Lomax distributions - Kullback Leibler divergence

Question

Does anyone know of a reference for an expression for the Kullback-Leibler divergence between two Lomax (Pareto II) distributions? Not really worried which way the Lomax is parameterized.

Answer 1

Using the parametrization from Wikipedia $$ p(x; \alpha, \lambda) = \frac\alpha\lambda \left( 1+\frac{x}\lambda \right)^{-(\alpha+1)} $$ with shape $\alpha>0$, scale $\lambda>0$ and define the Kullback.Leibler divergence as at Intuition on the Kullback–Leibler (KL) Divergence $$ \DeclareMathOperator{\KL}{KL} \KL(P || Q) = \int_{-\infty}^\infty p(x) \log \frac{p(x)}{q(x)} \; dx $$ we can see what maple gives us. First, assume equal scale parameters, then I get $$ \KL(p(\alpha_1,\lambda || p(\alpha_0,\lambda) = \frac{\ln \! \left(\alpha_1 \right) \alpha_1 -\ln \! \left(\alpha_0 \right) \alpha_1 +\alpha_0 -\alpha_1}{\alpha_1} $$ while the general case looks quite unhelpful, involving some special functions: $$ \KL(p(\alpha_1,\lambda_1 || p(\alpha_0,\lambda_0) = {{\frac {1}{\lambda_1}\,\Gamma \left( {\alpha_1}+1 \right) } \left( -{{\alpha_1}}^{2} G^{2, 2}_{2, 2}\left(1\, \Big\vert\,^{0, 1-{\alpha_1}}_{0, 0}\right) {\lambda_1}+{\alpha_1}\,{\lambda_0}\, G^{2, 2}_{2, 2}\left({\frac {{\lambda_0}}{{\lambda_1}}}\, \Big\vert\,^{-1, -{\alpha_1}}_{-1, -1}\right) {\alpha_0}+\ln \left( {\alpha_1} \right) {\lambda_1}\, \Gamma \left( {\alpha_1} + 1 \right) -\ln \left( {\alpha_0} \right) {\lambda_1}\,\Gamma \left( {\alpha_1} + 1 \right) +\ln \left( {\lambda_0} \right) {\lambda_1}\,\Gamma \left( {\alpha_1}+1 \right) -\ln \left( {\lambda_1} \right) {\lambda_1}\,\Gamma \left( {\alpha_1}+1 \right) -{\alpha_1}\, G^{2, 2}_{2, 2}\left(1\, \Big\vert\,^{0, 1-{\alpha_1}}_{0, 0}\right) {\lambda_1}+{\alpha_1}\,{\lambda_0}\, G^{2, 2}_{2, 2}\left({\frac {{\lambda_0}}{{\lambda_1}}}\, \Big\vert\,^{-1, -{\alpha_1}}_{-1, -1}\right) \right) } $$ where $G$ is the Meijer G function. That does not look too useful, so for practical use one can solve each concrete case with maple:

p := (x,alpha,lambda) -> (alpha/lambda) * (1+x/lambda)^(-(alpha+1))
LomaxKL := (alpha_0,alpha_1,lambda_0,lambda_1) ->  int( 
    p(x, alpha_1, lambda_1) * log( p(x, alpha_1, lambda_1)/ 
    p(x, alpha_0, lambda_0) ),  x=0..infinity)
LomaxKL(2,3,1,1)
                                       1
                       ln(3) - ln(2) - -
                                       3
LomaxKL(1/2,1/3,1,2) 
                   3  (1/3)

      -4 - ln(3) + - 2      ln(2)
                   2
       3  (1/3)   /   (2/3)      (1/3)    \
     - - 2      ln\4 2      - 8 2      + 4/
       4                                   

       3  (1/3)   /     (1/3)    (2/3)\
     + - 2      ln\1 + 2      + 2     /
       4                               

       3  (1/2)  (1/3)       /1  (1/2) /   (1/3)    \\
     - - 3      2      arctan|- 3      \2 2      + 1/|
       2                     \3                      /

       3  (1/3)  (1/2)   
     + - 2      3      Pi
       4                 

The last in mathematical notation is $$ -4-\ln \! \left(3\right) + \frac{3 \, 2^{\frac{1}{3}} \ln \! \left(2\right)}{2} - \\ \frac{3 \, 2^{\frac{1}{3}} \ln \! \left(4 \,2^{\frac{2}{3}} - 8 \, 2^{\frac{1}{3}} + \\ 4\right)}{4} + \frac{3 \, 2^{\frac{1}{3}} \ln \! \left(1 + \\ 2^{\frac{1}{3}} + 2^{\frac{2}{3}}\right)}{4} - \\ \frac{3 \sqrt{3}\, 2^{\frac{1}{3}} \arctan \! \left(\frac{\sqrt{3}\, \left(2 \,2^{\frac{1}{3}} + \\1\right)}{3}\right)}{2} + \\ \frac{3 \,2^{\frac{1}{3}} \sqrt{3}\, \pi}{4} $$