Solution Found!
Suppose that Y1, Y2, . . . , Yn denote a random sample of
Chapter 8, Problem 19E(choose chapter or problem)
Suppose that \(\mathrm{Y} 1, \mathrm{Y} 2, \ldots, \mathrm{Yn}\) denote a random sample of size n from a population with an
exponential distribution whose density is given by
\(f(y)=\left\{\begin{array}{ll} (1 / \theta) e^{-y / \theta}, & y>0 \\ 0, & \text { elsewhere } \end{array}\right.\)
If \(Y_{(1)}=\min \left(Y_{1}, Y_{2}, \ldots, Y_{n}\right)\) denotes the smallest-order statistic, show that \(\widehat{\theta}=n Y_{(1)}\) is an
unbiased estimator for \(\theta\) and find \(\operatorname{MSE}(\widehat{\theta})\). [Hint: Recall the results of Exercise 6.81.]
Equation Transcription:
{
Text Transcription:
Y1, Y2,...,Yn
f(y)=(1 / \theta) e^-y / \theta, & y>0 \\ 0, & elsewhere
Y(1)=min (Y1, Y2,...,Yn)
\widehat\theta=n Y_(1)
\theta
MSE(\widehat\theta)
Questions & Answers
QUESTION:
Suppose that \(\mathrm{Y} 1, \mathrm{Y} 2, \ldots, \mathrm{Yn}\) denote a random sample of size n from a population with an
exponential distribution whose density is given by
\(f(y)=\left\{\begin{array}{ll} (1 / \theta) e^{-y / \theta}, & y>0 \\ 0, & \text { elsewhere } \end{array}\right.\)
If \(Y_{(1)}=\min \left(Y_{1}, Y_{2}, \ldots, Y_{n}\right)\) denotes the smallest-order statistic, show that \(\widehat{\theta}=n Y_{(1)}\) is an
unbiased estimator for \(\theta\) and find \(\operatorname{MSE}(\widehat{\theta})\). [Hint: Recall the results of Exercise 6.81.]
Equation Transcription:
{
Text Transcription:
Y1, Y2,...,Yn
f(y)=(1 / \theta) e^-y / \theta, & y>0 \\ 0, & elsewhere
Y(1)=min (Y1, Y2,...,Yn)
\widehat\theta=n Y_(1)
\theta
MSE(\widehat\theta)
ANSWER:Step 1 of 3
Exponential density function is given by
For a random sample of size n
We need to show that,