### The joint PDF of the different classes of bivariate beta distribution

The joint PDF of the bivariate random variable in (3) is (see Libby and Novick (1982), Jones (2001) and Olkin and Liu (2003)),

f\left({z}_{1},{z}_{2}\right)=\frac{{z}_{1}^{{a}_{1}-1}{z}_{2}^{{a}_{2}-1}{\left(1-{z}_{1}\right)}^{{a}_{2}+b-1}{\left(1-{z}_{2}\right)}^{{a}_{1}+b-1}}{B\left({a}_{1},{a}_{2},b\right){\left(1-{z}_{1}{z}_{2}\right)}^{{a}_{1}+{a}_{2}+b}},\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}0<{z}_{1},{z}_{2}<1

(22)

where *B*(*a*_{1},*a*_{2},*b*) = Γ(*a*_{1})Γ(*a*_{2})Γ(*b*)/Γ(*a*_{1} + *a*_{2} + *b*) and the marginal distributions are {Z}_{1}\sim \mathcal{B}e({a}_{1},b) and {Z}_{2}\sim \mathcal{B}e({a}_{2},b). Note that (22) belongs to the three-parametric exponential family, where sufficient statistics for (*a*_{1},*a*_{2},*b*) are given by,

\left(\prod _{i=1}^{n}\frac{{y}_{1i}\left(1-{y}_{2i}\right)}{1-{y}_{1i}{y}_{2i}},\prod _{i=1}^{n}\frac{{y}_{2i}\left(1-{y}_{1i}\right)}{1-{y}_{1i}{y}_{2i}},\prod _{i=1}^{n}\frac{\left(1-{y}_{1i}\right)\left(1-{y}_{2i}\right)}{1-{y}_{1i}{y}_{2i}}\right).

For *n* = 1, the distributions of the sufficient statistics are:

\frac{1-{Z}_{1}{Z}_{2}}{{Z}_{1}\left(1-{Z}_{2}\right)}\sim \mathcal{B}2\left({a}_{2}+b,{a}_{1}\right)+1,

\frac{1-{Z}_{1}{Z}_{2}}{{Z}_{2}\left(1-{Z}_{1}\right)}\sim \mathcal{B}2\left({a}_{1}+b,{a}_{2}\right)+1,

and

\frac{1-{Z}_{1}{Z}_{2}}{\left(1-{Z}_{1}\right)\left(1-{Z}_{2}\right)}\sim \mathcal{B}2\left({a}_{1}+{a}_{2},b\right)+1,

where \mathcal{B}2(a,b) denotes beta distribution of the second kind.

The log-moments of (22) are:

\begin{array}{ll}\phantom{\rule{4.4em}{0ex}}E\left\{log\frac{{Z}_{1}\left(1-{Z}_{2}\right)}{1-{Z}_{1}{Z}_{2}}\right\}& =\psi \left({a}_{1}\right)-\psi \left({a}_{1}+{a}_{2}+b\right),\\ \phantom{\rule{4.4em}{0ex}}E\left\{log\frac{{Z}_{2}\left(1-{Z}_{1}\right)}{1-{Z}_{1}{Z}_{2}}\right\}& =\psi \left({a}_{2}\right)-\psi \left({a}_{1}+{a}_{2}+b\right),\\ \phantom{\rule{2em}{0ex}}E\left\{log\frac{\left(1-{Z}_{1}\right)\left(1-{Z}_{2}\right)}{1-{Z}_{1}{Z}_{2}}\right\}& =\psi (b)-\psi \left({a}_{1}+{a}_{2}+b\right).\end{array}

The joint PDF of the bivariate beta density (8) is givenby (see El-Bassiouny and Jones (2009)),

f\left({z}_{1},{z}_{2}\right)=k\frac{{z}_{1}^{{a}_{1}-1}{\left(1\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{z}_{1}\right)}^{A-{a}_{1}-1}{z}_{2}^{{a}_{2}-1}{\left(1\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{z}_{2}\right)}^{A-{a}_{2}-1}}{{\left(1\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{z}_{1}{z}_{2}\right)}^{A}}\times {}_{2}{F}_{1}\left[A,{a}_{4};A\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{a}_{2};\frac{{z}_{1}\left(1-{z}_{2}\right)}{1\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{z}_{1}{z}_{2}}\right],

(23)

where *A* = *a*_{1} + *a*_{2} + *a*_{3} + *a*_{4}, *k*^{-1} = *B*(*a*_{1},*a*_{3})*B*(*a*_{2},*a*_{1} + *a*_{3} + *a*_{4}) and _{2}*F*_{1}[..;.;] denote the Gauss hypergeometric function.

The expression for the *f*_{V,W}(*v*,*w*) function is given by (see Arnold and Ng 2011),

{f}_{V,W}(v,w)={\int}_{0}^{\infty}{\int}_{0}^{\infty}{\int}_{{u}_{4}/\left(w-{u}_{5}\right)}^{\left({u}_{4}+{u}_{5}\right)v}f\left(v,w,{u}_{3},{u}_{4},{u}_{5}\right){\mathit{\text{du}}}_{3}{\mathit{\text{du}}}_{4}{\mathit{\text{du}}}_{5},\phantom{\rule{2.77626pt}{0ex}}u,w>0,

(24)

where

\begin{array}{ll}\phantom{\rule{5.5pt}{0ex}}f\left(v,w,{u}_{3},{u}_{4},{u}_{5}\right)& =\frac{\left({u}_{3}+{u}_{5}\right)\left({u}_{4}+{u}_{5}\right)}{{\prod}_{i=1}^{5}\mathrm{\Gamma}\left({a}_{i}\right)}{\left[\left(v\left({u}_{4}+{u}_{5}\right)-{u}_{3}\right)\right]}^{{a}_{1}-1}\\ \phantom{\rule{1em}{0ex}}\times {\left[w\left({u}_{3}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{u}_{5}\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{u}_{4}\right]}^{{a}_{2}-1}\prod _{i=3}^{5}{u}_{i}^{{a}_{i}-1}exp\{-\phantom{\rule{0.3em}{0ex}}\left[{u}_{3}w\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{u}_{4}v\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{u}_{5}\left(v\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}w\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}1\right)\right]\},\end{array}

where *u*_{4}/*w* - *u*_{5} < *u*_{3} < (*u*_{4} + *u*_{5})*v*, *u*_{4},*u*_{5},*v*,*w* > 0.

### Description of the data set

The list of countries used in the analysis are the following:

Afghanistan, Guatemala, Pakistan, Albania, Guyana, Panama, Algeria, Haiti, Papua New Guinea, Argentina, Honduras, Paraguay, Armenia, Hong Kong, China (SAR), Peru, Australia, Hungary, Philippines, Austria, Iceland, Poland, Bahrain, India, Portugal, Bangladesh, Indonesia, Qatar, Belgium, Iran (Islamic Republic of), Romania, Belize, Ireland, Russian Federation, Benin, Israel, Rwanda, Bolivia (Plurinational State of), Italy, Saudi Arabia, Botswana, Jamaica, Senegal, Brazil, Japan, Sierra Leone, Brunei Darussalam, Jordan, Slovakia, Bulgaria, Kenya, Slovenia, Burundi, Korea (Republic of), South Africa, Cameroon, Kuwait, Spain, Canada, Lao PDR, Sri Lanka, Central African Republic, Latvia, Sudan, Chile, Lesotho, Swaziland, China, Liberia, Sweden, Colombia, Lithuania, Switzerland, Congo, Luxembourg, Syrian Arab Republic, Congo (Democratic Republic of), Malawi, Tajikistan, Costa Rica, Malaysia, Tanzania (United Republic of), Cote D’ivoire, Mali, Thailand, Cuba, Malta, Togo, Cyprus, Mauritania, Tonga, Denmark, Mauritius, Trinidad and Tobago, Dominican Republic, Mexico, Tunisia, Ecuador, Moldova (Republic of), Turkey, Egypt, Mongolia, Uganda, El Salvador, Morocco, Ukraine, Estonia, Mozambique, United Arab Emirates, Fiji, Myanmar, United Kingdom, Finland, Namibia, United States, France, Nepal, Uruguay, Gabon, Netherlands, Venezuela (Bolivarian R.), Gambia, New Zealand, VietNam, Germany, Nicaragua, Yemen, Ghana, Niger, Zambia, Greece, Norway, Zimbabwe.

Data on the health index can be retrieved from https://data.undp.org/dataset/Health-index/9v27-i7ic, data on the education index can be drawn from https://data.undp.org/dataset/Expected-Years-of-Schooling-of-children-years-/qnam-f624 for the variable expected years of schooling and https://data.undp.org/dataset/Mean-years-of-schooling-of-adults-years-/m67k-vi5c for the mean years of schooling. Finally, income data come from https://data.undp.org/dataset/Income-index/qt4g-yea9.