Estimation of Maximum
This example is described by saying that a sample of k observations is obtained from a uniform distribution on the integers, with the problem being to estimate the unknown maximum N. This problem is commonly known as the German tank problem, following the application of maximum estimation to estimates of German tank production during World War II.
The UMVU estimator for the maximum is given by
where m is the sample maximum and k is the sample size, sampling without replacement. This can be seen as a very simple case of maximum spacing estimation.
The formula may be understood intuitively as:
- "The sample maximum plus the average gap between observations in the sample",
the gap being added to compensate for the negative bias of the sample maximum as an estimator for the population maximum.
This has a variance of
so a standard deviation of approximately, the (population) average size of a gap between samples; compare above.
The sample maximum is the maximum likelihood estimator for the population maximum, but, as discussed above, it is biased.
If samples are not numbered but are recognizable or markable, one can instead estimate population size via the capture-recapture method.
Read more about this topic: Uniform Distribution (discrete)
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