A random variable X is distributed with pdf f ( x , θ ). T ( x ) is a sufficient statistic for θ ,, and S ( x ) is a minimal sufficient statistic for θ .. The following is stated in my notes without explanation: E θ ( T ∣ S ) = g ( S )for some function g (independent of θ .). E θ here refers to the fact that the expectation is a function of θ ..how can we more formally show this is true, using the definition of sufficiency?

Question

A random variable   X is distributed with pdf   f  (  x  ,      θ    ).   T  (  x  ) is a sufficient statistic for       θ    ,, and   S  (  x  ) is a minimal sufficient statistic for       θ    .. The following is stated in my notes without explanation:      E          θ        (  T  ∣  S  )  =  g  (  S  )for some function   g (independent of       θ    .).       E          θ       here refers to the fact that the expectation is a function of       θ    ..how can we more formally show this is true, using the definition of sufficiency?

Myah Charles · Accepted Answer

It is not true that every function of the data can be expressed as a function of the minimal sufficient statistic. For example, if       X    1    ,  …  ,      X    n    ∼  i.i.d.  N  ⁡  (  μ  ,  1  ) and   μ  ∈      R   indexes this family of distributions, then       X    1    +  ⋯  +      X    n   is sufficient for this family of distributions, in the sense that the conditional distribution of   (      X    1    ,  …  ,      X    n    ) given       X    1    +  ⋯  +      X    n   does not depend on   μ  ,, but clearly one cannot express       X    1   as a function of this sufficient statistic.But the definition implies the conditional distribution of   T given   S does not depend on   θ  ., and from that it follows that the conditional expectation of   T given   S does not depend on   θ  ..

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