A Poisson mean

Topic: Estimation · more questions on this topic
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Let X_1,\ldots,X_n be a random sample from \operatorname{Poisson}(\theta). Find the maximum likelihood estimator of \theta, show that it is unbiased, and find its mean squared error. Is it consistent?

Worked answer (click to show)

The log-likelihood is \sum x_i\log\theta-n\theta-\sum\log x_i!, with derivative \sum x_i/\theta-n. Setting it to zero gives \widehat\theta=\overline X; the second derivative -\sum x_i/\theta^2 is negative, so it is a maximum. (If every x_i=0 the likelihood e^{-n\theta} is largest at \theta=0, which is again \overline x.)

\mathbb{E}\overline X=\theta, so the estimator is unbiased, and its mean squared error is its variance, \operatorname{Var}\overline X=\theta/n, because a Poisson variance equals its mean. That tends to zero, so \overline X is consistent, by Chebyshev’s inequality or by the law of large numbers.