 Research
 Open Access
Bivariate betagenerated distributions with applications to wellbeing data
 José María Sarabia^{1}Email author,
 Faustino Prieto^{1} and
 Vanesa Jordá^{1}
https://doi.org/10.1186/21955832115
© Sarabia et al.; licensee Springer. 2014
Received: 28 February 2014
Accepted: 10 June 2014
Published: 30 June 2014
Abstract
The class of betagenerated distributions (Commun. Stat. Theory Methods 31:497–512, 2002; TEST 13:1–43, 2004) has received a lot of attention in the last years. In this paper, three new classes of bivariate betagenerated distributions are proposed. These classes are constructed using three different definitions of bivariate distributions with classical beta marginals and different covariance structures. We work with the bivariate beta distributions proposed in (J. Educ. Stat. 7:271–294, 1982; Metrika 54:215–231, 2001; Stat. Probability Lett. 62:407–412, 2003) for the first proposal, in (Stat. Methods Appl. 18: 465–481, 2009) for the second proposal and (J. Multivariate Anal. 102:1194–1202, 2011) for the third one. In each of these three classes, the main properties are studied. Some specific bivariate betagenerated distributions are studied. Finally, some empirical applications with wellbeing data are presented.
Mathematics Subject Classification (2000)
62E15; 60E05
Keywords
 Classical beta distribution
 Bivariate beta distribution
 Covariance structure
 GB1 and GB2 distributions
1 Introduction
In the recent statistical literature several methodologies of constructing bivariate and multivariate distributions based on marginal and conditional distributions have been proposed; see the works by Arnold et al. (1999; 2001), Kotz et al. (2000), Sarabia and GómezDéniz (2008) and Balakrishnan and Lai (2009) among others.
An important field of research focuses on the study of new classes of univariate distributions which contain the classical proposals, also allowing for more flexibility in fitting data. In this sense, the class of betagenerated (BG) distributions (Eugene et al. 2002; Jones 2004) has received an increasing amount of attention in recent years.
There are several reasons for studying classes of multivariate beta generated distributions. The two existing proposals of multivariate BG distributions present some drawbacks. The first proposal (Jones and Larsen 2004) is only valid for modeling data above the diagonal. The second proposal (Arnold et al. 2006) is defined in terms of the conditional distributions, and the corresponding marginal distributions do not follow, in general, beta generated distributions.
The three bivariate and multivariate models proposed in this paper present BG marginals, with high flexibility in the marginals and in the dependence structure. The marginal distributions of the first model share one of the shape parameters, and the structure of dependence satisfies TP2 condition (see Section 3). The marginals of the second model are free, and the different pairwise of marginals are associated (see Section 3). The third model is the more flexible, in the sense than all the marginals are free (they do not share any shape parameter) and the covariance structure admits correlations of any sign.
On the other hand, these classes of distributions present several fields of applicability. For example, bivariate beta generated distributions with classical beta marginals are natural choices to be used as prior distributions for the parameters of correlated binomial random variables (with any sign for the correlation) in Bayesian analysis (see Apostolakis and Moieni 1987; Arnold and Ng 2011).
In this work, three new classes of bivariate BG distributions are proposed. These classes are constructed using three alternative definitions of bivariate distributions with classical beta marginals and different covariance structures. We work with the bivariate beta distributions proposed by Libby and Novick (1982), Jones (2001) and Olkin and Liu (2003) for the first proposal, ElBassiouny and Jones (2009) for the second proposal and Arnold and Ng (2011) for the third one. For each of these three classes, the main properties are obtained. Some specific bivariate BG distributions are studied. Finally, some empirical applications with wellbeing data are presented.
The contents of this paper are as follows. In Section 2 we present some basic properties of the class of the BG distributions and a brief review about two multivariate extensions of the BG distribution. Section 3 considers and studies the three classes of bivariate BG distributions and their main properties as well as to introduce three specific bivariate distributions. A number of applications of these distributions to fit wellbeing data are presented in Section 4. Finally, some conclusions and future research directions are given in Section 5.
2 Some univariate and multivariate betagenerated distributions
2.1 The univariate class of betagenerated distributions
where B(a,b) = Γ(a)Γ(b)/Γ(a + b) denotes the classical beta function. A random variable X with PDF (1) will be denoted by $X\sim \mathcal{B}\mathcal{G}(a,b;F)$.
where I_{F(x)}(·,·) denotes the incomplete beta ratio.
An important number of new classes of distributions have been proposed using this methodology. Some representatives examples of BG distributions include the generalized beta of the first kind (GB1) proposed by McDonald (1984), the generalized beta of the second kind (GB2) proposed and studied by Venter (1983) and McDonald (1984), the logF distribution (BarndorffNielsen et al. 1982), the betanormal distribution (Eugene et al. 2002), the betaexponential distribution (Nadarajah and Kotz 2006) and the Skewt distribution (Jones 2004).
Some extensions of this family have been proposed by Alexander and Sarabia (2010), Alexander et al. (2012), Cordeiro and de Castro (2011) and Zografos (2011). Other alternative flexible families of distributions can be found in Alzaatreh et al. (2013, 2014) and Lee et al. (2013).
If a = i and b = n  i + 1 in (1), we obtain the PDF of the ith order statistic from F (Jones 2004). Below, we highlight some representative values of a and b,

If a = b = 1, g_{ F } = f.

If a = n and b = 1, we obtain the distribution of the maximum.

If a = 1 and b = n, we obtain the distribution of the minimum.

If a ≠ b, we obtain a family of skew distributions.
Parameters a and b control the tailweight of the distribution. Specifically, the a parameter controls lefthand tailweight and the b parameter controls the righthand tailweight of the distribution. On the other hand, a = b yields a symmetric subfamily, with a controlling tailweight. If a = b = 1 the BG family is always symmetric if the baseline function F(x) is symmetric. In this sense, the BG distribution accommodates several kind of tails. For example (see Jones 2004),

Potential tails: If f ∼ x^{(α+1)} and α > 0, when x→∞ g_{ F } ∼ x^{b α1},

Exponential tails: If f ∼ e^{α x} and β > 0, then g_{ F } ∼ e^{b β x} if x → ∞.
2.2 Two previous classes of multivariate extensions of betagenerated distributions
There are two proposals for multivariate extensions of BG distributions. The first proposal used the joint PDF of a subset of order statistics, and has been proposed by Jones and Larsen (2004). The second proposal used the so called Rosenblatt construction (Rosenblatt 1952), and has been proposed by Arnold et al. (2006). These two alternatives are not related with the multivariate BG distributions studied in this paper.
3 Three classes of bivariate betagenerated distributions
In this section we introduce three new classes of bivariate BG distributions. These three classes are constructed combining the basic stochastic representation (2) with three recent definitions of bivariate beta distributions proposed in the literature. The three definitions differ from each other in the marginal distributions and in the flexibility of the covariance structure. First, we need some previous notation. Let X be a random variable distributed as the classical gamma denoted by X ∼ G_{ a }, with PDF f(x) = [Γ(a)]^{1}x^{a1}e^{x}, with x ≥ 0 and a > 0. Then, if X_{1} ∼ G_{ a } and X_{2} ∼ G_{ b } are independent gamma random variables, the transformed random variable X = X_{1}/(X_{1} + X_{2}) is distributed as the classical beta distribution with parameters (a,b).
3.1 The first class of bivariate betagenerated distributions
The first class of bivariate betagenerated distribution is based on the following class of bivariate beta distribution.
This class was initially proposed by Libby and Novick (1982) and then studied by Jones (2001) and Olkin and Liu (2003).
Now, using (3) we define the following class of bivariate BG distributions.
where F_{1}(·),F_{2}(·) are genuine CDF.
3.1.1 Basic properties
Note that both marginals share the second shape parameter b. However, this fact does not make the model less flexible, since both baseline distributions F_{1} and F_{2} are different.
We use the definitions of total positivity of order 2 (T P_{2}) functions and reverse rule of order 2 (R R_{2}) functions, which are the following.
for all x ≤ u and y ≤ v.
The following result relates the local dependence function γ(x_{1},x_{2}) with the T P_{2} and R R_{2} (see Theorem 7.1 in Holland and Wang 1987).
Theorem 1. Let f(x_{1},x_{2}) be the joint PDF of (X_{1},X_{2}) with support on a set S where the set S = S_{1} × S_{2}. Then, f(x_{1},x_{2}) is T P_{2} (R R_{2}) if and only if γ(x_{1},x_{2}) ≥ 0(≤ 0).
and then X_{1} and X_{2} are T P_{2}. As a consequence, the linear correlation coefficient between X_{1} and X_{2} is always positive.
It can be proved (see Shaked 1977) that if the joint PDF f(x_{1},x_{2}) is T P_{2} (R R_{2}), then the conditional hazard rate of X_{1}X_{2} = x_{2} is decreasing (increasing) in x_{2}. A similar property holds for the other conditional distribution X_{2}X_{1} = x_{1}. As the PDF in (5) is T P_{2}, this property shows the monotonicity properties of the hazard rate functions of the conditional distributions of X_{1}X_{2} = x_{2} as a function of x_{2} and the X_{2}X_{1} = x_{1} as a function of X_{1}.
On the other hand, because X_{1} and X_{2} are increasing functions of independent random variables, X_{1} and X_{2} are associated random variables (Esary et al. 1967).
3.1.2 Extension to higher dimensions
where c^{1} = B(a_{1},…,a_{ m },b) = Γ(a_{1}) ⋯ Γ(a_{ m })Γ(b)/Γ(a_{1} + … + a_{ m } + b).
3.2 The second type of bivariate betagenerated distributions
The second type of bivariate BG distribution is motivated by the fact of having a bivariate distribution with arbitrary BG marginals. This second class is based on the following class of bivariate beta distribution, which was proposed by ElBassiouny and Jones (2009).
Now, we define the second class of BG distributions.
where F_{1}(·),F_{2}(·) are genuine CDF.
3.2.1 Basic properties
being _{2}F_{1}[..;.;] the Gauss confluent hypergeometric function.
where ${k}^{\prime \prime}=k\frac{B({a}_{1},{a}_{3})}{B({a}_{1},A{a}_{1})B({a}_{2},A{a}_{2})}$.
being ${a}_{2}({x}_{1})={k}^{\prime \prime}{\left[1{F}_{1}({x}_{1})\right]}^{A{a}_{1}{a}_{3}}$.
The second term in (11) is long and will not be included here.
The random variables X_{1} and X_{2} are associated and then the linear correlation coefficient is always nonnegative (see Definition I.11 and Proposition I.13 in Marshall and Olkin (2007)).
3.2.2 Multivariate extensions
3.3 The third type of bivariate betagenerated distributions
The next class of bivariate beta distribution is the more general class in the sense that the marginal distributions have arbitrary parameters and admits any sign for the linear correlation coefficient. The following definition was proposed by Arnold and Ng (2011).
where ${G}_{{a}_{i}}$, i = 1,2,3,4,5 are independent gamma random variables, where a_{ i } > 0, i = 1,2,3,4,5.
Now, we define the third class of BG distributions.
where F_{1}(·),F_{2}(·) are genuine CDF.
3.3.1 Basic properties
The marginal distributions of (12) are ${X}_{1}\sim \mathcal{B}\mathcal{G}({a}_{1}+{a}_{3},{a}_{4}+{a}_{5};{F}_{1})$ and ${X}_{2}\sim \mathcal{B}\mathcal{G}({a}_{2}+{a}_{4},{a}_{3}+{a}_{5};{F}_{2})$.
where f_{V,W}(·,·) is defined in equation (24) in the Appendix.
where k^{′′} = B(a_{1} + a_{3},a_{4} + a_{5}).
with ${b}_{2}({x}_{1})=\frac{{k}^{\prime \prime}}{{[1{F}_{1}({x}_{1})]}^{{a}_{4}+{a}_{5}+1}{F}_{1}{({x}_{1})}^{{a}_{1}+{a}_{3}1}}$.
where $h({x}_{1},{x}_{2})=\frac{{f}_{1}({x}_{1}){f}_{2}({x}_{2})}{[1{F}_{1}({x}_{1})][1{F}_{2}({x}_{2})]}$ and ${u}_{i}=\frac{{F}_{i}({x}_{i})}{1{F}_{i}({x}_{i})}$, i = 1,2.
In a similar way, the crossproduct moments can be obtained using formula (10), where now (Z_{1},Z_{2}) is the bivariate random variable with joint PDF given by (24).
The covariance structure of (12) is flexible and the sign of the linear correlation coefficient can be positive or negative.
3.3.2 Multivariate extensions
where ${G}_{{a}_{i}}$, ${G}_{{b}_{i}}$, i = 1,2,…,m and G_{ c } are independent gamma random variables.
3.4 Estimation
where we have used the notation ${\stackrel{\u0304}{F}}_{i}\left(\xb7;{\mathit{\tau}}_{i}\right)=1{F}_{i}\left(\xb7;{\mathit{\tau}}_{i}\right)$, i = 1,2.
This expression may be maximized either directly, e.g. using the Mathematica software function FindMaximum (see Wolfram Research, Inc. 2010), the SAS procedure NLMIXED (SAS Institute, Inc. 2010), the R software functions nlm or optim (R Development Core Team 2011), or the MATLAB function fmincon (The Mathworks, Inc. 2011), among others, which provides numerical algorithms for nonlinear optimization), or by solving the nonlinear equations obtained by differentiating expression (13).
where ${\stackrel{\u0307}{f}}_{j}{({x}_{\mathit{\text{ji}}})}_{{\mathit{\tau}}_{j}}=\partial {f}_{j}({x}_{\mathit{\text{ji}}};{\mathit{\tau}}_{j})/\partial {\mathit{\tau}}_{j}$ and ${\stackrel{\u0307}{F}}_{j}{({x}_{\mathit{\text{ji}}})}_{{\mathit{\tau}}_{j}}=\partial {F}_{j}({x}_{\mathit{\text{ji}}};{\mathit{\tau}}_{j})/\partial {\mathit{\tau}}_{j}$ are p_{ j } × 1 vectors, with j = 1,2 and ψ(s) = d logΓ(s)/d s is the digamma function.
For obtaining interval estimation and hypothesis testing on the model parameters, we need the observed information matrix. The (p_{1} + p_{2} + 3,p_{1} + p_{2} + 3) observed matrix J = J(θ) can be obtained by taking partial second derivatives in the score vector U(θ). Assuming conditions that are fulfilled for parameters in the interior of the parameter space (but not in the boundary), the distribution of $\sqrt{n}\left(\widehat{\mathit{\theta}}\mathit{\theta}\right)$ is asymptotically normal ${\mathcal{N}}_{{p}_{1}+{p}_{2}+3}\left(0,I{(\mathit{\theta})}^{1}\right)$, where I(θ) denotes the expected information matrix. As usual, we can substitute I(θ) by $J\left(\widehat{\mathit{\theta}}\right)$, that is, the observed information matrix evaluated at $\widehat{\mathit{\theta}}$ and then, the distribution ${\mathcal{N}}_{{p}_{1}+{p}_{2}+3}\left(0,J{(\widehat{\mathit{\theta}})}^{1}\right)$ can be used to construct approximate confidence intervals for the parameters.
The estimation of the other two models (9) and (12) requires a detailed study, which is beyond the scope of this paper and will be object of future research.
To finish this section, it should be mentioned that all the models proposed in this paper (4, 9 and 12) and their multivariate extensions can be enriched including location and scale parameters.
3.5 Some specific bivariate distributions
In this section we propose three specific bivariate BG models.
3.5.1 Bivariate BetaNormal distributions
where a_{1},a_{2},b > 0.
3.5.2 Bivariate GB1 income distributions
with 0 ≤ x_{1},x_{2} ≤ 1. The marginal distributions are X_{1} ∼ G B 1(p_{1},q,a_{1}) and X_{2} ∼ G B 1(p_{2},q,a_{2}). If we set a_{1} = a_{2} = 1 in (20), we obtain the bivariate beta proposed by Olkin and Liu (2003).
3.5.3 Bivariate GB2 income distributions
where the marginal distributions are X_{1} ∼ G B 2(p_{1},q,a_{1}) and X_{2} ∼ G B 2(p_{2},q,a_{2}).
4 Applications
To illustrate the methodology developed in this paper, we have fitted the bivariate BG model of the fist type defined in (4) to estimate the international distribution of wellbeing for the period 19802010. The estimation method is based on the formulation developed in Section 3.4. It should be worth noting that we have focused on three dimensions of wellbeing, namely income, health and education. Since these components present a positive correlation, the first type of BG distributions given by (4) is specially suitable in this case.
4.1 The data
We have used the most recent available data from International Human Development Indicators (UNDP 2012) on the HDI and its three components for the period 19802010 with five years intervals.
Note that we consider wellbeing as a multidimensional process which, in addition to economic variables, also involves social aspects such as health and education. In this context, the Human Development Index provides an excellent theoretical benchmark to make multidimensional assessments of wellbeing. Then, we have focused on three dimensions of quality of life: income, educational standards and health. In particular, we focus on the singledimensional indices of the HDI, which are three normalized variables placed on scale 10. This structure of the data is specially representative in this case since we consider Beta and GB1 marginals for the BG models.
Income is represented by Gross National Income per capita measured in PPP 2005 US dollars, to make incomes comparable across countries and over time. The health component is represented by life expectancy at birth, which is considered an indicator of the health level.The education index is made up of two indicators, expected years of schooling and mean years of schooling, which are aggregated using the geometric mean. The first educational variable informs about the number of years that a child of school entrance age can expect to receive if prevailing patterns of agespecific enrollment rates persist throughout the child’s life (UNDP 2012). The second indicator reports the average number of years of education received by people aged 25 and older, converted from education attainment levels using official durations of each level (Barro and Lee, 2013).
Originally, our sample comprised only 105 countries, covering less than the 75 percent of global population. We had nonavailable data for 26 countries for one or more years before 1995. In order to offer comparable results across periods and to not restricting the sample considerably, missing values have been estimated. The estimation of these missing values has been based on two complementary methodologies which jointly offered feasible and consistent results according to the sample: piecewise cubic Hermite interpolating polynomial (PCHI) and the average rate of change, which was used when PCHI offered unfeasible estimations or out of range results. The interpolated values have been obtained using the command pchip of the R package Signal, which uses the methodology described by Fritsch and Carlson (1980). After this procedure, our data set includes 132 countries whose indicators of income, health and education are available for eight points of time (see Appendix for details). Consequently, the sample covers over 90 percent of the world population during the whole period.
4.2 Fitted models and results
The bivariate data consist of three pairs of variables (income,education), (income,health) and (education, health).
We have fitted the class of models given by Equation (4) with three specifications for the baseline CDFs:

F_{ i }(x_{ i }) = x_{ i }, with 0 ≤ x_{ i } ≤ 1, i = 1,2 (classical beta marginals),

${F}_{i}\left({x}_{i}\right)={x}_{i}^{{a}_{i}}$, with 0 ≤ x_{ i } ≤ 1, a_{ i } > 0, i = 1,2 (GB1 marginals) and

${F}_{i}\left({x}_{i}\right)=\frac{1exp\left({a}_{i}{x}_{i}\right)}{1exp\left({a}_{i}\right)}$, with 0 ≤ x_{ i } ≤ 1, a_{ i } > 0, i = 1,2 (BG truncated exponential marginals).
The first model (with classical beta marginals) depends on 3 parameters, and the second and third models (with GB1 and BG truncated exponential marginals) are characterized by 5 parameters. The three models have been estimated by maximum likelihood using the equations given in Section 3.4. In total, we have fitted 7 × 3 × 3 = 63 different models. The initial estimates of the parameters have been obtained using Equations (17) to (19). In the case of the model with classical beta marginals, initial estimates are quite close to the ML estimators because they are based on sufficient statistics.
where $logL=\ell \left(\widehat{\mathit{\theta}}\right)$ is the loglikelihood of the model evaluated at the maximum likelihood estimates and d is the number of parameters. We chose the model with the smallest value of AIC statistic.
Parameter estimates for the BG models (with beta and GB1 marginals) fitted to the variables education & health by maximum likelihood (standard errors in parenthesis)
Variables: education & health  

Year  BG model with beta marginals  BG model with GB1 marginals  
${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\mathit{\xe2}}_{\mathbf{1}}$  ${\mathit{\xe2}}_{\mathbf{2}}$  
1980  2.6512  7.0544  3.4327  22.5595  10.5446  2.6651  0.1177  0.5381 
(0.2358)  (0.6334)  (0.3063)  (9.1385)  (2.4534)  (0.2890)  (0.0481)  (0.1299)  
1985  2.9384  7.5581  3.2621  11.1697  14.2985  2.6168  0.2527  0.4416 
(0.2614)  (0.6780)  (0.2906)  (3.6937)  (6.2408)  (0.2810)  (0.0818)  (0.1918)  
1990  2.9333  7.1301  2.8442  26.6132  6.6869  2.3141  0.1070  0.8566 
(0.2615)  (0.6410)  (0.2534)  (20.0869)  (1.7468)  (0.2473)  (0.0800)  (0.2243)  
1995  2.8508  6.1734  2.3377  17.3255  4.4878  2.0254  0.1606  1.1454 
(0.2551)  (0.5572)  (0.2085)  (7.0388)  (0.9738)  (0.2125)  (0.0649)  (0.2588)  
2000  2.7879  5.5716  1.9464  4.6151  5.7575  1.7644  0.5647  0.8677 
(0.2505)  (0.5051)  (0.1736)  (1.7265)  (2.8469)  (0.1811)  (0.2029)  (0.4191)  
2005  2.6539  5.0138  1.5705  13.1632  2.3387  1.5422  0.2096  1.9972 
(0.2398)  (0.4577)  (0.1401)  (7.3975)  (0.5126)  (0.1578)  (0.1173)  (0.4455)  
2010  2.6591  5.2296  1.4064  3.4113  3.7361  1.4084  0.7889  1.3821 
(0.2411)  (0.4794)  (0.1254)  (1.1784)  (1.4461)  (0.1418)  (0.2710)  (0.5291) 
Parameter estimates for the BG models (with beta and GB1 marginals) fitted to the variables education & income by maximum likelihood (standard errors in parenthesis)
Variables: education & income  

Year  BG model with beta marginals  BG model with GB1 marginals  
${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\mathit{\xe2}}_{\mathbf{1}}$  ${\mathit{\xe2}}_{\mathbf{2}}$  
1980  2.2431  3.3973  2.8809  2.1813  16.6110  2.6533  0.9603  0.2195 
(0.2008)  (0.3063)  (0.2591)  (0.5790)  (13.5230)  (0.2743)  (0.2243)  (0.1715)  
1985  2.8775  3.7222  3.1934  4.3394  10.2173  2.7715  0.6321  0.3586 
(0.2572)  (0.3338)  (0.2858)  (1.4677)  (5.6252)  (0.2881)  (0.1955)  (0.1857)  
1990  3.1691  3.6855  3.0715  6.4501  7.6217  2.6157  0.4702  0.4541 
(0.2834)  (0.3302)  (0.2746)  (2.7530)  (3.5058)  (0.2725)  (0.1880)  (0.1960)  
1995  3.3196  3.3074  2.7070  6.3618  6.2069  2.3322  0.4907  0.4965 
(0.2978)  (0.2967)  (0.2421)  (2.4599)  (2.2291)  (0.2418)  (0.1818)  (0.1698)  
2000  3.4652  3.0623  2.3826  5.8748  5.8042  2.0680  0.5420  0.4885 
(0.3119)  (0.2751)  (0.2131)  (2.2960)  (2.0882)  (0.2139)  (0.2052)  (0.1691)  
2005  3.6366  2.8670  2.0777  7.2438  3.4311  1.8787  0.4773  0.7694 
(0.3285)  (0.2580)  (0.1858)  (3.2031)  (0.8872)  (0.1918)  (0.2070)  (0.1937)  
2010  3.7596  2.7789  1.8944  3.4581  4.0863  1.8134  1.0309  0.6702 
(0.3406)  (0.2505)  (0.1693)  (1.0479)  (1.1725)  (0.1835)  (0.3034)  (0.1870) 
Parameter estimates for the BG models (with beta and GB1 marginals) fitted to the variables income & health by maximum likelihood (standard errors in parenthesis)
Variables: income & health  

Year  BG model with beta marginals  BG model with GB1 marginals  
${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\mathit{\xe2}}_{\mathbf{1}}$  ${\mathit{\xe2}}_{\mathbf{2}}$  
1980  3.4799  5.9823  2.9491  7.2415  4.5699  2.7798  0.4964  1.1929 
(0.3103)  (0.5360)  (0.2624)  (2.1497)  (1.0817)  (0.2855)  (0.1417)  (0.2674)  
1985  3.8895  7.7393  3.3336  19.9607  7.1397  2.7940  0.1918  0.9025 
(0.3458)  (0.6913)  (0.2959)  (11.3280)  (1.8739)  (0.2993)  (0.1072)  (0.2316)  
1990  3.7983  8.0223  3.1630  19.2882  10.1747  2.5143  0.18067  0.63504 
(0.3379)  (0.7173)  (0.2808)  (10.8390)  (3.4854)  (0.2721)  (0.1003)  (0.2148)  
1995  3.4082  7.5345  2.7867  12.4184  11.5787  2.2338  0.2457  0.5300 
(0.3039)  (0.6760)  (0.2478)  (4.6027)  (4.5656)  (0.2401)  (0.0901)  (0.2104)  
2000  3.0781  7.0973  2.3944  11.9092  7.5709  1.9876  0.2346  0.7667 
(0.2752)  (0.6396)  (0.2132)  (4.7419)  (2.2840)  (0.2109)  (0.0935)  (0.2342)  
2005  2.7891  6.8378  2.0250  19.9351  4.0164  1.7675  0.1323  1.4058 
(0.2501)  (0.6197)  (0.1805)  (19.3729)  (1.2603)  (0.1863)  (0.1265)  (0.4381)  
2010  2.4296  6.5389  1.6754  4.4081  4.2348  1.5941  0.5388  1.4186 
(0.2188)  (0.5974)  (0.1496)  (1.2981)  (1.4324)  (0.1627)  (0.1582)  (0.4772) 
Parameter estimates for the BG model (BG truncated exponential marginals) fitted to the variables education & health by maximum likelihood (standard errors in parenthesis)
Variables: education & health  

Year  Truncated exponential model  
${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\mathit{\xe2}}_{\mathbf{1}}$  ${\mathit{\xe2}}_{\mathbf{2}}$  
1980  3.4104  19.2541  1.3217  3.1388  4.1114 
(0.3671)  (4.0935)  (0.2372)  (0.4543)  (0.6456)  
1985  3.9159  17.3862  1.4386  2.7821  3.4638 
(0.4287)  (3.7250)  (0.2519)  (0.4325)  (0.6248)  
1990  4.1159  10.5428  1.6196  2.2094  2.1086 
(0.4795)  (2.0906)  (0.2652)  (0.4185)  (0.5804)  
1995  4.0476  6.7782  1.6363  1.7177  1.0854 
(0.5304)  (1.3450)  (0.2499)  (0.4271)  (0.5825)  
2000  3.8541  5.3537  1.5412  1.3568  0.5281 
(0.5772)  (1.1472)  (0.2193)  (0.4437)  (0.6087)  
2005  4.3824  3.8719  1.2821  1.6870  0.0000 
(0.7229)  (0.4862)  (0.1420)  (0.4704)  (0.0427)  
2010  3.9681  4.1826  1.1884  1.3861  0.0001 
(0.7878)  (1.0311)  (0.1607)  (0.5223)  (0.6912) 
Parameter estimates for the BG model (BG truncated exponential marginals) fitted to the variables education & income by maximum likelihood (standard errors in parenthesis)
Variables: education & income  

Year  Truncated exponential model  
${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\mathit{\xe2}}_{\mathbf{1}}$  ${\mathit{\xe2}}_{\mathbf{2}}$  
1980  2.3477  6.0323  1.5187  1.7859  2.8785 
(0.2801)  (0.9499)  (0.2891)  (0.5446)  (0.6098)  
1985  3.1220  6.4260  1.6681  1.8322  2.7940 
(0.3847)  (0.9671)  (0.3094)  (0.5189)  (0.5639)  
1990  3.5887  6.2829  1.5834  1.9604  2.8090 
(0.4533)  (0.9215)  (0.2838)  (0.5057)  (0.5350)  
1995  3.6986  5.9955  1.3892  1.9633  3.0090 
(0.4814)  (0.8708)  (0.2429)  (0.5081)  (0.5264)  
2000  3.7836  5.7931  1.2656  1.8579  3.0539 
(0.5115)  (0.8442)  (0.2131)  (0.5086)  (0.5110)  
2005  3.9060  5.3306  1.1572  1.7335  2.9354 
(0.5793)  (0.8152)  (0.1925)  (0.5332)  (0.5233)  
2010  3.3507  5.8944  1.1977  0.9970  2.8693 
(0.5127)  (0.9557)  (0.1937)  (0.5579)  (0.5168) 
Parameter estimates for the BG model (BG truncated exponential marginals) fitted to the variables income & health by maximum likelihood (standard errors in parenthesis)
Variables: income & health  

Year  Truncated exponential model  
${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\mathit{\xe2}}_{\mathbf{1}}$  ${\mathit{\xe2}}_{\mathbf{2}}$  
1980  5.2910  5.6187  2.1441  1.7075  0.6332 
(0.7765)  (1.0281)  (0.3450)  (0.4689)  (0.5700)  
1985  6.0810  8.9083  2.0766  2.1172  1.3880 
(0.7943)  (1.6258)  (0.3272)  (0.4063)  (0.5318)  
1990  6.6056  9.9881  1.7975  2.5828  1.7600 
(0.8013)  (1.7123)  (0.2787)  (0.3801)  (0.5043)  
1995  6.6121  8.9254  1.5662  2.8884  1.7273 
(0.8033)  (1.4908)  (0.2377)  (0.3774)  (0.4970)  
2000  6.2010  7.4495  1.4527  2.8150  1.3495 
(0.7880)  (1.2753)  (0.2108)  (0.3768)  (0.4997)  
2005  6.5460  5.5399  1.3561  2.9102  0.6186 
(0.9084)  (0.9529)  (0.1881)  (0.3852)  (0.5150)  
2010  6.6966  4.1236  1.1760  3.1741  0.0000 
(1.0480)  (0.7957)  (0.1720)  (0.4275)  (0.6583) 
Confidence intervals (95%) for the BG models (with beta and GB1 marginals) fitted to the variables education & health by maximum likelihood
Variables: education & health  

Year  Limit  BG model with beta marginals  BG model with GB1 marginals  
${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\mathit{\xe2}}_{\mathbf{1}}$  ${\mathit{\xe2}}_{\mathbf{2}}$  
1980  Lower  2.1890  5.8129  2.8324  4.6480  5.7359  2.0987  0.0234  0.2835 
Upper  3.1134  8.2959  4.0330  40.4710  15.3533  3.2315  0.2120  0.7927  
1985  Lower  2.4261  6.2292  2.6925  3.9300  2.0665  2.0660  0.0924  0.0657 
Upper  3.4507  8.8870  3.8317  18.4094  26.5305  3.1676  0.4130  0.8175  
1990  Lower  2.4208  5.8737  2.3475  12.7571  3.2632  1.8294  0.0498  0.4170 
Upper  3.4458  8.3865  3.3409  65.9835  10.1106  2.7988  0.2638  1.2962  
1995  Lower  2.3508  5.0813  1.9290  3.5295  2.5792  1.6089  0.0334  0.6382 
Upper  3.3508  7.2655  2.7464  31.1215  6.3964  2.4419  0.2878  1.6526  
2000  Lower  0.6984  2.8142  0.3379  7.9680  16.3910  0.3195  0.1146  0.3637 
Upper  3.2789  6.5616  2.2867  7.9990  11.3374  2.1194  0.9624  1.6891  
2005  Lower  2.1839  4.1167  1.2959  1.3359  1.3340  1.2329  0.0203  1.1240 
Upper  3.1239  5.9109  1.8451  27.6623  3.3434  1.8515  0.4395  2.8704  
2010  Lower  2.1865  4.2900  1.1606  1.1016  0.9017  1.1305  0.2577  0.3451 
Upper  3.1317  6.1692  1.6522  5.7210  6.5705  1.6863  1.3201  2.4191 
Confidence Intervals (95%) for the BG models (with beta and GB1 marginals) fitted to the variables education & income by maximum likelihood
Variables: education & health  

Year  Limit  BG model with beta marginals  BG model with GB1 marginals  
${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\mathit{\xe2}}_{\mathbf{1}}$  ${\mathit{\xe2}}_{\mathbf{2}}$  
1980  Lower  1.8495  2.7970  2.3731  1.0465  9.8941  2.1157  0.5207  0.1166 
Upper  2.6367  3.9976  3.3887  3.3161  43.1161  3.1909  1.3999  0.5556  
1985  Lower  2.3734  3.0680  2.6332  1.4627  0.8081  2.2068  0.2489  0.0054 
Upper  3.3816  4.3764  3.7536  7.2161  21.2427  3.3362  1.0153  0.7226  
1990  Lower  2.6136  3.0383  2.5333  1.0542  0.7503  2.0816  0.1017  0.0699 
Upper  3.7246  4.3327  3.6097  11.8460  14.4931  3.1498  0.8387  0.8383  
1995  Lower  2.7359  2.7259  2.2325  1.5404  1.8379  1.8583  0.1344  0.1637 
Upper  3.9033  3.8889  3.1815  11.1832  10.5759  2.8061  0.8470  0.8293  
2000  Lower  1.0808  0.8424  0.5077  13.4885  12.1203  0.4423  0.1112  0.0826 
Upper  4.0765  3.6015  2.8003  10.3750  9.8971  2.4872  0.9442  0.8199  
2005  Lower  2.9927  2.3613  1.7135  0.9657  1.6922  1.5028  0.0716  0.3897 
Upper  4.2805  3.3727  2.4419  13.5219  5.1700  2.2546  0.8830  1.1491  
2010  Lower  3.0920  2.2879  1.5626  1.4042  2.5649  1.4537  0.4362  0.3037 
Upper  4.4272  3.2699  2.2262  5.5120  7.1611  2.1731  1.6256  1.0367 
Confidence intervals (95%) for the BG models (with beta and GB1 marginals) fitted to the variables income & health by maximum likelihood
Variables: education & health  

Year  Limit  BG model with beta marginals  BG model with GB1 marginals  
${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\widehat{\mathit{p}}}_{\mathbf{1}}$  ${\widehat{\mathit{p}}}_{\mathbf{2}}$  $\widehat{\mathit{q}}$  ${\mathit{\xe2}}_{\mathbf{1}}$  ${\mathit{\xe2}}_{\mathbf{2}}$  
1980  Lower  2.8717  4.9317  2.4348  3.0281  2.4498  2.2202  0.2187  0.6688 
Upper  4.0881  7.0329  3.4634  11.4549  6.6900  3.3394  0.7741  1.7170  
1985  Lower  3.2117  6.3844  2.7536  2.2422  3.4669  2.2074  0.0183  0.4486 
Upper  4.5673  9.0942  3.9136  42.1636  10.8125  3.3806  0.4019  1.3564  
1990  Lower  3.1360  6.6164  2.6126  1.9562  3.3433  1.9810  0.0159  0.2140 
Upper  4.4606  9.4282  3.7134  40.5326  17.0061  3.0476  0.3773  1.0560  
1995  Lower  2.8126  6.2095  2.3010  3.3971  2.6301  1.7632  0.0691  0.1176 
Upper  4.0038  8.8595  3.2724  21.4397  20.5273  2.7044  0.4223  0.9424  
2000  Lower  0.8471  4.5394  0.5105  56.4722  17.2919  0.4192  0.0219  0.1796 
Upper  3.6175  8.3509  2.8123  21.2033  12.0475  2.4010  0.4179  1.2257  
2005  Lower  2.2989  5.6232  1.6712  18.0358  1.5462  1.4024  0.1156  0.5471 
Upper  3.2793  8.0524  2.3788  57.9060  6.4866  2.1326  0.3802  2.2645  
2010  Lower  2.0008  5.3680  1.3822  1.8638  1.4273  1.2752  0.2287  0.4833 
Upper  2.8584  7.7098  1.9686  6.9524  7.0423  1.9130  0.8489  2.3539 
AIC statistics obtained by maximum likelihood for the BG models with beta (3 parameters) and GB1 marginals (5 parameters) fitted to pairs of the variables: Education, Health and Income
Education & health  Education & income  Income & health  

Year  Model (3 par)  Model (5 par)  Model (3 par)  Model (5 par)  Model (3 par)  Model (5 par) 
1980  292.88  352.59  174.12  180.00  279.48  288.67 
1985  309.52  354.03  206.89  217.35  332.58  354.54 
1990  304.40  340.63  213.99  228.01  344.03  380.03 
1995  291.33  313.41  206.65  217.94  332.39  370.94 
2000  291.77  296.28  207.64  217.52  325.00  354.22 
2005  296.81  306.00  210.98  213.52  323.60  340.72 
2010  330.82  327.52  215.63  213.91  317.16  318.24 
AIC statistics obtained by maximum likelihood for the BG model (BG truncated exponential maginals) with 5 parameters fitted to pairs of the variables: education, health and income
Education & health  Education & income  Income & health  

Year  Model (5 par)  Model (5 par)  Model (5 par) 
1980  333.49  188.88  286.86 
1985  342.58  223.42  350.60 
1990  323.42  233.30  375.72 
1995  300.82  230.08  373.12 
2000  295.86  233.28  363.92 
2005  303.35  233.61  364.73 
2010  332.99  237.16  359.70 
5 Conclusions
The main conclusions of this paper are the following. Three different classes of bivariate BG distributions have been presented. These classes have been constructed using three different definitions of bivariate beta distributions, proposed by Libby and Novick (1982), Jones (2001) and Olkin and Liu (2003) for the first proposal, ElBassiouny and Jones (2009) for the second proposal and Arnold and Ng (2011) for the third proposal. The main properties of these three classes have been studied. Three specific bivariate BG distributions have been obtained. Finally, an empirical application with wellbeing data has been presented.
The future research about bivariate BG distributions moves in three directions. The first line research is to propose specific models for their practical use in statistical modeling. The study of these possible models in any dimension could be an interesting field of research. Secondly, we propose to study statistical inference methodologies for bivariate (and, more generally, multivariate) BG distributions in (9) and (12). Finally, we propose to establish a model competition between BG distributions in (4), (9) and (12) for different choices of F_{1} and F_{2}.
Appendix
The joint PDF of the different classes of bivariate beta distribution
where $\mathcal{B}2(a,b)$ denotes beta distribution of the second kind.
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.
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/Healthindex/9v27i7ic, data on the education index can be drawn from https://data.undp.org/dataset/ExpectedYearsofSchoolingofchildrenyears/qnamf624 for the variable expected years of schooling and https://data.undp.org/dataset/Meanyearsofschoolingofadultsyears/m67kvi5c for the mean years of schooling. Finally, income data come from https://data.undp.org/dataset/Incomeindex/qt4gyea9.
Declarations
Acknowledgements
The authors thank to Ministerio de Economía y Competitividad (project ECO201015455) for partial support of this work. The authors thank also the comments by the attendants of the first ICOSDA meeting celebrated at Mount Pleasant, MI USA. We are grateful to the EditorsinChief and the reviewers for careful reading and for their comments and suggestions which greatly improved the paper.
Authors’ Affiliations
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