MLE for an exponential rate

Question

You observe n independent waiting times x1, \dots, xn (say, seconds between quote updates) modeled as 	ext{Exponential}(\lambda) with density \lambda e^{-\lambda x}. Derive the MLE of the rate \lambda, verify it is a maximum, and comment on its bias.

Accepted Answer

Likelihood. L(\lambda) = \prod{i=1}^n \lambda e^{-\lambda xi} = \lambda^n e^{-\lambda \sumi xi}, \qquad \ell(\lambda) = n\ln\lambda - \lambda\sumi xi. First-order condition. \ell'(\lambda) = \frac{n}{\lambda} - \sumi xi = 0 \implies \hat{\lambda} = \frac{n}{\sumi xi} = \frac{1}{\bar{x}}. Second-order check. \ell''(\lambda) = -\dfrac{n}{\lambda^2} < 0, so this is the global maximum. Bias. The MLE of the mean 	heta = 1/\lambda is \hat{	heta} = \bar{x}, which is unbiased with variance 	heta^2/n. But the rate estimator \hat{\lambda} = 1/\bar{x} is biased: \sumi xi is 	ext{Gamma}(n, \lambda), …

MLE for an exponential rate

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