MLE for a shifted exponential threshold and rate

Question

Message latencies have a hard physical floor $a$ (nothing arrives faster) followed by an exponential tail, modeled as a shifted exponential with density

$f(x \mid a, \lambda) = \lambda\, e^{-\lambda(x - a)}, \qquad x \ge a,$

with both the threshold $a$ and the rate $\lambda > 0$ unknown. Five latencies (in ms) are 5, 8, 6, 12, 9.

Derive the maximum likelihood estimators of $a$ and $\lambda$ and compute them.

Accepted Answer

The likelihood is L(a, \lambda) = \lambda^n \exp\!\big(-\lambda\sumi (xi - a)\big) but only for a \le \mini xi; if a exceeds any observation the density is zero there and the likelihood collapses to zero. Estimate a first. For any fixed \lambda 0, the exponent is -\lambda\big(\sumi xi - n a\big), which increases as a increases. So we push a as high as allowed, right up to the constraint: \hat a = \mini xi. Now fix a = \hat a and maximize over \lambda: with \ell(\lambda) = n\ln\lambda - \lambda\sumi (xi - \hat a), \ell'(\lambda) = \frac{n}{\lambda} - \sumi (xi - \hat a) = 0 \;\Longrightarrow\;…

MLE for a shifted exponential threshold and rate

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