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Let $X_1, X_2, ..., X_n$ be a random sample from a distribution with p.d.f., $$f(x;\theta)=\theta^2xe^{-x\theta} ; 0<x<\infty, \theta>0$$ Obtain minimum variance unbiased estimator of $\theta$ and examine whether it is attained?

MY WORK:

Using MLE i have found the estimator for $\theta=\frac{2}{\bar{x}}$ Or as $$X\sim Gamma(2, \theta)$$So $E(X)=2\theta$ $E(\frac{x}{2})=\theta$ so can i take $\frac {x}{2}$ as unbiased estimator of $\theta$. I'm stuck and confused need some help.

Thank u.

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  • $\begingroup$ Crossposted at math.SE as per comment suggesting that over there: math.stackexchange.com/questions/28779/… $\endgroup$ – cardinal Mar 24 '11 at 12:29
  • $\begingroup$ @amul28 You probably want to limit your search to linear estimators; otherwise, this question might be difficult to answer. $\endgroup$ – whuber Mar 24 '11 at 14:05
  • $\begingroup$ @whuber, or rather, functions of linear estimators. I've posted an answer on math.SE where this question originated. $\endgroup$ – cardinal Mar 24 '11 at 17:44
  • $\begingroup$ @cardinal I don't follow. Since $X_1, X_2, \ldots, X_n$ are individually linear estimators, the most general function of a set of linear estimators is just--any function of the data. You seem to be making no effective distinction. Do you mean functions of a single linear form in the data? $\endgroup$ – whuber Mar 24 '11 at 17:46
  • $\begingroup$ @whuber, just slightly sloppy wording on my part, as I suppose it could be interpreted in at least two ways. "Functions of a linear estimator" would probably have been clearer. $\endgroup$ – cardinal Mar 24 '11 at 17:51

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