91Ó°ÊÓ

\({X_1},{X_2}...{X_n}\)denote iid random variables from the uniform distribution with parameter\(\left[ {a,b} \right]\)where b is known and the value of a is unknown.

\(\left( {b > a} \right)\),\(T = \min \left\{ {{X_1},...{X_n}} \right\}\)

The pdf of X is given as

\({f_x} = \left\{ \begin{array}{l}\frac{1}{{b - a}}\;\;\;\;\;\;\;b > a\\0\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{array} \right.\)

Let us consider the statistic\(T = \min \left\{ {{X_1},...{X_n}} \right\}\)

The estimator T is sufficient for a if

\(f\left( {{x_1}...{x_n}|T = t} \right) = \frac{{f\left( {{x_1}} \right)f\left( {{x_1}} \right)...f\left( {{x_1}} \right)}}{{{g_1}\left( {a,t} \right)}}\) is independent of a

Defining a function\(g\left( {a,t} \right)\)

\(g\left( {a,t} \right) = \left\{ \begin{array}{l}\frac{1}{{{{\left( {b - a} \right)}^n}}}\;\;\;if\;a \le \min \left\{ {{X_1}...{X_n}} \right\}\\0\;\;\;\;\;\;\;\;\;\;\;if\;a > \min \left\{ {{X_1}...{X_n}} \right\}\end{array} \right.\)

On rearranging the joint function as

\(\begin{array}{c}f\left( {{x_1}...{x_n}|T = t} \right) = \frac{{f\left( {{x_1}} \right)f\left( {{x_1}} \right)...f\left( {{x_1}} \right)}}{{{g_1}\left( {a,t} \right)}}\\ = \frac{{\frac{1}{{b - a}} \times ...\frac{1}{{b - a}}n\;times}}{{\frac{1}{{{{\left( {b - a} \right)}^n}}}}}\end{array}\)

\(\begin{array}{c}f\left( {{x_1}...{x_n}|T = t} \right) = \frac{1}{{{{\left( {b - a} \right)}^n}}} \times \frac{{{{\left( {b - a} \right)}^n}}}{1}\\ = 1\end{array}\)

Hence we observe the conditional joint distribution gives the estimator T is independent of unknown paramenter a.