Given: Let X be a sample from P ∈ P , δ 0 ( X ) be a decision rule (which may be randomized) in a problem with R k as the action space, and T be a sufficient statistic for P ∈ P . For any Borel A ⊂ R k , define δ 1 ( T , A ) = E [ δ 0 ( X , A ) | T ]Let L ( P , a ) be a loss function. Show that ∫ L ( P , a ) d δ 1 ( X , a ) = E [ ∫ L ( P , a ) d δ 0 ( X , a ) | T ] My idea of proving this is to show that this holds for a simple function L and generalize that to non-negative functions L by using the conditional Montone Convergence Theorem. But I can't really show the equality for a simple function L. That is, if we take L = ∑ i = 1 n c i 1 A i can I show that the given result indeed holds true?

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Given: Let   X be a sample from   P  ∈      P  ,       δ          0        (  X  ) be a decision rule (which may be randomized) in a problem with             R        k   as the action space, and   T be a sufficient statistic for   P  ∈      P  . For any Borel   A  ⊂            R        k  , define      δ    1    (  T  ,  A  )  =  E  [      δ    0    (  X  ,  A  )      |    T  ]Let   L  (  P  ,  a  ) be a loss function. Show that      ∫    L    (    P    ,    a    )    d          δ      1        (    X    ,    a    )    =    E    [    ∫    L    (    P    ,    a    )    d          δ      0        (    X    ,    a    )          |        T    ]  My idea of proving this is to show that this holds for a simple function   L and generalize that to non-negative functions   L by using the conditional Montone Convergence Theorem. But I can&#039;t really show the equality for a simple function   L. That is, if we take       L    =          ∑              i        =        1                    n                    c      i                      1                              A          i                     can I show that the given result indeed holds true?

Kroatujon3 · Accepted Answer

You can prove this for simple functions   L  =      ∑          i      =      1        n        c    i              1                      A        i             by using linearity of conditional expectation as follows:            ∫            ∑          i      =      1        n        c    i              1                      A        i              d      δ    1    (  X  ,  a  )  =                    ∑                  i          =          1                n                  c    i        δ    1    (  X  ,      A    i    ) by definition    =                    ∑                  i          =          1                n                    c        i                    E                    [                  δ          0                (        X        ,                  A          i                )                          |                        T        ]                =      E        [                            ∑                      i            =            1                    n                          c          i                          δ          0                (        X        ,                  A          i                )                                      |                                  T              ]   by linearity of conditional expectation    =      E        [                  ∫                    ∑              i        =        1            n              c      i                      1                              A          i                      d          δ      0        (    X    ,    a    )                      |                  T    ]      =      E        [                  ∫              L        d          δ      0        (    X    ,    a    )                      |                  T    ]   as desired.

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