Table 1.

KLD and KL distance computed when ${\mathbf{X}}^1$ and ${\mathbf{X}}^2$ are two random vectors following central MCDs with pdfs $f_{{\mathbf{X}}^1}\left(\mathit{x}\right|{\mathbf{\Sigma }}_1,p)$ and $f_{{\mathbf{X}}^2}\left(\mathit{x}\right|{\mathbf{\Sigma }}_2,p)$.

$$\begin{array}{ll} \mathrm{KL}\left({\mathbf{X}}^1\right|\left|{\mathbf{X}}^2\right)& \\ \mathrm{(106)}& =-\frac{1}{2}log\prod _{i=1}^p{\lambda }_i+\frac{1+p}{2}\left[log{\lambda }_p-\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(a,\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2},a+\frac{1}{2}}};a+\frac{1+p}{2};1-\frac{{\lambda }_1}{{\lambda }_p},\dots ,1-\frac{{\lambda }_{p-1}}{{\lambda }_p},1-\frac{1}{{\lambda }_p}\right)\right\}{|}_{a=0}\right]\hfill \mathrm{(107)}& =-\frac{1}{2}log\prod _{i=1}^p{\lambda }_i-\frac{1+p}{2}{\lambda }_p^{\frac{p}{2}}\prod _{i=1}^{p-1}{\lambda }_i^{-\frac{1}{2}}\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(\frac{1+p}{2},\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2},a+\frac{1}{2}}};a+\frac{1+p}{2};1-\frac{{\lambda }_p}{{\lambda }_1},\dots ,1-\frac{{\lambda }_p}{{\lambda }_{p-1}},1-{\lambda }_p\right)\right\}{|}_{a=0}\hfill \mathrm{(108)}& =-\frac{1}{2}log\prod _{i=1}^p{\lambda }_i+\frac{1+p}{2}\left[log{\lambda }_1-\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(a,\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2},a+\frac{1}{2}}};a+\frac{1+p}{2};1-\frac{{\lambda }_p}{{\lambda }_1},\dots ,1-\frac{{\lambda }_2}{{\lambda }_1},1-\frac{1}{{\lambda }_1}\right)\right\}{|}_{a=0}\right]\hfill \mathrm{(109)}& =-\frac{1}{2}log\prod _{i=1}^p{\lambda }_i-\frac{1+p}{2}\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(a,\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2}}};a+\frac{1+p}{2};1-{\lambda }_1,\dots ,1-{\lambda }_p\right)\right\}{|}_{a=0}\hfill \mathrm{(110)}& =-\frac{1}{2}log\prod _{i=1}^p{\lambda }_i-\frac{1+p}{2}\prod _{i=1}^p{\lambda }_i^{-\frac{1}{2}}\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(\frac{1+p}{2},\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2}}};a+\frac{1+p}{2};1-\frac{1}{{\lambda }_1},\dots ,1-\frac{1}{{\lambda }_p}\right)\right\}{|}_{a=0}\hfill \end{array}$$

$$\begin{array}{ll} \mathrm{KL}\left({\mathbf{X}}^2\right|\left|{\mathbf{X}}^1\right)\hfill & \\ \mathrm{(111)}& =\frac{1}{2}log\prod _{i=1}^p{\lambda }_i-\frac{1+p}{2}\left[log{\lambda }_p+\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(a,\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2},a+\frac{1}{2}}};a+\frac{1+p}{2};1-\frac{{\lambda }_p}{{\lambda }_1},\dots ,1-\frac{{\lambda }_p}{{\lambda }_{p-1}},1-{\lambda }_p\right)\right\}{|}_{a=0}\right]\hfill \mathrm{(112)}& =\frac{1}{2}log\prod _{i=1}^p{\lambda }_i-\frac{1+p}{2}{\lambda }_p^{-\frac{p}{2}}\prod _{i=1}^{p-1}{\lambda }_i^{\frac{1}{2}}\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(\frac{1+p}{2},\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2},a+\frac{1}{2}}};a+\frac{1+p}{2};1-\frac{{\lambda }_1}{{\lambda }_p},\dots ,1-\frac{{\lambda }_{p-1}}{{\lambda }_p},1-\frac{1}{{\lambda }_p}\right)\right\}{|}_{a=0}\hfill \mathrm{(113)}& =\frac{1}{2}log\prod _{i=1}^p{\lambda }_i-\frac{1+p}{2}\left[log{\lambda }_1+\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(a,\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2},a+\frac{1}{2}}};a+\frac{1+p}{2};1-\frac{{\lambda }_1}{{\lambda }_p},\dots ,1-\frac{{\lambda }_1}{{\lambda }_2},1-{\lambda }_1\right)\right\}{|}_{a=0}\right]\hfill \mathrm{(114)}& =\frac{1}{2}log\prod _{i=1}^p{\lambda }_i-\frac{1+p}{2}\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(a,\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2}}};a+\frac{1+p}{2};1-\frac{1}{{\lambda }_1},\dots ,1-\frac{1}{{\lambda }_p}\right)\right\}{|}_{a=0}\hfill \mathrm{(115)}& =\frac{1}{2}log\prod _{i=1}^p{\lambda }_i-\frac{1+p}{2}\prod _{i=1}^p{\lambda }_i^{\frac{1}{2}}\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(\frac{1+p}{2},\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2}}};a+\frac{1+p}{2};1-{\lambda }_1,\dots ,1-{\lambda }_p\right)\right\}{|}_{a=0}\hfill \end{array}$$

$$\begin{array}{lll}& d_{\mathrm{KL}}({\mathbf{X}}^1,{\mathbf{X}}^2)\hfill & \\ =& \frac{1+p}{2}[log{\lambda }_p-\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(a,\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2},a+\frac{1}{2}}};a+\frac{1+p}{2};1-\frac{{\lambda }_1}{{\lambda }_p},\dots ,1-\frac{{\lambda }_{p-1}}{{\lambda }_p},1-\frac{1}{{\lambda }_p}\right)\right\}{|}_{a=0}-{\lambda }_p^{-\frac{p}{2}}\prod _{i=1}^{p-1}{\lambda }_i^{\frac{1}{2}}& \\ \mathrm{(116)}& & \times \frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(\frac{1+p}{2},\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2},a+\frac{1}{2}}};a+\frac{1+p}{2};1-\frac{{\lambda }_1}{{\lambda }_p},\dots ,1-\frac{{\lambda }_{p-1}}{{\lambda }_p},1-\frac{1}{{\lambda }_p}\right)\right\}{|}_{a=0}]\hfill =& -\frac{1+p}{2}[\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(a,\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2}}};a+\frac{1+p}{2};1-{\lambda }_1,\dots ,1-{\lambda }_p\right)\right\}{|}_{a=0}+\prod _{i=1}^p{\lambda }_i^{\frac{1}{2}}\frac{\partial }{\partial a}\{F_D^{\left(p\right)}(\frac{1+p}{2},\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2}}};a+\frac{1+p}{2};& \\ \mathrm{(117)}& & 1-{\lambda }_1,\dots ,1-{\lambda }_p\left)\right\}{|}_{a=0}]\hfill =& -\frac{1+p}{2}[\prod _{i=1}^p{\lambda }_i^{-\frac{1}{2}}\frac{\partial }{\partial a}\left\{F_D^{\left(p\right)}\left(\frac{1+p}{2},\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2}}};a+\frac{1+p}{2};1-\frac{1}{{\lambda }_1},\dots ,1-\frac{1}{{\lambda }_p}\right)\right\}{|}_{a=0}+\frac{\partial }{\partial a}\{F_D^{\left(p\right)}(a,\underset{p}{\underbrace{\frac{1}{2},\dots ,\frac{1}{2}}};a+\frac{1+p}{2};& \\ \mathrm{(118)}& & 1-\frac{1}{{\lambda }_1},\dots ,1-\frac{1}{{\lambda }_p}\left)\right\}{|}_{a=0}]\hfill \end{array}$$