Following this question, I have the PDF of a gamma-exponential compound distribution as
f(y)=αβα(y+β)(α+1)
For my application I need the entropy of this distribution. So far I've evaluated this numerically using quadrature, but it's too slow. Is there a closed-form expression for the entropy of this distribution? I've attempted to derive it myself by evaluating
∫∞−∞f(y)log(f(y))dy
as a sum of two improper integrals ∫0−∞...dy+∫∞0...dy but I get lost when trying to evaluate the infinite limits:
∫f(y)log(f(y))dy=βα(β+y)−α(1+α−α⋅log(αβα(β+y)−1−α))α (according to Wolfram)
∫∞−∞f(y)log(f(y))dy=lim
(Skipped a step or two there.)
But here I get stuck because I'm not sure how to evaluate the limits when the logarithms are part of the function, as I haven't found (here or here) a clear limits of logarithms rule that I can apply here.
Sorry my typesetting is ugly. I'd love to know if there's a known entropy for this distribution or if somebody can help me past where I'm stuck in the derivation of it. Thanks
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