Editing Estimation theory (section)

===Maximum of a uniform distribution===
{{main|German tank problem}}

One of the simplest non-trivial examples of estimation is the estimation of the maximum of a uniform distribution. It is used as a hands-on classroom exercise and to illustrate basic principles of estimation theory. Further, in the case of estimation based on a single sample, it demonstrates philosophical issues and possible misunderstandings in the use of [[maximum likelihood]] estimators and [[likelihood functions]].

Given a [[discrete uniform distribution]] <math>1,2,\dots,N</math> with unknown maximum, the [[UMVU]] estimator for the maximum is given by
<math display="block">\frac{k+1}{k} m - 1 = m + \frac{m}{k} - 1</math>
where ''m'' is the [[sample maximum]] and ''k'' is the [[sample size]], sampling without replacement.<ref name="Johnson">{{citation | last=Johnson | first=Roger | title=Estimating the Size of a Population | year=1994 | journal=Teaching Statistics | volume=16 | issue=2 (Summer) | doi=10.1111/j.1467-9639.1994.tb00688.x | pages = 50–52 }}</ref><ref name="Johnson2">{{citation | last=Johnson | first=Roger | contribution=Estimating the Size of a Population | title=Getting the Best from Teaching Statistics | year=2006 | url=http://www.rsscse.org.uk/ts/gtb/contents.html | contribution-url=http://www.rsscse.org.uk/ts/gtb/johnson.pdf | url-status=dead | archive-url=https://web.archive.org/web/20081120085633/http://www.rsscse.org.uk/ts/gtb/contents.html | archive-date=November 20, 2008 }}</ref> This problem is commonly known as the [[German tank problem]], due to application of maximum estimation to estimates of German tank production during [[World War II]].

The formula may be understood intuitively as;
{{block indent | em = 1.5 | text = "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.{{NoteTag|The sample maximum is never more than the population maximum, but can be less, hence it is a [[biased estimator]]: it will tend to ''underestimate'' the population maximum.}}

This has a variance of<ref name="Johnson" />
<math display="block">\frac{1}{k}\frac{(N-k)(N+1)}{(k+2)} \approx \frac{N^2}{k^2} \text{ for small samples } k \ll N</math>
so a standard deviation of approximately <math>N/k</math>, the (population) average size of a gap between samples; compare <math>\frac{m}{k}</math> above. This can be seen as a very simple case of [[maximum spacing estimation]].

The sample maximum is the [[maximum likelihood]] estimator for the population maximum, but, as discussed above, it is biased.