Editing Sufficient statistic (section)

===Uniform distribution (with two parameters)===

If <math>X_1,...,X_n</math> are independent and [[Uniform distribution (continuous)|uniformly distributed]] on the interval <math>[\alpha, \beta]</math> (where <math>\alpha</math> and <math>\beta</math> are unknown parameters), then <math>T(X_1^n)=\left(\min_{1 \leq i \leq n}X_i,\max_{1 \leq i \leq n}X_i\right)</math> is a two-dimensional sufficient statistic for <math>(\alpha\, , \, \beta)</math>.

To see this, consider the joint [[probability density function]] of <math>X_1^n=(X_1,\ldots,X_n)</math>. Because the observations are independent, the pdf can be written as a product of individual densities, i.e.

:<math>\begin{align}
f_{X_1^n}(x_1^n)
  &= \prod_{i=1}^n \left({1 \over \beta-\alpha}\right) \mathbf{1}_{ \{ \alpha \leq x_i \leq \beta \} }
  = \left({1 \over \beta-\alpha}\right)^n \mathbf{1}_{ \{ \alpha \leq x_i \leq \beta, \, \forall \, i = 1,\ldots,n\}} \\
  &= \left({1 \over \beta-\alpha}\right)^n \mathbf{1}_{ \{ \alpha \, \leq \, \min_{1 \leq i \leq n}X_i \} } \mathbf{1}_{ \{ \max_{1 \leq i \leq n}X_i \, \leq \, \beta \} }.
\end{align}</math>

The joint density of the sample takes the form required by the Fisher–Neyman factorization theorem, by letting

:<math>\begin{align}
h(x_1^n)= 1, \quad
g_{(\alpha, \beta)}(x_1^n)= \left({1 \over \beta-\alpha}\right)^n \mathbf{1}_{ \{ \alpha \, \leq \, \min_{1 \leq i \leq n}X_i \} } \mathbf{1}_{ \{ \max_{1 \leq i \leq n}X_i \, \leq \, \beta \} }.
\end{align}</math>

Since <math>h(x_1^n)</math> does not depend on the parameter <math>(\alpha, \beta)</math> and <math>g_{(\alpha \, , \, \beta)}(x_1^n)</math> depends only on <math>x_1^n</math> through the function <math>T(X_1^n)= \left(\min_{1 \leq i \leq n}X_i,\max_{1 \leq i \leq n}X_i\right),</math>

the Fisher–Neyman factorization theorem implies <math>T(X_1^n) = \left(\min_{1 \leq i \leq n}X_i,\max_{1 \leq i \leq n}X_i\right)</math> is a sufficient statistic for <math>(\alpha\, , \, \beta)</math>.