Question

Suppose that

𝑇1 =(𝑋1 −𝜇)2

𝑇2 =1

2[(𝑋1 −𝜇)2 +(𝑋2 −𝜇)2]

And

𝑇3 =1

2(𝑋1 −𝑋2)2

Are estimators for 𝜎2.

i) Show whether or not 𝑇3 is an unbiased estimator of 𝜎2. (4)

ii) Which estimator is the most efficient between 𝑇1 and 𝑇2? (6)

Answer 1

i)

$E(\sigma^2)=\frac{n-1}{n}\sigma^2$

$E(T_3)=E(\frac{(X_1-X_2)^2}{2})=\frac{1}{2}(2(\mu^2+\sigma^2)-2\mu^2)=\sigma^2/2$

Since $E(\sigma^2)\neq E(T_3)$ , T3 is a biased estimator of 𝜎2.

ii)

$E(T_1)=$ $E(T_1)=E()=$$E(T_1)=(n-1)\sigma^2$

$E(T_2)=\frac{1}{2}\cdot2(n-1)\sigma^2=(n-1)\sigma^2$

So, 𝑇1 and 𝑇2 are same efficient estimators.