91Ó°ÊÓ

A Pareto distribution for which both parameters \({x_0}\) and \(\alpha \) are unknown. We need to show that \({T_1} = \min \left\{ {{X_1},...,{X_n}} \right\}\) and \({T_2} = \prod\limits_{i = 1}^n {{x_i}} \) 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 pareto distribution is

\(\begin{array}{c}f\left( {x;\alpha } \right) = \frac{{\alpha {x_0}}}{{{x^{\alpha + 1}}}}\,\,\,\,,x \ge {x_0}\\ = 0\,\,\,\,\,x < {x_0}\end{array}\)

The joint pdf of pareto distribution is

\(\begin{array}{c}f\left( {x;{x_0},\alpha } \right) = \prod\limits_{i = 1}^n {\left[ {\frac{{\alpha {x_0}^\alpha }}{{{x^{\alpha + 1}}}}} \right]} \\ = \frac{{{{\left( {\alpha {x_0}^\alpha } \right)}^n}}}{{{{\left( {\prod\limits_{i = 1}^n {{x_i}} } \right)}^{\alpha + 1}}}}\end{array}\)

\(f\left( {x;{x_0},\alpha } \right) = t_2^{ - \left( {\alpha + 1} \right)}{\alpha ^n}x_0^{\alpha n}\)

When \({t_1} = \min \left\{ {{x_1},...,{x_n}} \right\}\) and \({t_2} = \prod\limits_{i = 1}^n {{x_i}} \) then by using the Fisher-Neyman Factorization Theorem

\(\begin{array}{c}f\left( {x;{x_0},\alpha } \right) = \frac{{{{\left( {\alpha {x_0}^\alpha } \right)}^n}}}{{{{\left( {\prod\limits_{i = 1}^n {{x_i}} } \right)}^{\alpha + 1}}}}g\left( {{x_0},\min \left\{ {{x_1},{x_2},{x_3},......{x_n}} \right\}} \right)\\ = \frac{{{{\left( {\alpha {x_0}^\alpha } \right)}^n}}}{{{{\left( {{t_2}} \right)}^{^{\alpha + 1}}}}}g\left( {{x_0},{t_1}} \right)\end{array}\)

Where g is the indicator function such that

\(\begin{array}{c}g\left( {x,y} \right) = 1\,\,\,\,\,,x \le y\\ = 0\,\,\,\,\,x > y\end{array}\)

Now,

\(\nu \left[ {{r_1}\left( x \right){r_2}\left( x \right);{x_0},\alpha } \right] = t_2^{ - \left( {\alpha + 1} \right)}{\alpha ^n}x_0^{\alpha n}g\left( {{x_0},{t_1}} \right)\)and \(u\left( x \right) = 1\)where

\(\begin{array}{c}g\left( {{x_0},{t_1}} \right) = 1\,\,\,\,\,,{x_0} \le {t_1}\\ = 0\,\,\,\,\,,{x_0} > {t_1}\end{array}\)

It follows that\({T_1} = \min \left\{ {{X_1},...,{X_n}} \right\}\) and \({T_2} = \prod\limits_{i = 1}^n {{x_i}} \) are jointly sufficient statistics.