Accepted Answer

The likelihood \lambda^n \exp(-\lambda\sumi (xi - a)) is valid only for a \le \mini xi and increases in a, so the threshold MLE is the smallest observation: \hat a = \mini xi = 10. Then \hat\lambda = \dfrac{n}{\sumi (xi - \hat a)}. The excesses xi - 10 are 0, 4, 1, 10, 5, summing to 20, so \hat\lambda = \frac{5}{20} = \frac{1}{4} = 0.25. Takeaway. Threshold equals the minimum; rate equals the reciprocal of the mean excess above it.

Shifted exponential MLE for sensor floors

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