$\begingroup$I want to know $(\frac{\Gamma'(\alpha)}{\Gamma(\alpha)})'$. My goal is to show $\alpha $ times this derivative of digamma is greater than 1.
The background of question is to show $\bar{x}$ is not asymptotically efficient for Gamma($\alpha$,1), because the ratio of Var$\bar{x}$ and Cramer-Rao Lower Bound is greater than 1. I can show that this ratio is $\alpha $ times this derivative of digamma. But I don't know how to show it is >1.
$\endgroup$31 Answer
$\begingroup$First note that by definition of the polygamma function: $$ \alpha \partial_\alpha\frac{\Gamma^\prime(\alpha)}{\Gamma(\alpha)}=\alpha \partial_\alpha^2\log\Gamma(\alpha)=\alpha\psi^{(1)}(\alpha). $$ Now calling on this proof detailing inequalities of polygamma functions we find for all $\alpha>0$: $$ \frac{1}{\alpha}+\frac{1}{2\alpha^2}\leq\psi^{(1)}(\alpha)\leq \frac{1}{\alpha}+\frac{1}{\alpha^2}. $$ Multiplying the entire inequality by $\alpha$ and only considering the lower bound then gives $$ \alpha\psi^{(1)}(\alpha)\geq1+\frac{1}{2\alpha}>1, $$ which is the desired result.
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