91Ó°ÊÓ

The uniform distribution on the interval [θ, θ + 3], where the value of θ is unknown. We need to show that \({T_1} = \min \left\{ {{X_1},...,{X_n}} \right\}\,\,\,{\rm{and}}\,\,\,{T_2} = \max \left\{ {{X_1},...,{X_n}} \right\}\)are jointly sufficient statistics.

Fisher-Neyman Factorization Theorem

\({X_1},...,{X_n}\) be i.i.dr.v. with pdf \(f\left( {x;\theta } \right)\) and let \(T = r\left( {{X_1},...,{X_n}} \right)\) be a statistic .T is sufficient statistic for \(\theta \)iff

\(f\left( {x;\theta } \right) = u\left( x \right)\nu \left[ {r\left( x \right);\theta } \right]\) where \(u\,\,\,{\rm{and}}\,\,\,\nu \) are non-negative functions

The pdf of the distribution is

\(f\left( {x;\theta } \right) = \frac{1}{\theta }\)

The joint pdf of the uniform distribution on the interval [θ, θ + 3],is

\(f\left( {x;\theta } \right) = \frac{1}{{{3^n}}}\) . When \({T_1} = \min \left\{ {{X_1},...,{X_n}} \right\}\,\,\,{\rm{and}}\,\,\,{T_2} = \max \left\{ {{X_1},...,{X_n}} \right\}\)by the Fisher-Neyman Factorization Theorem we get

\(\nu \left[ {{r_1}\left( x \right),{r_2}\left( x \right);\theta } \right] = h\left( {\theta ,{t_1}} \right)h\left( {{t_2},\theta + 3} \right)\) and \(u\left( x \right) = \frac{1}{{{3^n}}}\)where

\(\begin{aligned}{}h\left( {y,z} \right) &= 1\,\,\,\,,y \le z\\ &= 0\,\,\,\,,y > z\end{aligned}\)

This implies \({T_1} = \min \left\{ {{X_1},...,{X_n}} \right\}\,\,\,{\rm{and}}\,\,\,{T_2} = \max \left\{ {{X_1},...,{X_n}} \right\}\)are jointly sufficient statistics.