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My question is related to empirical Bayes and large-scale inference. It is explained that the local Bayes false discovery rate (fdr) is greater than the Bayes false discovery rate (FDR). It is argued that the FDR is the tangent that connects the origin and the point and fdr the slope of the secant at that point.

But, I am stack in solving the next "simple" problem. Given $F_1(z) = F_0(z)^\alpha$ for $\alpha < 1$ show that:

$log\left ( \frac{fdr(z)}{1-fdr(z)} \right ) = log\left ( \frac{FDR(z)}{1-FDR(z)} \right ) + log\left ( \frac{1}{\alpha} \right ) $

From the figure seem pretty clear that indeed fdr must be greater than the FDR.

enter image description here

But I'd find the answer to the problem very reassuring.

---------- EDIT ---------

I want to add two more equations, given in large-scale inference, that make my question more complete:

the distribution of the mixture:

$F(Z) = \pi _0F_0(Z)+(1-\pi _0)F_1(Z)$

The pdf of the mixtures:

$f(z) = \pi _0f_0(z)+(1-\pi _0)f_1(z)$

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