Solution Found!
Suppose that Y1, Y2, Y3 denote a random sample from an
Chapter 8, Problem 8E(choose chapter or problem)
Suppose that \(Y_{1}, Y_{2}, Y_{3}\) denote a random sample from an exponential distribution with density function
f(y)=\left\{\begin{array}{ll} \left(\frac{1}{\theta}\right) e^{-y / \theta} & y>0 \\ 0, & \text { elsewhere }
\end{array}\right.
Consider the following five estimators of \(\theta\):
\(\widehat{\theta}_{1}=Y_{1}\)
\(\widehat{\theta}_{2}=\frac{Y_{1}+Y_{2}}{2}\)
\(\widehat{\theta}_{3}=\frac{Y_{1}+2 Y_{2}}{3}\)
\(\widehat{\theta}_{4}=\min \left(Y_{1}, Y_{2}, Y_{3}\right)\)
\(\widehat{\theta}_{5}=\bar{Y}\)
Which of these estimators are unbiased?Among the unbiased estimators, which has the smallest variance?
Equation Transcription:
{
Text Transcription:
Y1, Y2,Y3
f(y)=(\frac 1\theta) e^-y / \theta y>0 0, & elsewhere
\theta
\widehattheta_1=Y_1
\widehat\theta_2=\frac Y_1+Y_2 2
\widehat\theta_3=\frac Y_1+2 Y_2 3
\widehat\theta_4=\min t(Y_1, Y_2, Y_3)
\widehat\theta_5=\bar Y
Questions & Answers
QUESTION:
Suppose that \(Y_{1}, Y_{2}, Y_{3}\) denote a random sample from an exponential distribution with density function
f(y)=\left\{\begin{array}{ll} \left(\frac{1}{\theta}\right) e^{-y / \theta} & y>0 \\ 0, & \text { elsewhere }
\end{array}\right.
Consider the following five estimators of \(\theta\):
\(\widehat{\theta}_{1}=Y_{1}\)
\(\widehat{\theta}_{2}=\frac{Y_{1}+Y_{2}}{2}\)
\(\widehat{\theta}_{3}=\frac{Y_{1}+2 Y_{2}}{3}\)
\(\widehat{\theta}_{4}=\min \left(Y_{1}, Y_{2}, Y_{3}\right)\)
\(\widehat{\theta}_{5}=\bar{Y}\)
Which of these estimators are unbiased?Among the unbiased estimators, which has the smallest variance?
Equation Transcription:
{
Text Transcription:
Y1, Y2,Y3
f(y)=(\frac 1\theta) e^-y / \theta y>0 0, & elsewhere
\theta
\widehattheta_1=Y_1
\widehat\theta_2=\frac Y_1+Y_2 2
\widehat\theta_3=\frac Y_1+2 Y_2 3
\widehat\theta_4=\min t(Y_1, Y_2, Y_3)
\widehat\theta_5=\bar Y
ANSWER:Step 1 of 4
Given:
The five estimators of 
