Show that T ( X ) = ( R , V ) = ( X ( n ) − X ( 1 ) , X ( n ) + X ( 1 ) 2 ) is a minimal sufficient statistic for θ.

Question

Show that   T  (  X  )  =  (  R  ,  V  )  =      (          X              (        n        )              −          X              (        1        )              ,                            X                      (            n            )                          +                  X                      (            1            )                              2        )  is a minimal sufficient statistic for   θ.

Jaxson White · Accepted Answer

Any invertible function of a minimal sufficient statistic is also minimal sufficient. Such a function is given by  M  (  x  ,  y  )  =  (  y  −  x  ,  (  x  +  y  )      /    2  )which has the inverse      M          −      1        (  x  ,  y  )  =  (  y  −  x      /    2  ,  y  +  x      /    2  )  .This can also be conceptualized as a   2  ×  2 matrix:  M  =      [                            −          1                          1                                      1                      /                    2                          1                      /                    2                      ]    .

Show that T(X)=(R,V)=(X_(n)−X_(1), (X(n)+X(1))/2) is a minimal sufficient statistic for θ.

Answered question

Answer & Explanation

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