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Ch 7.3 - 30E
Chapter 7, Problem 30E(choose chapter or problem)
Suppose that \(\hat{\Theta}_{1}\), \(\hat{\Theta}_{2}\) and \(\hat{\Theta}_{3}\) are estimators of \(\theta\). We know that \(E\left(\hat{\Theta}_{1}\right)=E\left(\hat{\Theta}_{2}\right)=\theta\), \(E\left(\hat{\Theta}_{3}\right) \neq 0\), \(V\left(\hat{\Theta}_{1}\right)=12\), \(V\left(\hat{\Theta}_{2}\right)=10\) and \(E\left(\hat{\Theta}_{3}-\theta\right)^{2}=6\). Compare these three estimators. Which do you prefer? Why?
Equation Transcription:
Text Transcription:
Theta hat_1
Theta hat_2
Theta hat_3
theta
E(Theta hat_1)=E(Theta hat_2)=theta
E(Theta hat_3)not=0
V(Theta hat_1)=12
V(Theta hat_2)=10
E(Theta hat_3-theta)^2=6
Questions & Answers
QUESTION:
Suppose that \(\hat{\Theta}_{1}\), \(\hat{\Theta}_{2}\) and \(\hat{\Theta}_{3}\) are estimators of \(\theta\). We know that \(E\left(\hat{\Theta}_{1}\right)=E\left(\hat{\Theta}_{2}\right)=\theta\), \(E\left(\hat{\Theta}_{3}\right) \neq 0\), \(V\left(\hat{\Theta}_{1}\right)=12\), \(V\left(\hat{\Theta}_{2}\right)=10\) and \(E\left(\hat{\Theta}_{3}-\theta\right)^{2}=6\). Compare these three estimators. Which do you prefer? Why?
Equation Transcription:
Text Transcription:
Theta hat_1
Theta hat_2
Theta hat_3
theta
E(Theta hat_1)=E(Theta hat_2)=theta
E(Theta hat_3)not=0
V(Theta hat_1)=12
V(Theta hat_2)=10
E(Theta hat_3-theta)^2=6
ANSWER:
Solution:
Step 1 of 2:
Let ,
,
are unbiased estimator of the parameter
.
We have find which estimator is better by using the given information.