Let X 1 , ⋯ , X n be iid from a uniform distribution U [ θ − 1 2 , θ + 1 2 ] with θ ∈ R unknown. Show that the statistic T ( X ) = ( X ( 1 ) , X ( n ) ) is minimal sufficient but not complete.

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Let       X    1    ,  ⋯  ,      X    n   be iid from a uniform distribution   U  [  θ  −      1    2    ,  θ  +      1    2    ] with   θ  ∈      R   unknown. Show that the statistic   T  (      X    )  =  (      X          (      1      )        ,      X          (      n      )        ) is minimal sufficient but not complete.

Teagan Raymond · Accepted Answer

HINTFor   g  (      X          (      1      )        ,      X          (      n      )        )  =      X          (      n      )        −      X          (      1      )        ,, can you show  E  [  g  (      X          (      1      )        ,      X          (      n      )        )  ]is independent of   θ  ??

Let X_1,⋯,X_n be iid from a uniform distribution U[theta−1/2,theta+1/2] with theta in R unknown. Show that the statistic T(X)=(X_(1),X_(n)) is minimal sufficient but not complete.

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