3. Fisher information

Question 1

Score function

Answer 1

S_@(x)= l’(@,x) = f’(@,x)/f(@,x)

Question 2

E[S] and V[S]

Answer 2

E[S] = 0 and V[S] = E[S^2]

Question 3

Fisher Information

Answer 3

is the variance of the score function, for CR regular models (with stricter conditions) it is equal to -E[l’’(@,x)]

Question 4

Additivity

Answer 4

The FI of two independent r.v. is additive, therefore for a sample of IID r.v. we have that I_X=nI_x

Question 5

FI and sufficiency

Answer 5

The FI of a sample is always greater or equal to the FI of a statistic, it is equal only for sufficient statistic, while it’s null for ancillary stat

Question 6

Cramer Rao regularity

Answer 6

1. The parameter space Theta is R or an open interval of R
2. The first (second*) derivative of fX exists and it’s finite (for every x in X and @ in Theta)
3. The first (second) derivative of the integral is the integral of the first (second) derivative
4. The variance of the score function is between 0 and +infinite (but non zero)