Suppose X ∼ N ( μ , σ 2 ). I know that T ( X ) = ( X ¯ , S 2 ) is a complete sufficient statistic for μ , σ 2 if μ , σ 2 are unknown. But if μ , σ 2 is known, is S 2 still a complete statistic of σ 2 ?

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Suppose   X  ∼  N  (  μ  ,      σ    2    ). I know that   T  (  X  )  =  (            X      ¯        ,      S    2    ) is a complete sufficient statistic for   μ  ,      σ    2   if   μ  ,      σ    2   are unknown. But if   μ  ,      σ    2   is known, is       S    2   still a complete statistic of       σ    2  ?

Szulikto · Accepted Answer

S    2    =      1          n      −      1            Σ    i    (      X    i    −            X      ¯        n        )    2  being      Σ    i    (      X    i    −            X      ¯        n        )    2    =      Σ    i    (      X    i    −      μ    0        )    2    −  n  (            X      ¯        n    −      μ    0        )    2  Let&#039;s define the following funcion  g  (      S    2    )  =      Σ    i    (      X    i    −      μ    0        )    2    −      n    2    (            X      ¯        n    −      μ    0        )    2  where,   ∀      σ    2        E    [  g  (      S    2    )  ]  =  n      σ    2    −      n    2              σ      2        n    =  0but evidently  P  [  g  (      S    2    )  =  0  ]  ≠  1this shows that       S    2   is not a complete statistic for       σ    2   when   μ  =      μ    0   is a known value

Suppose X∼N(mu,sigma^2). I know that T(X)=(bar X,S^2) is a complete sufficient statistic for mu,sigma^2 if mu,sigma^2 are unknown. But if mu is known, is S^2 still a complete statistic of sigma^2?

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