2. The Exponential family

Question 1

One parameter EXPO FAM

Answer 1

f(x)=a(@)g(x)exp{b(@)R(x)}

Question 2

E[R(x)] and V[R(x)]

Answer 2

E[R(x)]= -c’(@)/b’(@) and V[R(x)]= -c’’(@)/b’(@)*b’’(@)

Question 3

Property of closure under random sampling

Answer 3

Given a sample of r.v. with IID f(x) all belonging to the EXPO FAM, the distribution of the sample belongs to the EXPO FAM too

Question 4

Sufficiency in the EXPO FAM and Minimal and Complete

Answer 4

By the N.F. Factorization th. R(x) is the SUFF, and it belongs to the EXPO FAM, the statistic is also minimal and complete if the distribution is full-rank

Question 5

Multi-parameter EXPO FAM

Answer 5

f(x)=a(@)g(x)exp{SUM(b(@)R(x))}

Question 6

Minimal EXPO FAM

Answer 6

If both b_j(@) and R_j(X) are linearly independent

Question 7

Full rank EXPO FAM

Answer 7

If it’s minimal and the parameter space is exactly of the dimension of the vector of b_j(@)

Question 8

Efficiency in the EXPO FAM

Answer 8

An efficient estimator exists only for the EXPO FAM, the goal must be to estimate the expected value of R(x) or linear transformations