Estimating A Poisson Probability
A far more extreme case of a biased estimator being better than any unbiased estimator arises from the Poisson distribution. Suppose that X has a Poisson distribution with expectation λ. Suppose it is desired to estimate
(For example, when incoming calls at a telephone switchboard are modeled as a Poisson process, and λ is the average number of calls per minute, then e−2λ is the probability that no calls arrive in the next two minutes.)
Since the expectation of an unbiased estimator δ(X) is equal to the estimand, i.e.
the only function of the data constituting an unbiased estimator is
To see this, note that when decomposing e−λ from the above expression for expectation, the sum that is left is a Taylor series expansion of e−λ as well, yielding e−λe−λ = e−2λ (see Characterizations of the exponential function).
If the observed value of X is 100, then the estimate is 1, although the true value of the quantity being estimated is very likely to be near 0, which is the opposite extreme. And, if X is observed to be 101, then the estimate is even more absurd: It is −1, although the quantity being estimated must be positive.
The (biased) maximum likelihood estimator
is far better than this unbiased estimator. Not only is its value always positive but it is also more accurate in the sense that its mean squared error
is smaller; compare the unbiased estimator's MSE of
The MSEs are functions of the true value λ. The bias of the maximum-likelihood estimator is:
Read more about this topic: Bias Of An Estimator, Examples
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