Shifted exponential MLE for rainfall over a threshold

Question

Daily rainfall on rainy days follows a shifted exponential with density $f(x \mid a, \lambda) = \lambda e^{-\lambda(x - a)}$ for $x \ge a$ , both parameters unknown. Six days record 2, 5, 3, 8, 4, 2 (millimeters).

Estimate $a$ and $\lambda$ by maximum likelihood.

Accepted Answer

The threshold MLE is the sample minimum, \hat a = 2. The excesses xi - 2 are 0, 3, 1, 6, 2, 0, summing to 12 over n = 6, so \hat\lambda = \frac{n}{\sumi (xi - \hat a)} = \frac{6}{12} = \frac{1}{2} = 0.5. Takeaway. The minimum sets the floor; the mean amount above the floor sets the reciprocal of the rate.

Shifted exponential MLE for rainfall over a threshold

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