Exponential family: f ( x | θ ) = c ( x ) d ( θ ) exp ⁡ [ a ( θ ) b ( x ) ] T = ∑ b ( x i )Show that T is a sufficient statistic

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Exponential family:  f  (  x      |    θ  )  =  c  (  x  )  d  (  θ  )  exp  ⁡  [  a  (  θ  )  b  (  x  )  ]  T  =  ∑  b  (      x    i    )Show that   T is a sufficient statistic

Sage Braun · Accepted Answer

The point of sufficiency is not quite what you said. Let&#039;s say you have a density   f  (  x      |    θ  ) and that it factors like this:  f  (  x      |    θ  )  =  h  (  x  )  ⋅  g  (  θ  ,  T  (  x  )  )In other words, the density factors into a part which involves only data (the function   h), and into a part which depends on both the parameter   θ and a function of the data   T  (  x  ) (the function   g). In this case the function   T is said to be a sufficient statistic for   θ.The intuition behind this is that since   h  (  x  ) can be factored out from   g, then it has no &quot;interaction&quot; with   θ so any information contained in   h  (  x  ) will not affect your ability to estimate   θ. The only part of the data that matters is   T  (  x  ), since it interacts with   θ through the function   g.In the exponential case, the factorization is   h  (  x  )  =  c  (  x  ) and   g  (  θ  ,  T  (  x  )  )  =  d  (  θ  )  exp  ⁡  (  a  (  θ  )  ,  b  (  x  )  ). Thus   g depends on   x solely through the function   b  (  x  ).

Exponential family: f(x|theta)=c(x)d(theta)exp[a(theta)b(x)], T = sum b(x_i) Show that T is a sufficient statistic

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