8. The Cramér-Rao lower bound and efficiency

Question 1

CRLB

Answer 1

V[T] <= (E’[T])^2/ I^X

• the larger the FI, the higher the precision we estimate @ with
• if T is unbiased we have V[T] <= (g’(@))^2/ I^X, then if we are estimating @, it’s the reciprocal of the FI

Question 2

Proof of th CRLB

Answer 2

Derive the expected value of T and show it’s equal to the expected value of TS (score function), knowing that the exp. value of S is 0, write, it’s equal to the covariance of T and S, by the Cauchy-Schwartz inequality Cov^2 <= V(T)V(S)

Question 3

Condition to achieve the CRLB

Answer 3

IIF T is unbiased and the score function is linear in T f(@)(T-g(@))
PROOF: using the cauchy-schwarz inequality S=q(@)+r(@)T(x) and finding the expected value, then substituting q(@)
For the EXPO FAM: only the SUFF stat is the efficient estimator for the expected value of R(x)

Question 4

Efficiency

Answer 4

An estimator is more efficient then another if it has a lower variance, in absolute, it is efficient if it attains the CRLB.
The unbiased estimator attaining the CRLB is the UMVUE (the converse is not generally true)