Sufficiency and the exponential family

Question

State the one-parameter exponential-family form, explain how it hands you a sufficient statistic for free, and apply it to show T = \sumi Xi is sufficient for the rate of an exponential distribution.

Accepted Answer

Exponential-family form. A one-parameter family can be written f(x; 	heta) = h(x)\,\exp\!\big(\eta(	heta)\,t(x) - A(	heta)\big), for some functions h, \eta, t, A. For an i.i.d. sample the joint density is f(x; 	heta) = \Big(\prodi h(xi)\Big)\exp\!\Big(\eta(	heta)	extstyle\sumi t(xi) - nA(	heta)\Big). This already is a Fisher-Neyman factorization: the 	heta-part depends on the data only through T = \sumi t(xi), and \prodi h(xi) has no 	heta. So T = \sumi t(Xi) is sufficient, read straight off the form with no extra work. This is exactly why exponential-family members always admit a lo…

Sufficiency and the exponential family

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