Exercise2.11 Let Y1, ...,Yn be anormalsamplewhereboth and are unknown.Considera familyofestimatorsfor 2 of theform 2

Exercise2.11 Let Y1, ...,Yn be anormalsamplewhereboth μ and σ are unknown.Considera familyofestimatorsfor σ2 of theform

σˆ 2 an =

Σn i=1(Yi−¯Y )2 an

, where an may dependon n butnotonthedata.

1. Do theMLE ˆ σ2 = 1n

Σn i=1(Yi −¯Y )2 (see Example2.7)andtheUMVUE s2 = 1 n−1 Σn i=1(Yi −¯Y )2

(see Example2.20)belongtothisfamily?

2. What istheMSEof σˆ 2 an?

3. Compare theMLE σˆ 2 and theUMVUE s2 by MSE.

4. Find anestimatorwiththeminimalMSEwithinthefamily.

(Hint: recallthat Σn i=1(Yi−¯Y )2 ∼ σ2χ2 n−1.)