Suppose that Y1, Y2, . . . , Yn constitute a

Chapter 9, Problem 104SE

(choose chapter or problem)

Suppose that \(Y_{1}, Y_{2}, \ldots, Y_{n}\) constitute a random sample from the density function

  \(f(y \mid \theta)=\left\{\begin{array}{ll}  e^{-(y-\theta)}, & y>\theta \\  0, & \text { elsewhere }

  \end{array}\right.\)

                             

where \(\theta\) is an unknown, positive constant.
a Find an estimator \(\widehat{\theta}_{1}\) for \(\theta\) by the method of moments.
b Find an estimator  \(\widehat{\theta}_{2}\) for \(\theta\) by the method of maximum likelihood.
c Adjust \(\widehat{\theta}_{1} \text { and } \widehat{\theta}_{2}\) so that they are unbiased. Find the efficiency of the adjusted  \(\widehat{\theta}_{1}\) relative to the adjusted \(\widehat{\theta}_{2}\).

Equation Transcription:

 {

Text Transcription:

Y1, Y2,...,Yn

f(y \mid \theta)={e^-(y-\theta), & y>\theta \\  0, elsewhere

\theta

\widehat\theta}_1

\theta

\widehat\theta}_2

\theta

\widehat\theta_1 and  \widehat\theta_2

\widehat\theta}_1

\widehat\theta}_2

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