Suppose that Y1, Y2, . . . , Yn denote a random sample of

Chapter 8, Problem 19E

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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)

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,

       

 

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