Answer to Question #253062 in Statistics and Probability for lucky

Question #253062

Suppose that

𝑇1 =(𝑋1 βˆ’πœ‡)2

𝑇2 =1

2[(𝑋1 βˆ’πœ‡)2 +(𝑋2 βˆ’πœ‡)2]

And

𝑇3 =1

2(𝑋1 βˆ’π‘‹2)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)


1
Expert's answer
2021-10-20T17:48:05-0400

i)

E(Οƒ2)=nβˆ’1nΟƒ2E(\sigma^2)=\frac{n-1}{n}\sigma^2


E(T3)=E((X1βˆ’X2)22)=12(2(ΞΌ2+Οƒ2)βˆ’2ΞΌ2)=Οƒ2/2E(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(Οƒ2)β‰ E(T3)E(\sigma^2)\neq E(T_3) , T3 is a biased estimator of 𝜎2.


ii)

E(T1)=E(T_1)= E(T1)=E()=E(T_1)=E()=E(T1)=(nβˆ’1)Οƒ2E(T_1)=(n-1)\sigma^2


E(T2)=12β‹…2(nβˆ’1)Οƒ2=(nβˆ’1)Οƒ2E(T_2)=\frac{1}{2}\cdot2(n-1)\sigma^2=(n-1)\sigma^2


So, 𝑇1 and 𝑇2 are same efficient estimators.


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