91Ó°ÊÓ

\({X_1},...{X_n}\) be the random sample of size n from the distribution with pdf \(f\left( {x|\theta } \right)\) where \(\theta \) is unknown parameter and belong to the space \(\Omega \)

Let \(T = r\left( {{X_1},...{X_n}} \right)\) and \({T'} = r\left( {{X_1},...{X_n}} \right)\) be the two statistic such that \({T'}\) is one-one fuction of T.

Hence the value of T can be determined from the value of T without knowing the values of sample \({X_1},...{X_n}\)

From the factorization criterion for a sufficient statistic, it is known that the statistic T will be sufficient for \(\theta \)if and only if\({f_n}\left( {x|\theta } \right)\)can be factored as

\({f_n}\left( {x|\theta } \right) = u\left( x \right)v\left[ {r\left( x \right),\theta } \right]\),................(1)

where u and v are non- negative functions.

Similarly the function \({T'}\) to be sufficient statistic for \(\theta \) following equation must be true.

\({f_n}\left( {x|\theta } \right) = u\left( x \right)v\left[ {{r'}\left( x \right),\theta } \right]\).................(2)

But \(r\left( x \right)\) can be expressed as a function of \({r'}\left( x \right)\)and vice versa.

Hence factorization in equation (1) and equation (2) must be similar in which v is a function of \({r'}\left( x \right)\)and \(\theta \).

Hence \({T'}\) be the sufficient statistic if and only if T is sufficient statistic for \(\theta \).