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1 vote

The approximated function of $\mathbb{E}\left\{ \ln\left(1+X\right)\right\}$, where $X\sim\operatorname{Gamma}\left(\kappa\ge1,\theta>0\right)$

To set notation define $$ G(\kappa, \theta) = \mathbb{E}_{X \sim \mathsf{Gamma}(\kappa, \theta)}\Big[\psi(X)\Big], \quad \psi(z) = \log(1 + z). $$ Explicit formula for the Jensen gap: By Taylor's ...
Drew Brady's user avatar
1 vote

The approximated function of $\mathbb{E}\left\{ \ln\left(1+X\right)\right\}$, where $X\sim\operatorname{Gamma}\left(\kappa\ge1,\theta>0\right)$

We can get the bound as $$ \mathbb{E}[\ln(1+X)]\le\ln(1+\mathbb{E}[X]) + \ln\left(\kappa\right)-\psi\left(\kappa\right). $$ This is because $$ \mathbb{E}\left[\ln\left(1+X\right)\right]-\ln\left(1+\...
Lee White's user avatar
2 votes

The approximated function of $\mathbb{E}\left\{ \ln\left(1+X\right)\right\}$, where $X\sim\operatorname{Gamma}\left(\kappa\ge1,\theta>0\right)$

I don't know if the following is sharp enough to answer your question, but it is an improvement at least. One can pretty easily get the bound $$\mathbb{E}[\ln(1+X)]\ge\psi(\kappa)+\ln(\theta)+\ln(1+\...
Mark Schultz-Wu's user avatar

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