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For an exponential distribution

$$X \sim \exp(\lambda) = \lambda \ \exp(-\lambda\ x),\ x>0$$

Does there exist an Best Unbiased Estimator (BUE) for $\lambda$ i.e. can it achieve the lower bound for variance given by Cramer-Rao Lower Bound $\lambda^2\, /\,n$?

Lee David Chung Lin
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Nagabhushan S N
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  • Thanks. Sorry, I posted the question a bit wrong. I want to know if the estimator can have the variance given by Cramer-Rao Lower Bound. I've updated the question. – Nagabhushan S N Apr 15 '19 at 19:50
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    For a sample of $n$ observations, variance of the minimum variance unbiased estimator (UMVUE) of $\lambda$ (shown here) does not attain the Cramer-Rao bound. Using the Gamma distribution of $\sum X_i$, it can be shown that variance of UMVUE is $\lambda^2/(n-2)$ for $n>2$, which is strictly greater than $\lambda^2/n$. But UMVUE remains the best estimator within the unbiased class; its variance need not attain the lower bound. – StubbornAtom May 25 '20 at 21:08
  • Thanks! If you can write it as an answer, I'll accept it – Nagabhushan S N May 26 '20 at 06:43

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