1. Sufficiency

Question 1

Statistic

Answer 1

Given a random sample, with given distribution, any function T=T(X) that is a function solely of X and not of the unknown parameters

Question 2

Sufficient statistic

Answer 2

T is sufficient for the statistical model IFF the conditional distribution of X|T does not depend on @

Question 3

Neyman-Fisher Factorization th.

Answer 3

T is sufficient IFF the distribution can be written using two non-negative functions: f_X=g(t,@)*h(x), where t=T(x)

• the functions are not unique
• if the support depends on @, the sufficient statistic will be a function of order statistics

Question 4

Proof of the Factorization th.

Answer 4

f_X = Pr{X=x} = Pr{X=x|T=t} = Pr{X=x|T=t}Pr{T|t} = h(x)g(t,@)

Question 5

Transformations of SUFF.

Answer 5

If T is sufficient, then given T=r(T), for r an arbitrary function, then T is sufficient
PROOF: g(t,@)=g(r(t*).@)=g’(t,@)

Question 6

Minimal sufficiency

Answer 6

A sufficient stat. is said to be minimal if T is a function of any other sufficient stat T* i.e. the partition induced by the MIN SUFF is the one with the least number of elements

Question 7

Lehmann-Scheffé #1

Answer 7

Let T be a statistic, it is MIN SUFF if the ratio of distributions of two sample realizations does not depend on @ IFF T(x)=T(y)