Editing Sufficient statistic (section)

===Uniform distribution===
{{see also|German tank problem}}
If ''X''<sub>1</sub>, ...., ''X''<sub>''n''</sub> are independent and [[uniform distribution (continuous)|uniformly distributed]] on the interval [0,''θ''], then ''T''(''X'') = max(''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>) is sufficient for θ — the [[sample maximum]] is a sufficient statistic for the population maximum.

To see this, consider the joint [[probability density function]] of ''X''&nbsp;&nbsp;(''X''<sub>1</sub>,...,''X''<sub>''n''</sub>). Because the observations are independent, the pdf can be written as a product of individual densities

:<math>\begin{align}
f_{\theta}(x_1,\ldots,x_n)
  &= \frac{1}{\theta}\mathbf{1}_{\{0\leq x_1\leq\theta\}} \cdots
     \frac{1}{\theta}\mathbf{1}_{\{0\leq x_n\leq\theta\}} \\[5pt]
  &= \frac{1}{\theta^n} \mathbf{1}_{\{0\leq\min\{x_i\}\}}\mathbf{1}_{\{\max\{x_i\}\leq\theta\}}
\end{align}</math>

where '''1'''<sub>{''...''}</sub> is the [[indicator function]].  Thus the density takes form required by the Fisher–Neyman factorization theorem, where ''h''(''x'')&nbsp;=&nbsp;'''1'''<sub>{min{''x<sub>i</sub>''}≥0}</sub>, and the rest of the expression is a function of only ''θ'' and ''T''(''x'')&nbsp;=&nbsp;max{''x<sub>i</sub>''}.

In fact, the [[minimum-variance unbiased estimator]] (MVUE) for ''θ'' is

:<math> \frac{n+1}{n}T(X). </math>

This is the sample maximum, scaled to correct for the [[bias of an estimator|bias]], and is MVUE by the [[Lehmann–Scheffé theorem]]. Unscaled sample maximum ''T''(''X'') is the [[maximum likelihood estimator]] for ''θ''.