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胃 be a parameter, and let X be discrete with p.f. \(f\left( {x|\theta } \right)\) conditional on 胃. Let T = r(X) be a statistic. We need to prove that T is sufficient if and only if, for every t and every x such that t = r(x), the likelihood function from observing T = t is proportional to the likelihood function from observing X = x

Fisher-Neyman Factorization Theorem

\({X_1},...,{X_n}\) be i.i.d r.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 likelihood function is

\(\begin{align}g\left( {t;\theta } \right) &= \sum\limits_{r\left( x \right) = t}^{} {u\left( x \right)\nu \left( {r\left( x \right);\theta } \right)} \\ &= \sum\limits_{r\left( x \right) = t}^{} {u\left( x \right)\nu } \left( {t;\theta } \right)\\ &= \nu \left( {t;\theta } \right)\sum\limits_{r\left( x \right) = t}^{} {u\left( x \right)} \end{align}\)

The likelihood function is proportional to \(\nu \left( {r\left( x \right);\theta } \right) = \nu \left( {t;\theta } \right)\).

Let us assume there exists a function h such that

\(f\left( {x;\theta } \right) = h\left( x \right)g\left( {t;\theta } \right)\)

Now, by the Fisher-Neyman Factorization Theorem

\(\begin{align}\nu \left( {r\left( x \right);{x_0}} \right) &= g\left( {t;\theta } \right)\\u\left( x \right) &= h\left( x \right)\end{align}\)

Hence we arrives at the proof of T is sufficient if and only if, for every t and every x such that t = r(x), the likelihood function from observing T = t is proportional to the likelihood function from observing X = x