Uniform Distribution
For more details on this topic, see German tank problem.For sampling without replacement from a uniform distribution with one or two unknown endpoints (so with N unknown, or with both M and N unknown), the sample maximum, or respectively the sample maximum and sample minimum, are sufficient and complete statistics for the unknown endpoints; thus an unbiased estimator derived from these will be UMVU estimator.
If only the top endpoint is unknown, the sample maximum is a biased estimator for the population maximum, but the unbiased estimator (where m is the sample maximum and k is the sample size) is the UMVU estimator; see German tank problem for details.
If both endpoints are unknown, then the sample range is a biased estimator for the population range, but correcting as for maximum above yields the UMVU estimator.
If both endpoints are unknown, then the mid-range is an unbiased (and hence UMVU) estimator of the midpoint of the interval (here equivalently the population median, average, or mid-range).
The reason the sample extrema are sufficient statistics is that the conditional distribution of the non-extreme samples is just the distribution for the uniform interval between the sample maximum and minimum – once the endpoints are fixed, the values of the interior points add no additional information.
Read more about this topic: Sample Maximum And Minimum, Applications, Estimation
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