Let θ ′ , θ ∈ Θ such that θ ′ ≠ θ.Prove that T is a sufficient statistic if and only if f ( x , θ ′ ) f ( x , θ ) is a function dependent only on T ( x ).

Question

Let       θ    ′  ,   θ  ∈  Θ such that       θ    ′    ≠  θ.Prove that   T is a sufficient statistic if and only if            f      (      x      ,              θ        ′            )              f      (      x      ,      θ      )      is a function dependent only on   T  (  x  ).

fonfonvkdpf · Accepted Answer

From the factorisation theorem we know that   T is sufficient if and only if:      f    θ    (  x  )  =  h  (  x  )      g    θ    (  T  (  x  )  )  .So we then have:      R                  θ        ′            ,      θ        (  x  )  ≡                    f        θ        ′            (      x      )                      f        θ            (      x      )        =            h      (      x      )              g        θ        ′            (      T      (      x      )      )              h      (      x      )              g        θ            (      T      (      x      )      )        =                    g        θ        ′            (      T      (      x      )      )                      g        θ            (      T      (      x      )      )        ,which depends on   x only through   T  (  x  ).

Let theta′, theta in Theta such that theta′=!theta. I want to prove that T is a sufficient statistic if and only if f(x,theta′)/f(x,theta) is a function dependent only on T(x).

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