If $Y_1, Y_2, \ldots , Y_n$ denote a random sample from an exponential distribution with mean $θ$, then $E(Y_i)=θ$ and $V(Y_i)=θ^2$. Thus, $E(\bar Y)=θ$ and $V(\bar Y)=θ^2/n$, or $σ_Y=θ/\sqrt{n}$. Suggest an unbiased estimator for θ and provide an estimate for the standard error of your estimator.

With these $Y_i$ and $\bar Y$, I don't know where to start with, should I do $E(\hat \theta)=\theta$?

  • 1
    $\begingroup$ The question is answered in the question. $\endgroup$ – André Nicolas Feb 9 '16 at 5:23
  • $\begingroup$ that's kind of what I thought, so I should use $E(\bar Y)=\theta$? @AndréNicolas $\endgroup$ – mathmathmath Feb 9 '16 at 5:24
  • $\begingroup$ Yes, Well, we could be silly and use $Y_1$, which would be technically correct but throws away information (it has larger variance than $\bar{Y}$ if $n\gt1$). $\endgroup$ – André Nicolas Feb 9 '16 at 5:27
  • $\begingroup$ the standard error is just $\theta / \sqrt(n)$? @AndréNicolas $\endgroup$ – mathmathmath Feb 9 '16 at 5:31
  • $\begingroup$ what is the point of this problem then? @AndréNicolas $\endgroup$ – mathmathmath Feb 9 '16 at 5:33

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