Suppose $(X_1,X_2,\ldots,X_n)\sim \operatorname{Gamma}(a,b)$, independent and identically distributed with pdf:

$$f(x)=\frac{b^a}{\Gamma(a)}x^{a-1}e^{-bx},\quad x>0$$

Find the UMVUE of $\frac{\sqrt a}{b}$.

What eludes me is finding an unbiased estimator of $\frac{\sqrt a}{b}$

  • $\begingroup$ I am not sure about the unbiased estimator, but the parameter of interest is the standard deviation. $\endgroup$ – BGM Jul 5 '18 at 6:16
  • $\begingroup$ Are both $a$ and $b$ unknown? $\endgroup$ – StubbornAtom Jul 5 '18 at 7:05
  • $\begingroup$ Yes, $a$ and $b$ are both unknown $\endgroup$ – Momo Jul 5 '18 at 11:08
  • $\begingroup$ I thought about this a while, but could not find an unbiased estimator. Not to mention, if the UMVUE exists, it has to be a function of the complete sufficient statistic $(\sum \ln X_i,\sum X_i)$. Are you sure about the problem statement? Is there any reference? $\endgroup$ – StubbornAtom Jul 5 '18 at 13:09
  • $\begingroup$ It was asked here a while ago, but i could not find it any longer. $\endgroup$ – Momo Jul 5 '18 at 23:03

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