Sunday, 8 December 2013

limits - Entropy of gamma-exponential compound distribution



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+ααlog(αβα(β+y)1α))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


Answer



As commented in the original question, this is a Pareto distribution with parameter \alpha (it's shifted, but that's irrelevant - you can consider instead the variable z=y+\beta). It's entropy is then



H(y)=\log\left(\frac{\beta}{\alpha}\right)+1 + 1/\alpha



as computed here


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