7. MSE and the UMVUE

Question 1

MSE

Answer 1

Mean Square Error, most popular criteria for evaluating an estimator, which represents the risk under quadratic loss
MSE = E[(T-@)^2] = V[T] + B^2[T]

Question 2

Unbiased

Answer 2

E[T]=g(@)

Question 3

UMVUE

Answer 3

(optimality) uniformly minimum variance unbiased estimator:

E[T]=g(@) + V[T] =< V[T] for any T unbiased estimator

Question 4

Unbiasedness vs Variability

Answer 4

accuracy vs precision

Question 5

Rao-Blackwell th.

Answer 5

Given U, an unbiased estimator for g(@) and T a SUFF stat for @, then E[U|T] is an unbiased estimator with variance smaller or equal to U.
Any unbiased estimator can be improved using the SUFF

Question 6

Proof of Rao Blackwell

Answer 6

1. T is a statistic
2. it is unbiased (using the tower property)
3. the variance is less or equal
4. it’s only equal if T*=U with probability 1

Question 7

Lehman-Scheffé #2

Answer 7

Given T a COMP SUFF for @ and T=h(T) and unbiased estimator for g(@), then T is the UMVUE for g(@):

Question 8

The UMVUE is always a function of the min suff

Answer 8

Since the UMVUE must be @=E[@|T] where T is a suff stat, then it’s a function of a suff stat + a suff stat is always a function of the min suff

Question 9

Proof of Lehmann-Scheffé #2

Answer 9

1. UNIQUENESS: let’s suppose both A and B are functions to T and unbiased for g(@) then E[ A-B]= E[A] - E[B]= 0.
   Notice that their difference is also a function of T, combining this with completeness of T we get that Pr{A=B}=1
2. MINIMAL VARIANCE
   For an unbiased estimator U, by RB E[U|T] is unbiased, and with lower variance. Since by the previous point, there exists only one unbiased estimator function of T with these characteristics, and U is arbitrarily chosen, then the variance of such estimator is minimal and then it’s the UMVUE