MLE for a Poisson rate

Question

You observe n independent counts x1, \dots, xn (say, trades arriving per second) modeled as 	ext{Poisson}(\lambda) with unknown rate \lambda. Derive the maximum likelihood estimator of \lambda, verify it is a maximum, and state its bias and variance.

Accepted Answer

Likelihood. Each count contributes e^{-\lambda}\lambda^{xi}/xi!, so L(\lambda) = \prod{i=1}^n \frac{e^{-\lambda}\lambda^{xi}}{xi!}, \qquad \ell(\lambda) = -n\lambda + \left(\sumi xi\right)\ln\lambda - \sumi \ln(xi!). First-order condition. \ell'(\lambda) = -n + \frac{\sumi xi}{\lambda} = 0 \implies \hat{\lambda} = \frac{1}{n}\sumi xi = \bar{x}. Second-order check. \ell''(\lambda) = -\dfrac{\sumi xi}{\lambda^2} < 0, so \ell is concave and \hat{\lambda} = \bar{x} is the global maximum. Properties. - Unbiased: \mathbb{E}[\hat{\lambda}] = \frac{1}{n}\sumi \mathbb{E}[xi] = \lambda. - Variance: \op…

MLE for a Poisson rate

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