Family of bivariate distributions on the unit square: theoretical properties and applications

Abstract

We introduce the bivariate unit-log-symmetric model based on the bivariate log-symmetric distribution (BLS) defined in Vila et al. [25] as a flexible family of bivariate distributions over the unit square. We then study its mathematical properties such as stochastic representations, quantiles, conditional distributions, independence of the marginal distributions and marginal moments. Maximum likelihood estimation method is discussed and examined through Monte Carlo simulation. Finally, the proposed model is used to analyze some soccer data sets.

1. Introduction

Bivariate distributions over the unit-square have been discussed in detail in the literature; see, e.g. Barreto-Souza and Lemonte [4] and Özbilen and Genç [20]. Many of them are based on the beta distribution and its generalizations; see Arnold and Ng [2] and Nadarajah et al. [18]. Models of this type have been studied since the 1980s. Some other distributions on the unit square are based on generalized arcsine and inverse Gaussian distributions. A recent model, the bivariate unit-sinh-normal distribution, is based on the bivariate Birnbaum-Saunders distribution; see Martinez-Flóres et al. [16]. Bivariate distributions over the unit square arise naturally in comparing indices, rates or proportions in the interval $(0,1)$.

In this paper, we study the bivariate unit-log-symmetric (BULS) distribution defined over the unit-square, obtained as a modification of the bivariate log-symmetric (BLS) distribution introduced by Vila et al. [25]. The definitions of BLS and BULS distributions are given in Section 2, along with some special cases of the BULS. In Section 3, we discuss some properties of the new model, including a stochastic representation, marginal quantiles, and the conditional distributions of BULS. We derive compact formulas for the conditional densities, using the distribution functions of normal, Student-t, hyperbolic, Laplace and slash distributions. One of the uses of having closed formulas for the conditional densities (of the BULS model), for example, is in studying Heckman-type selection models (see Heckman [11]) when the selection variables have bounded support. In addition, we derive the distribution of the squared Mahalanobis distance of a random vector $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}$ with BULS distribution, and present a necessary condition for the independence of the components of $\mathit{W}$ and formulas for the moments of $W_1$ and $W_2$. In Section 4, the log-likelihood function and the likelihood equations for the BULS distribution are presented. In Section 5, we carry out a Monte Carlo simulation study to evaluate the performance of the ML estimators by means of their bias, root mean square error and coverage probability. In Section 6, we present two applications to soccer data. Specifically, in Section 6.1, we model the vector $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}$, where $W_1$ represents the time elapsed until a first kick goal (of any team) and the time elapsed until a goal of any type of the home team, and show that specific BULS distributions are suitable for modelling $\mathit{W}$. In Section 6.2, we consider the data of 2022 FIFA World Cup wherein the components of the vector $\mathit{W}$ represent the pass completion proportions of medium passes (14 to 18 meters) and long passes (longer than 37 meters). We then demonstrate that these data can also be fitted well by BULS distributions.

2. Bivariate unit-log-symmetric model

In this section, we describe the bivariate unit-log-symmetric model (BULS). To define this model, we first need to describe the bivariate log-symmetric distribution (BLS) defined in Vila et al. [25].

2.1. BLS family of distributions

Following Vila et al. [25], a continuous random vector $\mathit{T}=(T_1,T_2{)}^{\mathrm{\top }}$ is said to have a bivariate log-symmetric (BLS) distribution if its joint probability density function (PDF) is given by

$$f_{T_1,T_2}(t_1,t_2;\mathit{\theta })=\frac{1}{t_1{t}_2{\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2}{Z}_{g_c}}{g}_c\left(\frac{{\widetilde{t_1}}^2-2\rho \widetilde{t_1}\widetilde{t_2}+{\widetilde{t_2}}^2}{1-{\rho }^2}\right),t_1,t_2>0,$$ (1)

where $\widetilde{t_i}=\log (t_i/{\eta }_i{)}^{1/{\sigma }_i},{\eta }_i=\exp ({\mu }_i),i=1,2$, with $\mathit{\theta }=({\eta }_1,{\eta }_2,{\sigma }_1,{\sigma }_2,\rho {)}^{\mathrm{\top }}$ being the parameter vector, ${\mu }_i\in \mathbb{R}$, ${\sigma }_i>0$, i=1,2 and $\rho \in (-1,1)$. Furthermore, $Z_{g_c}>0$ is the partition function, that is,

$$Z_{g_c}={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\frac{1}{t_1{t}_2{\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2}}{g}_c\left(\frac{{\widetilde{t_1}}^2-2\rho \widetilde{t_1}\widetilde{t_2}+{\widetilde{t_2}}^2}{1-{\rho }^2}\right)\mathrm{d}{t}_1\mathrm{d}{t}_2=\pi {\int }_0^{\mathrm{\infty }}{g}_c(u)\mathrm{d}u,$$ (2)

and $g_c$ is a scalar function referred to as the density generator (see Fang et al. [8]). The second integral in (2) is a consequence of a change of variables; for more details, see Proposition 3.1 of Vila et al. [25]. When a random vector $\mathit{T}$ is BLS distributed, with parameter vector $\mathit{\theta }$, we denote it by $\mathit{T}\sim \mathrm{BLS}(\mathit{\theta },g_c)$.

2.2. BULS family of distributions

We say that a continuous random vector $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}$ has a bivariate unit-log-symmetric (BULS) distribution with parameter vector $\mathit{\theta }=({\eta }_1,{\eta }_2,{\sigma }_1,{\sigma }_2,\rho {)}^{\mathrm{\top }}$, denoted by $\mathit{W}\sim \mathrm{BULS}(\mathit{\theta },g_c)$, if its PDF is, for $0<w_1,w_2<1$, given by

$$f_{W_1,W_2}(w_1,w_2;\mathit{\theta })=\frac{1}{(1-w_1)t_1{\sigma }_1(1-w_2)t_2{\sigma }_2\sqrt{1-{\rho }^2}{Z}_{g_c}}{g}_c\left(\frac{{\tilde{w}}_1^2-2\rho {\tilde{w}}_1{\tilde{w}}_2+{\tilde{w}}_2^2}{1-{\rho }^2}\right),$$ (3)

where ${\tilde{w}}_i=\log (t_i/{\eta }_i{)}^{1/{\sigma }_i},t_i=-\log (1-w_i),{\eta }_i=\exp ({\mu }_i),i=1,2$, with ${\sigma }_i>0$, i=1,2, $\rho \in (-1,1)$, and $Z_{g_c}$ and $g_c$ are as given in (2). We shall prove later that the BULS PDF in (3) is obtained by taking $W_i=1-\exp (-T_i)$, i=1,2, with $(T_1,T_2{)}^{\mathrm{\top }}\sim \mathrm{BLS}(\mathit{\theta },g_c)$.

Table 1 presents some examples of bivariate unit-log-symmetric distributions.

Table 1.

Partition functions $(Z_{g_c})$ and density generators $(g_c)$ for some BULS distributions.

Distribution	$Z_{g_c}$	$g_c$	Parameter
Bivariate unit-log-normal	$2\pi$	$\exp (-x/2)$	–
Bivariate unit-log-Student-t	$\frac{\mathrm{\Gamma }(\nu /2)\mathrm{\nu \pi }}{\mathrm{\Gamma }((\nu +2)/2)}$	$(1+\frac{x}{\nu }{)}^{-(\nu +2)/2}$	$\nu >0$
Bivariate unit-log-hyperbolic	$\frac{2\pi (\nu +1)\exp (-\nu )}{{\nu }^2}$	$\exp (-\nu \sqrt{1+x})$	$\nu >0$
Bivariate unit-log-Laplace	π	$K_0(\sqrt{2x})$	–
Bivariate unit-log-slash	$\frac{\pi }{q}{2}^{\frac{2-q}{2}}$	$x^{-\frac{q+2}{2}}\gamma (\frac{q+2}{2},\frac{x}{2})$	q>0

In Table 1, $\mathrm{\Gamma }(t)={\int }_0^{\mathrm{\infty }}{x}^{t-1}\exp (-x)\mathrm{d}x$, t>0, is the complete gamma function, $K_{\lambda }(u)=(1/2)(u/2{)}^{\lambda }{\int }_0^{\mathrm{\infty }}{t}^{-\lambda -1}\exp (-t-u^2/4t)\mathrm{d}t$, u>0, is the modified Bessel function of the third kind with index λ (see Appendix of Kotz et al. [14]), and $\gamma (s,x)={\int }_0^x{t}^{s-1}\exp (-t)\mathrm{d}t$ is the lower incomplete gamma function.

Let $\mathit{T}=(T_1,T_2{)}^{\mathrm{\top }}\sim \mathrm{BLS}(\mathit{\theta },g_c)$. From (3), it is clear that the random vector $\mathit{X}=(X_1,X_2{)}^{\mathrm{\top }}$, with

$$X_i=\log (T_i)=\log [-\log (1-W_i)],i=1,2,$$ (4)

has a bivariate elliptically symmetric (BSY) distribution (see p. 592 in Balakrishnan and Lai [3]); that is, the PDF of $\mathit{X}$ is

$$f_{X_1,X_2}(x_1,x_2;{\mathit{\theta }}_{\ast })=\frac{1}{{\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2}{Z}_{g_c}}{g}_c\left(\frac{{\widetilde{x_1}}^2-2\rho \widetilde{x_1}\widetilde{x_2}+{\widetilde{x_2}}^2}{1-{\rho }^2}\right),-\mathrm{\infty }<x_1,x_2<\mathrm{\infty },$$ (5)

where $\widetilde{x_i}=(x_i-{\mu }_i)/{\sigma }_i,i=1,2$, with ${\mathit{\theta }}_{\ast }=({\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2,\rho )$ being the parameter vector and $Z_{g_c}$ is the partition function defined in (2). In this case, we shall use the notation $\mathit{X}\sim \mathrm{BSY}({\mathit{\theta }}_{\ast },g_c)$.

It is a simple task to observe that the joint cumulative distribution function (CDF) of $\mathit{W}\sim \mathrm{BULS}(\mathit{\theta },g_c)$, denoted by $F_{W_1,W_2}(w_1,w_2;\mathit{\theta })$, is given by

$$\begin{array}{rl}{F}_{W_1,W_2}(w_1,w_2;\mathit{\theta })& =F_{T_1,T_2}(-\log (1-w_1),-\log (1-w_2);\mathit{\theta })\\ & =F_{X_1,X_2}(\log [-\log (1-w_1)],\log [-\log (1-w_2)];{\mathit{\theta }}_{\ast }),\end{array}$$

wherein $F_{T_1,T_2}(t_1,t_2;\mathit{\theta })$ and $F_{X_1,X_2}(x_1,x_2;{\mathit{\theta }}_{\ast })$ denote the CDFs of $\mathit{T}\sim \mathrm{BLS}(\mathit{\theta },g_c)$ and $\mathit{X}\sim \mathrm{BES}({\mathit{\theta }}_{\ast },g_c)$, respectively.

Remark 2.1

As the components of $\mathit{W}$ are defined as increasing functions of the components of $\mathit{X}$, by invariance under monotone transformations property of copulas, both will have the same associated copula (for a review of copula theory, see, e.g. Nelsen [19]). Therefore, the analysis of dependence measures for the variables $W_1$ and $W_2$ that depend only on the copula is equivalent to the analysis of those for $X_1$ and $X_2$. For the normal copula and Student's t copula, some dependency measures such as Kendall's tau, Spearman's rho, Blest's measure of rank correlation, and coefficients of tail dependence were all studied by Roncalli [21].

3. Some basic properties of the model

In this section, some mathematical properties of the bivariate unit-log-symmetric distribution are established.

3.1. Stochastic representation

Proposition 3.1

The random vector $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}$ has a BULS distribution if

$$\begin{array}{l}{W}_1=1-\exp [-{\eta }_1\exp ({\sigma }_1{Z}_1)],\\ W_2=1-\exp [-{\eta }_2\exp ({\sigma }_2\rho Z_1+{\sigma }_2\sqrt{1-{\rho }^2}{Z}_2)],\end{array}$$

where $Z_1=\mathrm{RD}{U}_1$ and $Z_2=R\sqrt{1-D^2}{U}_2$ with $U_1$, $U_2$, R, and D being mutually independent random variables, $\rho \in (-1,1)$, ${\eta }_i=\exp ({\mu }_i)$, and $\mathbb{P}(U_i=-1)=\mathbb{P}(U_i=1)=1/2$, i=1,2. The random variable D is positive and has PDF $f_D(d)=2/(\pi \sqrt{1-d^2}),d\in (0,1).$ Further, the positive random variable R has its PDF as $f_R(r)=2r{g}_c(r^2)/{\int }_0^{\mathrm{\infty }}{g}_c(u)\mathrm{d}u,r>0.$

Proof.

It is well-known that (see Proposition 3.2 of Vila et al. [25]) the random vector $\mathit{T}=(T_1,T_2{)}^{\mathrm{\top }}$ has a BLS distribution if

$$\begin{array}{l}{T}_1={\eta }_1\exp ({\sigma }_1{Z}_1),\\ T_2={\eta }_2\exp ({\sigma }_2\rho Z_1+{\sigma }_2\sqrt{1-{\rho }^2}{Z}_2).\end{array}$$ (6)

Moreover, from (4), $W_i=1-\exp (-T_i)$, i=1,2. Hence, the result.

The following lemma provides a slight simplification in the representation of Proposition 3.1. This result plays a role in the next subsections, since all the probabilistic characteristics that depend on the distribution of $\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2$ will be simplified since it has the same distribution as $Z_2$.

Lemma 3.2

For a Borel subset B of $(-\mathrm{\infty },\mathrm{\infty })$, we have

$$\mathbb{P}(\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2\in B)=\mathbb{P}(Z_2\in B).$$

In other words, $\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2$ and $Z_2$ have the same distribution.

Proof.

It is clear that the density of $\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2$ is related to the joint density $f_{Z_1,Z_2}$ by

$$f_{\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2}(s_2)=\frac{1}{\sqrt{1-{\rho }^2}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{Z_1,Z_2}(z,\frac{s_2-\mathrm{\rho z}}{\sqrt{1-{\rho }^2}})\mathrm{d}z.$$ (7)

From Equation (13) of Saulo et al. [23], the joint PDF of $Z_1$ and $Z_2$ is given by

$$f_{Z_1,Z_2}(x,y)=\frac{1}{Z_{g_c}}{g}_c(x^2+y^2),-\mathrm{\infty }<x,y<\mathrm{\infty },$$ (8)

and so the integral in (7) is

$$=\frac{1}{\sqrt{1-{\rho }^2}{Z}_{g_c}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{g}_c(z^2+{\left(\frac{s_2-\mathrm{\rho z}}{\sqrt{1-{\rho }^2}}\right)}^2)\mathrm{d}z.$$ (9)

Using the identity

$$z^2+{\left(\frac{s_2-\mathrm{\rho z}}{\sqrt{1-{\rho }^2}}\right)}^2={\displaystyle \frac{z^2-2\mathrm{\rho z}{s}_2+s_2^2}{1-{\rho }^2}}={\left(\frac{z-\rho s_2}{\sqrt{1-{\rho }^2}}\right)}^2+s_2^2$$

the integral in (9) can be expressed as

$$\frac{1}{\sqrt{1-{\rho }^2}{Z}_{g_c}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{g}_c({\left(\frac{z-\rho s_2}{\sqrt{1-{\rho }^2}}\right)}^2+s_2^2)\mathrm{d}z.$$

Making the change of variable $s_1=(z-\rho s_2)/\sqrt{1-{\rho }^2}$, the above integral becomes

$$\frac{1}{Z_{g_c}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{g}_c(s_1^2+s_2^2)\mathrm{d}{s}_1={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{Z_1,Z_2}(s_1,s_2)\mathrm{d}{s}_1,$$

where, in the last line, we have used (8). Hence,

$$f_{\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2}(s_2)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{Z_1,Z_2}(s_1,s_2)\mathrm{d}{s}_1=f_{Z_2}(s_2).$$ (10)

Now, from (10), it is clear that $\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2$ and $Z_2$ are equal in distribution.

3.2. Marginal quantiles

Given $p\in (0,1)$, let $Q_{W_i}(p)$ be the p-quantile of $W_i$, for i=1,2. By using the stochastic representation in Proposition 3.1, for $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$, we have

$$\begin{array}{rl}p=\mathbb{P}(W_1⩽Q_{W_1}(p))& =\mathbb{P}(1-\exp [-{\eta }_1\exp ({\sigma }_1{Z}_1)]⩽Q_{W_1}(p))\\ & =\mathbb{P}(Z_1⩽\log {(-\frac{\log (1-Q_{W_1}(p))}{{\eta }_1})}^{1/{\sigma }_1})\end{array}$$

and

$$\begin{array}{rl}p=\mathbb{P}(W_2⩽Q_{W_2}(p))& =\mathbb{P}(1-\exp [-{\eta }_2\exp ({\sigma }_2\rho Z_1+{\sigma }_2\sqrt{1-{\rho }^2}{Z}_2)]⩽Q_{W_2}(p))\\ & =\mathbb{P}(\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2⩽\log {(-\frac{\log (1-Q_{W_2}(p))}{{\eta }_2})}^{1/{\sigma }_2}).\end{array}$$

Hence, the p-quantiles $Q_{Z_1}(p)$ and $Q_{Z_2}(p)$ of $Z_1$ and $Z_2$, respectively, are given by

$$\log {(-\frac{\log (1-Q_{W_1}(p))}{{\eta }_1})}^{1/{\sigma }_1}=Q_{Z_1}(p)$$

and

$$\log {(-\frac{\log (1-Q_{W_2}(p))}{{\eta }_2})}^{1/{\sigma }_2}=Q_{\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2}(p)=Q_{Z_2}(p),$$

where in the last equality we have used the fact that $\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2$ and $Z_2$ have the same distribution (see Lemma 3.2). Hence, the p-quantiles $Q_{W_1}(p)$ and $Q_{W_2}(p)$ are given by

$$\begin{array}{rl}{Q}_{W_1}(p)& =1-\exp [-{\eta }_1\exp ({\sigma }_1{Q}_{Z_1}(p))],\\ Q_{W_2}(p)& =1-\exp [-{\eta }_2\exp ({\sigma }_2{Q}_{Z_2}(p))],\end{array}$$

respectively.

3.3. Conditional distributions

Before enunciating and proving the main result (Theorem 3.4) of this subsection, we establish the following technical lemma which gets used subsequently.

Lemma 3.3

If $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$, then the PDF of $W_2|(W_1=w_1)$ is given by

$$f_{W_2}(w_2|W_1=w_1)=\frac{1}{(1-w_2)t_2{\sigma }_2\sqrt{1-{\rho }^2}}{f}_{Z_2}(\frac{1}{\sqrt{1-{\rho }^2}}({\tilde{w}}_2-\rho {\tilde{w}}_1)|Z_1={\tilde{w}}_1),$$ (11)

where ${\tilde{w}}_i$, i=1,2, and $t_2$ are as defined in (3), and $Z_1$ and $Z_2$ are as given in Proposition 3.1.

Proof.

If $W_1=w_1$, then $Z_1=\log [-\log (1-w_1)/{\eta }_1{]}^{1/{\sigma }_1}={\tilde{w}}_1$. So, the conditional distribution of $W_2$, given $W_1=w_1$, is the same as the distribution of

$$1-\exp [-{\eta }_2\exp ({\sigma }_2\rho {\tilde{w}}_1+{\sigma }_2\sqrt{1-{\rho }^2}{Z}_2)]|W_1=w_1.$$

Consequently,

$$\begin{array}{rl}& F_{W_2}(w_2|W_1=w_1)\\ & =\mathbb{P}(1-\exp [-{\eta }_2\exp ({\sigma }_2\rho {\tilde{w}}_1+{\sigma }_2\sqrt{1-{\rho }^2}{Z}_2)]⩽w_2|W_1=w_1)\\ & =\mathbb{P}(Z_2⩽\frac{1}{\sqrt{1-{\rho }^2}}({\tilde{w}}_2-\rho {\tilde{w}}_1)|Z_1={\tilde{w}}_1).\end{array}$$

Then, by differentiating $F_{W_2}(w_2|W_1=w_1)$ with respect to $w_2$, (11) is readily obtained.

The following result provides a simple formula for determining the conditional distribution of $W_1$, given $W_2\in B$, whenever the marginal and conditional distributions of $(Z_1,Z_2{)}^{\mathrm{\top }}$ are known. This result is essential for studying Heckman-type selection models (see Heckman [11]) when the selection variables have unitary support.

Theorem 3.4

For a Borel subset B of $(0,1)$, let us define the following Borel set:

$$B_r=\frac{1}{\sqrt{1-r^2}}\log {(-\frac{\log (1-B)}{{\eta }_2})}^{1/{\sigma }_2}-\frac{r}{\sqrt{1-r^2}}{\tilde{w}}_1,-1<r<1,$$ (12)

where ${\tilde{w}}_1$ is as in (3). If $\mathit{W}\sim \mathrm{BULS}(\mathit{\theta },g_c)$, then the PDF of $W_1|(W_2\in B)$ is given by

$$f_{W_1}(w_1|W_2\in B)=\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{Z_1}({\tilde{w}}_1)\frac{\mathbb{P}(Z_2\in B_{\rho }|Z_1={\tilde{w}}_1)}{\mathbb{P}(Z_2\in B_0)},$$

in which $t_1$ is as in (3), $B_r$ is as in (12), and $Z_1$ and $Z_2$ are as given in Proposition 3.1.

Proof.

Let B be a Borel subset of $(0,1)$. Note that

$$f_{W_1}(w_1|W_2\in B)=f_{W_1}(w_1)\frac{{\int }_B{f}_{W_2}(w_2|W_1=w_1)\mathrm{d}{w}_2}{\mathbb{P}(W_2\in B)}.$$

As $f_{W_1}(w_1)=f_{Z_1}({\tilde{w}}_1)/[(1-w_1)t_1{\sigma }_1]$ and $\mathbb{P}(W_2\in B)=\mathbb{P}(\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2\in B_0)$, where $B_0$ is as given in (12) with r=0, the term on the right-hand side of the above identity becomes

$$\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{Z_1}({\tilde{w}}_1)\frac{{\int }_B{f}_{W_2}(w_2|W_1=w_1)\mathrm{d}{w}_2}{\mathbb{P}(\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2\in B_0)}.$$

By using the formula for $f_{W_2}(w_2|W_1=w_1)$ provided in Lemma 3.3, the above expression becomes

$$\begin{array}{rl}& \frac{1}{(1-w_1)t_1{\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2}}{f}_{Z_1}({\tilde{w}}_1)\\ & \times \frac{{\int }_B\frac{1}{(1-w_2)t_2}{f}_{Z_2}(\frac{1}{\sqrt{1-{\rho }^2}}{\tilde{w}}_2-\frac{\rho }{\sqrt{1-{\rho }^2}}{\tilde{w}}_1|Z_1={\tilde{w}}_1)\mathrm{d}{w}_2}{\mathbb{P}(\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2\in B_0)},\end{array}$$

where ${\tilde{w}}_i$ and $t_i$, i=1,2, are as in (3). Finally, by applying the change of variable $z=({\tilde{w}}_2-\rho {\tilde{w}}_1)/\sqrt{1-{\rho }^2}$, the above expression is

$$=\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{Z_1}({\tilde{w}}_1)\frac{{\int }_{B_{\rho }}{f}_{Z_2}(z|Z_1={\tilde{w}}_1)\mathrm{d}z}{\mathbb{P}(\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2\in B_0)}.$$

We have thus proved that

$$f_{W_1}(w_1|W_2\in B)=\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{Z_1}({\tilde{w}}_1)\frac{{\int }_{B_{\rho }}{f}_{Z_2}(z|Z_1={\tilde{w}}_1)\mathrm{d}z}{\mathbb{P}(\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2\in B_0)}.$$

Finally, by combining the above identity with Lemma 3.2, the required result is obtained.

Using Theorem 3.4, for each generator ($g_c$) in Table 1, we present closed formulas for the conditional densities of $W_1|(W_2\in B)$ corresponding to bivariate unit-log-normal (Corollary 3.5), bivariate unit-log-Student-t (Corollary 3.6), bivariate unit-log-hyperbolic (Corollary 3.7), bivariate unit-log-Laplace (Corollary 3.8) and bivariate unit-log-slash (Corollary 3.9) distributions.

Corollary 3.5 Gaussian generator —

Let $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$ and $g_c(x)=\exp (-x/2)$ be the generator of the bivariate unit-log-normal distribution. Then, for each Borel subset B of $(0,1)$, the PDF of $W_1|(W_2\in B)$ is given by (for $0<w_1<1$)

$$f_{W_1}(w_1|W_2\in B)=\frac{1}{(1-w_1)t_1{\sigma }_1}\varphi ({\tilde{w}}_1){\displaystyle \frac{\mathrm{\Phi }(B_{\rho })}{\mathrm{\Phi }(B_0)}},$$

where $\mathrm{\Phi }(C)={\int }_C\varphi (x)\mathrm{d}x$ and $\varphi (x)$ is the standard normal PDF. Further, ${\tilde{w}}_1$ and $t_1$ are as in (3), and $B_r$ is as in (12).

Proof.

It is well-known that the bivariate log-normal distribution has a stochastic representation as in (6), where $Z_1\sim N(0,1)$ and $Z_2\sim N(0,1)$, and $Z_2|(Z_1=x)\sim N(0,1)$ (see Abdous [1]). Hence, $\mathbb{P}(Z_2\in B_0)=\mathrm{\Phi }(B_0)$ and $\mathbb{P}(Z_2\in B_{\rho }|Z_1={\tilde{w}}_1)=\mathrm{\Phi }(B_{\rho }).$ Then, by applying Theorem 3.4, the required result follows.

Corollary 3.6 Student-t generator —

Let $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$ and $g_c(x)=(1+(x/\nu ){)}^{-(\nu +2)/2}$, $\nu >0$, be the generator of the bivariate unit-log-Student-t distribution with ν degrees of freedom. Then, for each Borel subset B of $(0,1)$, the PDF of $W_1|(W_2\in B)$ is given by (for $0<w_1<1$)

$$f_{W_1}(w_1|W_2\in B)=\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{\nu }({\tilde{w}}_1){\displaystyle \frac{F_{\nu +1}\left(\sqrt{\frac{\nu +1}{\nu +{\tilde{w}}_1^2}}{B}_{\rho }\right)}{F_{\nu }(B_0)}},$$

where $F_{\nu }(C)={\int }_C{f}_{\nu }(x)\mathrm{d}x$ and $f_{\nu }(x)$ is the standard Student-t PDF with ν degrees of freedom.

Proof.

It is well-known that the bivariate log-Student-t distribution has a stochastic representation as in (6), where $Z_1\sim t_{\nu }$ and $Z_2\sim t_{\nu }$ (Student-t with ν degrees of freedom), and (see Corollary 3.7 of Vila et al. [25])

$$Z_2|(Z_1=x)\sim \sqrt{\frac{\nu +x^2}{\nu +1}}{t}_{\nu +1}.$$

Hence, $\mathbb{P}(Z_2\in B_0)=F_{\nu }(B_0)$ and

$$\mathbb{P}(Z_2\in B_{\rho }|Z_1={\tilde{w}}_1)=F_{\nu +1}\left(\sqrt{\frac{\nu +1}{\nu +{\tilde{w}}_1^2}}{B}_{\rho }\right).$$

By applying Theorem 3.4, the required result follows.

Corollary 3.7 Hyperbolic generator —

Let $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$ and $g_c(x)=\exp (-\nu \sqrt{1+x})$ be the generator of the bivariate unit-log-hyperbolic distribution. Then, for each Borel subset B of $(0,1)$, the PDF of $W_1|(W_2\in B)$ is given by (for $0<w_1<1$)

$$f_{W_1}(w_1|W_2\in B)=\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{\mathrm{GH}}({\tilde{w}}_1;3/2,\nu ,1){\displaystyle \frac{F_{\mathrm{GH}}(B_{\rho };1,\nu ,\sqrt{1+{\tilde{w}}_1^2})}{F_{\mathrm{GH}}(B_0;3/2,\nu ,1)}},$$

where $F_{\mathrm{GH}}(C;\lambda ,\alpha ,\delta )={\int }_C{f}_{\mathrm{GH}}(x;\lambda ,\alpha ,\delta )\mathrm{d}x$ and $f_{\mathrm{GH}}(x;\lambda ,\alpha ,\delta )$ is the generalized hyperbolic (GH) PDF (see Definition A.1 in the Appendix).

Proof.

It is well-known that the bivariate log-hyperbolic distribution has a stochastic representation as in (6), where $Z_1\sim \mathrm{GH}(3/2,\nu ,1)$ and $Z_2\sim \mathrm{GH}(3/2,\nu ,1)$ (see Subsection 2.1 of Deng and Yao [7]). Moreover, the distribution of $Z_2$, given $Z_1=x$, is $\mathrm{GH}(1/2,\sqrt{2},|x|)$ (Proposition A.2). Then, $\mathbb{P}(Z_2\in B_0)=F_{\mathrm{GH}}(B_0;3/2,\nu ,1)$ and $\mathbb{P}(Z_2\in B_{\rho }|Z_1={\tilde{w}}_1)=F_{\mathrm{GH}}(B_{\rho };1,\nu ,\sqrt{1+{\tilde{w}}_1^2})$. By applying Theorem 3.4, the required result follows.

Corollary 3.8 Laplace generator —

Let $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$ and $g_c(x)=K_0(\sqrt{2x})$ be the generator of the bivariate unit-log-Laplace distribution. Then, for each Borel subset B of $(0,1)$, the PDF of $W_1|(W_2\in B)$ is given by (for $0<w_1<1$)

$$f_{W_1}(w_1|W_2\in B)=\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{\mathrm{L}}({\tilde{w}}_1){\displaystyle \frac{F_{\mathrm{GH}}(B_{\rho };\frac{1}{2},\sqrt{2},|{\tilde{w}}_1|)}{F_{\mathrm{L}}(B_0)}},$$

where $F_{\mathrm{L}}(C)={\int }_C{f}_{\mathrm{L}}(x)\mathrm{d}x$ and $f_{\mathrm{L}}(x)=\exp (-\sqrt{2}|x|)/\sqrt{2}$ is the Laplace PDF with scale parameter $1/\sqrt{2}$, and $F_{\mathrm{GH}}$ is as defined in Corollary 3.7.

Proof.

It is well-known that the bivariate log-Laplace distribution has a stochastic representation as in (6), where $Z_1\sim \mathrm{Laplace}(0,1/\sqrt{2})$ and $Z_2\sim \mathrm{Laplace}(0,1/\sqrt{2})$ (see Subsection 5.1.4 of Kotz et al. [14]). Further, the distribution of $Z_2$, given $Z_1=x$, is $\mathrm{GH}(1/2,\sqrt{2},|x|)$ (Proposition A.3). Hence, $\mathbb{P}(Z_2\in B_0)=F_{\mathrm{L}}(B_0)$ and $\mathbb{P}(Z_2\in B_{\rho }|Z_1={\tilde{w}}_1)=F_{\mathrm{GH}}(B_{\rho };1/2,\sqrt{2},|{\tilde{w}}_1|).$ By applying Theorem 3.4, the required result follows.

Corollary 3.9 Slash generator —

Let $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$ and $g_c(x)=x^{-(q+2)/2}\gamma ((q+2)/2,x/2)$, be the generator of the bivariate unit-log-slash distribution. Then, for each Borel subset B of $(0,1)$, the PDF of $W_1|(W_2\in B)$ is given by (for $0<w_1<1$)

$$f_{W_1}(w_1|W_2\in B)=\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{\mathrm{SL}}({\tilde{w}}_1;q){\displaystyle \frac{F_{\mathrm{ESL}}(B_{\rho };{\tilde{w}}_1,q+1)}{F_{\mathrm{SL}}(B_0;q)}},$$

where $F_{\mathrm{SL}}(C;q)={\int }_C{f}_{\mathrm{SL}}(x;q)\mathrm{d}x$ and $f_{\mathrm{SL}}(x;q)=q{\int }_0^1{t}^q\varphi (\mathrm{tx})\mathrm{d}t$ is the classical slash PDF, and $F_{\mathrm{ESL}}(C;a,q)={\int }_C{f}_{\mathrm{ESL}}(x;a,q)\mathrm{d}x$, where $f_{\mathrm{ESL}}(x;a,q)$ is the generalized hyperbolic (ESL) PDF (see Definition A.4 in the Appendix).

Proof.

It is well-known that the bivariate log-slash distribution has a stochastic representation as in (6), where $Z_1\sim \mathrm{SL}(q)$ and $Z_2\sim \mathrm{SL}(q)$ (see Section 2 of Wang and Genton [26]). Moreover, the distribution of $Z_2$, given $Z_1=x$, is $\mathrm{ESL}(x,q+1)$ (Proposition A.5). Hence, $\mathbb{P}(Z_2\in B_0)=F_{\mathrm{SL}}(B_0;q)$ and $\mathbb{P}(Z_2\in B_{\rho }|Z_1={\tilde{w}}_1)=F_{\mathrm{ESL}}(B_{\rho };{\tilde{w}}_1,q+1).$ By applying Theorem 3.4, the required result follows.

Table 2 below presents some examples of conditional PDFs corresponding to all the bivariate unit-log-symmetric distributions presented earlier in Table 1.

Table 2.

Conditional densities of $W_1|(W_2\in B)$ and density generators $(g_c)$ for some BULS distributions.

Distribution	$g_c$	$f_{W_1}(w_1|W_2\in B)$
Bivariate unit-log-normal	$\exp (-x/2)$	$\frac{1}{(1-w_1)t_1{\sigma }_1}\varphi ({\tilde{w}}_1)\frac{\mathrm{\Phi }(B_{\rho })}{\mathrm{\Phi }(B_0)}$
Bivariate unit-log-Student-t	$(1+\frac{x}{\nu }{)}^{-(\nu +2)/2}$	$\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{\nu }({\tilde{w}}_1)\frac{F_{\nu +1}(\sqrt{\frac{\nu +1}{\nu +{\tilde{w}}_1^2}}{B}_{\rho })}{F_{\nu }(B_0)}$
Bivariate unit-log-hyperbolic	$\exp (-\nu \sqrt{1+x})$	$\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{\mathrm{GH}}({\tilde{w}}_1;3/2,\nu ,1)\frac{F_{\mathrm{GH}}(B_{\rho };1,\nu ,\sqrt{1+{\tilde{w}}_1^2})}{F_{\mathrm{GH}}(B_0;3/2,\nu ,1)}$
Bivariate unit-log-Laplace	$K_0(\sqrt{2x})$	$\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{\mathrm{L}}({\tilde{w}}_1)\frac{F_{\mathrm{GH}}(B_{\rho };\frac{1}{2},\sqrt{2},|{\tilde{w}}_1|)}{F_{\mathrm{L}}(B_0)}$
Bivariate unit-log-slash	$x^{-\frac{q+2}{2}}\gamma (\frac{q+2}{2},\frac{x}{2})$	$\frac{1}{(1-w_1)t_1{\sigma }_1}{f}_{\mathrm{SL}}({\tilde{w}}_1;q)\frac{F_{\mathrm{ESL}}(B_{\rho };{\tilde{w}}_1,q+1)}{F_{\mathrm{SL}}(B_0;q)}$

3.4. Independence

Proposition 3.10

Let $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$. If $\rho =0$ and the density generator $g_c$ in (3) is such that

$$g_c(x^2+y^2)=g_{c_1}(x^2)g_{c_2}(y^2),\mathrm{\forall }(x,y)\in {\mathbb{R}}^2,$$ (13)

for some density generators $g_{c_1}$ and $g_{c_2}$, then $W_1$ and $W_2$ are independent.

Proof.

The proof follows the same steps as the proof of Proposition 3.11 of Vila et al. [25]. For the sake of completeness, however, we present it here.

Let $\rho =0$. From (13), the joint density (3) of $(W_1,W_2)$ is such that

$$f_{W_1,W_2}(w_1,w_2;\mathit{\theta })=\frac{Z_{g_{c_1}}{Z}_{g_{c_2}}}{Z_{g_c}}{f}_1(w_1;{\mu }_1,{\sigma }_1)f_2(w_2;{\mu }_2,{\sigma }_2),\mathrm{\forall }(w_1,w_2)\in (0,1)\times (0,1),$$ (14)

where $f_i(w_i;{\mu }_i,{\sigma }_i)=g_{c_i}({\widetilde{w_i}}^2)/[(1-w_i)t_i{\sigma }_i{Z}_{g_{c_i}}],0<w_i<1$, $Z_{g_{c_i}}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{g}_{c_i}({z_i}^2)\mathrm{d}{z}_i,i=1,2,$ and $\widetilde{w_i}$ and $t_i$ are as in (3). Integrating (14) in terms of $w_1$ and $w_2$, we obtain

$$\frac{Z_{g_{c_1}}{Z}_{g_{c_2}}}{Z_{g_c}}=1,$$

and consequently, $Z_{g_c}=Z_{g_{c_1}}{Z}_{g_{c_2}}$. Therefore,

$$f_{W_1,W_2}(w_1,w_2;\mathit{\theta })=f_1(w_1;{\mu }_1,{\sigma }_1)f_2(w_2;{\mu }_2,{\sigma }_2),\mathrm{\forall }(w_1,w_2)\in (0,1)\times (0,1).$$

Moreover, it is easy to verify that $f_1$ and $f_2$ are PDFs corresponding to univariate symmetric random variables (see Vanegas and Paula [24]). Then, $W_1$ and $W_2$ are statistically independent, and even more, $f_i=f_{W_i}$, for i=1,2 (see Proposition 2.5 of James [13]).

Remark 3.11

In Table 1, the density generator of the bivariate unit-log-normal is the unique one that satisfies (13).

3.5. Marginal moments and correlation function

For $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$, $0<W_i<1$, it is clear that $0⩽\mathbb{E}(W_i^r)⩽1$, for any r>0 and i=1,2. Therefore, the positive moments of $W_i$ always exist.

In general, for any $r\in \mathbb{R}$, the moments of $W_i$, i=1,2, admit the following representations:

$$\begin{array}{rl}{\mu }_1(r)\equiv \mathbb{E}(W_1^r)& ={\displaystyle \mathbb{E}{\{1-\exp [-{\eta }_1\exp ({\sigma }_1{Z}_1)]\}}^r,}\\ {\mu }_2(r)\equiv \mathbb{E}(W_2^r)& ={\displaystyle \mathbb{E}(1-\exp \{-{\eta }_2\exp ({\sigma }_2[\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2])\}{)}^r}\\ & ={\displaystyle \mathbb{E}{\{1-\exp [-{\eta }_2\exp ({\sigma }_2{Z}_2)]\}}^r,}\end{array}$$ (15)

where in the last equality we have used the fact that $\rho Z_1+\sqrt{1-{\rho }^2}{Z}_2$ and $Z_2$ have the same distribution (see Lemma 3.2). Here, $Z_1$ and $Z_2$ are as given in Proposition 3.1.

Closed-forms expressions for the product moments can be derived as follows. Law of total expectation gives

$${\mu }_{1,2}\equiv \mathbb{E}(W_1{W}_2)={\int }_0^1{w}_1[{\int }_0^1{w}_2{f}_{W_2}(w_2|W_1=w_1)\mathrm{d}{w}_2]f_{W_1}(w_1)\mathrm{d}{w}_1.$$ (16)

Lemma 3.3 provides a formula for the PDF of $W_2|(W_1=w_1)$. Moreover, $f_{W_1}(w_1)=f_{Z_1}({\tilde{w}}_1)/[(1-w_1)t_1{\sigma }_1]$. By using these formulas in (16), the product moments ${\mu }_{1,2}$ is equal to

$$\begin{array}{rl}& \frac{1}{{\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2}}{\int }_0^1\frac{w_1}{(1-w_1)t_1}[{\int }_0^1\frac{w_2}{(1-w_2)t_2}\\ & f_{Z_2}(\frac{1}{\sqrt{1-{\rho }^2}}({\tilde{w}}_2-\rho {\tilde{w}}_1)|Z_1={\tilde{w}}_1)\mathrm{d}{w}_2]f_{Z_1}({\tilde{w}}_1)\mathrm{d}{w}_1,\end{array}$$

where ${\tilde{w}}_i$ and $t_i$, for i=1,2, are as defined in (3).

The conditional and unconditional distributions of vector $(Z_1,Z_2{)}^{\mathrm{\top }}$ corresponding to bivariate models of Table 1 are well-known (see proofs of Corollaries 3.5–3.9). Therefore, at least numerically, the respective product moments can be determined.

For illustrative purposes, we now present the product moments and correlation function formula only for the bivariate unit-log-normal model. In this case, $Z_1\sim N(0,1)$, $Z_2\sim N(0,1)$ and $Z_2|(Z_1=x)\sim N(0,1)$ (see proof of Corollary 3.5). So, using the last formula above for ${\mu }_{1,2}$, we have

$$\begin{array}{rl}{\mu }_{1,2}& =\frac{1}{{\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2}}{\int }_0^1\frac{w_1}{(1-w_1)t_1}\left[{\int }_0^1\frac{w_2}{(1-w_2)t_2}\varphi (\frac{1}{\sqrt{1-{\rho }^2}}({\tilde{w}}_2-\rho {\tilde{w}}_1))\mathrm{d}{w}_2\right]\\ & \varphi ({\tilde{w}}_1)\mathrm{d}{w}_1.\end{array}$$

Consequently, the correlation function between $W_1$ and $W_2$, denoted by ${\rho }_{\mathit{W}}$, can be expressed as

$${\rho }_{\mathit{W}}=\frac{\begin{array}{c}\frac{1}{{\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2}}{\int }_0^1\frac{w_1}{(1-w_1)t_1}\left[{\int }_0^1\frac{w_2}{(1-w_2)t_2}\varphi (\frac{1}{\sqrt{1-{\rho }^2}}({\tilde{w}}_2-\rho {\tilde{w}}_1))\mathrm{d}{w}_2\right]\varphi ({\tilde{w}}_1)\mathrm{d}{w}_1\\ -{\mu }_1(1){\mu }_2(1)\end{array}}{\sqrt{[{\mu }_1(2)-{\mu }_1^2(1)][{\mu }_2(2)-{\mu }_2^2(1)]}},$$ (17)

where ${\mu }_1(r)$ and ${\mu }_2(r)$ are as given in (15), and ${\tilde{w}}_i$ and $t_i$, for i=1,2, are as defined in (3).

Table 3 shows some values of the correlation function in (17) for different choices of ρ in the special case when ${\mu }_1={\mu }_2=0$ and ${\sigma }_1={\sigma }_2=1$.

Table 3.

Correlation values for different choices of ρ.

ρ	−0.9000	−0.7500	−0.5000	−0.2500	−0.1000	0.0000	0.1000	0.2500	0.5000	0.7500	0.9000
${\rho }_{\mathit{W}}$	−0.9413	−0.8406	−0.6678	−0.5275	−0.4481	−0.3929	−0.2880	−0.1463	0.1683	0.5339	0.7648

Remark 3.12

Given the first two moments of BULS model, some useful bounds for ${\rho }_{\mathit{W}}$ can be provided. For example,

1. Theorem 1 of Hössjer and Sjölander [12] gives

   $$\begin{array}{rl}& -\frac{min\{{\mu }_1(1){\mu }_2(1),[1-{\mu }_1(1)][1-{\mu }_2(1)]\}}{\sqrt{[{\mu }_1(2)-{\mu }_1^2(1)][{\mu }_2(2)-{\mu }_2^2(1)]}}⩽{\rho }_{\mathit{W}}\\ & ⩽\frac{min\{{\mu }_1(1)[1-{\mu }_2(1)],[1-{\mu }_1(1)]{\mu }_2(1)\}}{\sqrt{[{\mu }_1(2)-{\mu }_1^2(1)][{\mu }_2(2)-{\mu }_2^2(1)]}};\end{array}$$
2. Corollary 3 of Hössjer and Sjölander [12] provides

   $$\begin{array}{rl}& -\frac{1}{4\sqrt{[{\mu }_1(2)-{\mu }_1^2(1)][{\mu }_2(2)-{\mu }_2^2(1)]}}⩽{\rho }_{\mathit{W}}\\ & ⩽\frac{1}{4\sqrt{[{\mu }_1(2)-{\mu }_1^2(1)][{\mu }_2(2)-{\mu }_2^2(1)]}}.\end{array}$$

3.6. Squared mahalanobis distance

Given a probability distribution F on ${\mathbb{R}}^2$ with mean $\mathit{\mu }=({\mu }_1,{\mu }_2{)}^{\mathrm{\top }}$, ${\mu }_i\in \mathbb{R}$, and positive-definite covariance matrix $\mathbf{\Sigma }$, and given two points $\mathit{x}$ and $\mathit{y}$ in ${\mathbb{R}}^2$, the squared Mahalanobis distance between them with respect to F is

$$d^2(\mathit{x},\mathit{y})\equiv d^2(\mathit{x},\mathit{y};F)=(\mathit{x}-\mathit{y}{)}^{\mathrm{\top }}{\mathbf{\Sigma }}^{-1}(\mathit{x}-\mathit{y}).$$

Let $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$ with location (mean) vector ${\mathit{\mu }}_{\mathit{W}}=({\mu }_1(1),{\mu }_2(1))$ and covariance matrix

$$\begin{array}{r}{\mathbf{\Sigma }}_{\mathit{W}}=\left(\begin{array}{cc}{\sigma }_{W_1}^2& {\rho }_{\mathit{W}}{\sigma }_{W_1}{\sigma }_{W_2}\\ {\rho }_{\mathit{W}}{\sigma }_{W_1}{\sigma }_{W_2}& {\sigma }_{W_2}^2\end{array}\right),\end{array}$$

where ${\sigma }_{W_1}^2\equiv {\mu }_1(2)-{\mu }_1^2(1)$, ${\sigma }_{W_2}^2\equiv {\mu }_2(2)-{\mu }_2^2(1)$, ${\mu }_1(r)$ and ${\mu }_2(r)$ are the moments given in (15), and ${\rho }_{\mathit{W}}$ is as given in (17). The (random) squared Mahalanobis distance of $\mathit{W}$ from ${\mathit{\mu }}_{\mathit{W}}$ is then

$$\begin{array}{rl}{d}^2\equiv d^2(\mathit{W},{\mathit{\mu }}_{\mathit{W}})& =(\mathit{W}-{\mathit{\mu }}_{\mathit{W}}{)}^{\mathrm{\top }}{\mathbf{\Sigma }}_{\mathit{W}}^{-1}(\mathit{W}-{\mathit{\mu }}_{\mathit{W}})\\ & =\frac{1}{1-{\rho }_{\mathit{W}}^2}[{\left(\frac{W_1-{\mu }_1(1)}{{\sigma }_{W_1}}\right)}^2-2{\rho }_{\mathit{W}}\left(\frac{W_1-{\mu }_1(1)}{{\sigma }_{W_1}}\right)\left(\frac{W_2-{\mu }_2(1)}{{\sigma }_{W_2}}\right)\\ & +{\left(\frac{W_2-{\mu }_2(1)}{{\sigma }_{W_2}}\right)}^2].\end{array}$$

The following two results provide formulas for the PDF and CDF of $d^2$. The respective proofs can be found in the Appendix (Section 1).

Proposition 3.13

If $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$, then the CDF of $d^2\equiv d^2(\mathit{W},{\mathit{\mu }}_{\mathit{W}})$, denoted by $F_{d^2}$, is given by

$$\begin{array}{rl}{F}_{d^2}(w)& =\frac{k}{Z_{g_c}}{\int }_{-\sqrt{w}}^{\sqrt{w}}[{\int }_{-\sqrt{w-s_1^2}}^{\sqrt{w-s_1^2}}\frac{1}{(1-w_{1,s})t_{1,s}(1-w_{2,s})t_{2,s}}\\ & g_c({\tilde{w}}_{1,s}^2+\frac{1}{1-{\rho }^2}({\tilde{w}}_{2,s}-\rho {\tilde{w}}_{1,s}{)}^2)\mathrm{d}{s}_2]\mathrm{d}{s}_1,\end{array}$$

with $Z_{g_c}$ being as in (5), $k={\sigma }_{W_1}{\sigma }_{W_2}\sqrt{1-{\rho }_{\mathit{W}}^2}/({\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2})$, and

$$\begin{array}{rl}& w_{1,s}={\mu }_1(1)+{\sigma }_{W_1}{s}_1,w_{2,s}={\mu }_2(1)+{\sigma }_{W_2}[{\rho }_{\mathit{W}}{s}_1+\sqrt{1-{\rho }_{\mathit{W}}^2}{s}_2],\\ & {\tilde{w}}_{i,s}=\log (t_{i,s}/{\eta }_i{)}^{1/{\sigma }_i},{\eta }_i=\exp ({\mu }_i),t_{i,s}=-\log (1-w_{i,s}),i=1,2.\end{array}$$ (18)

Proposition 3.14

If $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}\sim \mathrm{BULS}(\mathit{\theta },g_c)$, then the PDF of $d^2\equiv d^2(\mathit{W},{\mathit{\mu }}_{\mathit{W}})$, denoted by $f_{d^2}$, is given by

$$\begin{array}{rl}{f}_{d^2}(w)& =\frac{k}{2Z_{g_c}}{\int }_{-\sqrt{w}}^{\sqrt{w}}\frac{1/\sqrt{w-s_1^2}}{(1-w_{1,s})t_{1,s}(1-x_{s_1})[-\log (1-x_{s_1})]}\\ & g_c({\tilde{w}}_{1,s}^2+\frac{1}{1-{\rho }^2}{[{\log }^{1/{\sigma }_2}(-\frac{\log (1-x_{s_1})}{{\eta }_2})-\rho {\tilde{w}}_{1,s}]}^2)\mathrm{d}{s}_1,\end{array}$$

where $x_{s_1}={\mu }_2(1)+{\sigma }_{W_2}[{\rho }_{\mathit{W}}{s}_1+\sqrt{1-{\rho }_{\mathit{W}}^2}\sqrt{w-s_1^2}]$, $Z_{g_c}$ is as given in (5), $k={\sigma }_{W_1}{\sigma }_{W_2}\sqrt{1-{\rho }_{\mathit{W}}^2}/({\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2})$, and ${\tilde{w}}_{1,s}$ is as in (18).

4. Maximum likelihood estimation of model parameters

Let $\{(W_{1i},W_{2i}{)}^{\mathrm{\top }}:i=1,\dots ,n\}$ be a bivariate random sample of size n from the $\mathrm{BULS}(\mathit{\theta },g_c)$ distribution with PDF as in (3), and let $(w_{1i},w_{2i}{)}^{\mathrm{\top }}$ be the corresponding observations of $(W_{1i},W_{2i}{)}^{\mathrm{\top }}$. Then, the log-likelihood function for $\mathit{\theta }=({\eta }_1,{\eta }_2,{\sigma }_1,{\sigma }_2,\rho {)}^{\mathrm{\top }}$, without the additive constant, is given by

$$\begin{array}{rl}\ell (\mathit{\theta })& =-n\sum _{i=1}^2\log ({\sigma }_i)-\frac{n}{2}\log (1-{\rho }^2)\\ & +\sum _{i=1}^n\log g_c\left(\frac{{\tilde{w}}_{1i}^2-2\rho {\tilde{w}}_{1i}{\tilde{w}}_{2i}+{\tilde{w}}_{2i}^2}{1-{\rho }^2}\right),0<w_{1i},w_{2i}<1,\end{array}$$

where ${\tilde{w}}_{\mathrm{ki}}=\log (t_{\mathrm{ki}}/{\eta }_k{)}^{1/{\sigma }_k},t_{\mathrm{ki}}=-\log (1-w_{\mathrm{ki}})>0$ and ${\eta }_k=\exp ({\mu }_k),k=1,2;i=1,\dots ,n.$

In the case when a supremum $\hat{\mathit{\theta }}=(\widehat{{\eta }_1},\widehat{{\eta }_2},\widehat{{\sigma }_1},\widehat{{\sigma }_2},\hat{\rho }{)}^{\mathrm{\top }}$ exists, it must satisfy the following likelihood equations:

$$\begin{array}{rl}& \frac{\mathrm{\partial }\ell (\mathit{\theta })}{\mathrm{\partial }{\eta }_1}{|}_{\mathit{\theta }=\hat{\mathit{\theta }}}=0,\frac{\mathrm{\partial }\ell (\mathit{\theta })}{\mathrm{\partial }{\eta }_2}=0,\mathrm{\partial }\ell (\mathit{\theta })\mathrm{\partial }{\sigma }_1{|}_{\mathit{\theta }=\hat{\mathit{\theta }}}=0,\mathrm{\partial \ell }(\mathit{\theta })\mathrm{\partial }{\sigma }_2{|}_{\mathit{\theta }=\hat{\mathit{\theta }}}=0,\\ & \mathrm{\partial \ell }(\mathit{\theta })\mathrm{\partial \rho }{|}_{\mathit{\theta }=\hat{\mathit{\theta }}}=0,\end{array}$$ (19)

with

$$\begin{array}{rl}& \frac{\mathrm{\partial }\ell (\mathit{\theta })}{\mathrm{\partial }{\eta }_1}=\frac{2}{{\sigma }_1{\eta }_1(1-{\rho }^2)}\sum _{i=1}^n(\rho {\tilde{w}}_{2i}-{\tilde{w}}_{1i})G({\tilde{w}}_{1i},{\tilde{w}}_{2i}),\\ & \mathrm{\partial \ell }(\mathit{\theta })\mathrm{\partial }{\eta }_2=\frac{2}{{\sigma }_2{\eta }_2(1-{\rho }^2)}\sum _{i=1}^n(\rho {\tilde{w}}_{1i}-{\tilde{w}}_{2i})G({\tilde{w}}_{1i},{\tilde{w}}_{2i}),\\ & \mathrm{\partial \ell }(\mathit{\theta })\mathrm{\partial }{\sigma }_1=-\frac{n}{{\sigma }_1}+2{\sigma }_1(1-{\rho }^2)\sum _{i=1}^n{\tilde{w}}_{1i}(\rho {\tilde{w}}_{2i}-{\tilde{w}}_{1i})G({\tilde{w}}_{1i},{\tilde{w}}_{2i}),\\ & \mathrm{\partial \ell }(\mathit{\theta })\mathrm{\partial }{\sigma }_2=-\frac{n}{{\sigma }_2}+2{\sigma }_2(1-{\rho }^2)\sum _{i=1}^n{\tilde{w}}_{2i}(\rho {\tilde{w}}_{1i}-{\tilde{w}}_{2i})G({\tilde{w}}_{1i},{\tilde{w}}_{2i}),\\ & \mathrm{\partial \ell }(\mathit{\theta })\mathrm{\partial \rho }=\mathrm{n\rho }1-{\rho }^2-2(1-{\rho }^2{)}^2\sum _{i=1}^n(\rho {\tilde{w}}_{1i}-{\tilde{w}}_{2i})(\rho {\tilde{w}}_{2i}-{\tilde{w}}_{1i})G({\tilde{w}}_{1i},{\tilde{w}}_{2i}),\end{array}$$ (20)

where we have used the notation

$$G({\tilde{w}}_{1i},{\tilde{w}}_{2i})=g_c^{\prime }(x_{\rho ,i})g_c(x_{\rho ,i}),$$ (21)

with $x_{\rho ,i}=({\tilde{w}}_{1i}^2-2\rho {\tilde{w}}_{1i}{\tilde{w}}_{2i}+{\tilde{w}}_{2i}^2)/(1-{\rho }^2),i=1,\dots ,n.$

Observe that the likelihood equations in (19) can be written as

$$\begin{array}{rl}& \sum _{i=1}^n{\tilde{w}}_{1i}G({\tilde{w}}_{1i},{\tilde{w}}_{2i}){|}_{\mathit{\theta }=\hat{\mathit{\theta }}}=0,\\ & \sum _{i=1}^n({\tilde{w}}_{1i}^2-{\tilde{w}}_{2i}^2)G({\tilde{w}}_{1i},{\tilde{w}}_{2i}){|}_{\mathit{\theta }=\hat{\mathit{\theta }}}=0,\\ & \sum _{i=1}^n{\tilde{w}}_{2i}[2\rho {\tilde{w}}_{2i}-(1+{\rho }^2){\tilde{w}}_{1i}]G({\tilde{w}}_{1i},{\tilde{w}}_{2i}){|}_{\mathit{\theta }=\hat{\mathit{\theta }}}=-\frac{n\hat{\rho }(1-{\hat{\rho }}^2)}{2}.\end{array}$$

Any nontrivial root $\hat{\mathit{\theta }}$ of the above likelihood equations is an ML estimator in the loose sense. When the parameter value provides the absolute maximum of the log-likelihood function, it becomes the ML estimator in the strict sense.

In the following proposition, we discuss the existence of the ML estimator $\hat{\rho }$ when all other parameters are known.

Proposition 4.1

Let $g_c$ be a density generator such that

$$g_c^{\prime }(x)=r(x)g_c(x),-\mathrm{\infty }<x<\mathrm{\infty },$$ (22)

for some real-valued function $r(x)$ with $\underset{\rho \to \pm 1}{lim}r(x_{\rho ,i})=c\in (-\mathrm{\infty },0)$, where $x_{\rho ,i}$, $i=1,\dots ,n$, are as in (21). If the parameters ${\eta }_1,{\eta }_2,{\sigma }_1$ and ${\sigma }_2$ are all known, then (20) has at least one root in the interval $(-1,1)$.

Proof.

The proof of this result follows by a direct application of Intermediate value theorem. For more details, see Proposition 5.1 of Vila et al. [25].

By using Morse theory, Mäkeläinen et al. [15] established that, under some regularity conditions, there is a unique MLE for $\mathit{\theta }$. For the BULS model, no closed-form solution to the maximization problem is available, and the MLE can only be found by means of numerical optimization. Under mild regularity conditions (Cox and Hinkley [5] and Davison [6]), the asymptotic distribution of the ML estimator $\hat{\mathit{\theta }}$ of $\mathit{\theta }$ is as follows: $\sqrt{n}(\hat{\mathit{\theta }}-\mathit{\theta })\stackrel{\mathcal{D}}{⟶}N(\mathbf{0},I^{-1}(\mathit{\theta }))$, where $\mathbf{0}$ is the zero mean vector and $I^{-1}(\mathit{\theta })$ is the inverse expected Fisher information matrix. The main use of the last convergence is to construct confidence regions and to perform hypothesis testing for $\mathit{\theta }$ (see Davison [6]).

4.1. Residual analysis

To assess the goodness of fit and departures from the assumptions of the model, we consider the stochastic relation

$$\mathrm{Resid}(\mathit{W})=\frac{{\tilde{W}}_1^2-2\rho {\tilde{W}}_1{\tilde{W}}_2+{\tilde{W}}_2^2}{1-{\rho }^2},$$ (23)

where the random vector $\mathit{W}=(W_1,W_2{)}^{\mathrm{\top }}$ follows a BULS distribution, with ${\tilde{W}}_i=\log (T_i/{\eta }_i{)}^{1/{\sigma }_i},T_i=-\log (1-W_i),$ and ${\eta }_i=\exp ({\mu }_i),i=1,2$. The corresponding CDF and PDF of (23) are given respectively by

$$\begin{array}{rl}{F}_{\mathrm{Resid}(\mathit{W})}(x)& ={\displaystyle \frac{4}{Z_{g_c}}{\int }_0^{\sqrt{x}}[{\int }_0^{\sqrt{x-z_1^2}}{g}_c(z_1^2+z_2^2)\mathrm{d}{z}_2]\mathrm{d}{z}_1,x>0,}\\ f_{\mathrm{Resid}(\mathit{W})}(x)& =\frac{\pi }{Z_{g_c}}{g}_c(x),x>0,\end{array}$$

where $Z_{g_c}$ is as in (2).

For example, upon taking $g_c(x)=\exp (-x/2)$ and $Z_{g_c}=2\pi$ (see Table 1), we get $\mathrm{Resid}(\mathit{W})\sim {\chi }_2^2$ (chi-square with 2 degrees of freedom). Next, upon taking $g_c(x)=(1+(x/\nu ){)}^{-(\nu +2)/2}$ and $Z_{g_c}=\mathrm{\Gamma }(\nu /2)\mathrm{\nu \pi }/\mathrm{\Gamma }((\nu +2)/2)$ (see Table 1), we have $\mathrm{Resid}(\mathit{W})\sim 2F_{2,\nu }$, where $F_{2,\nu }$ denotes the F-distribution with 2 and ν degrees of freedom. Therefore, we can use the relation in (23) to check the goodness of fit, contrasting the empirical distribution against the theoretical one. Specifically, quantile-quantile (QQ) plots and the Kolmogorov-Smirnov test can then be used to assess the fit.

5. Simulation study

In this section, we carry out a Monte Carlo simulation study for evaluating the performance of the ML estimators of the parameters of BULS distribution. For illustrative purposes, we only present the results for the bivariate unit-log-normal model. The simulation scenario considered is as follows: 1,000 Monte Carlo replications, sample size $n\in (25,100,500,700)$, vector of true parameters $({\eta }_1,{\eta }_2,{\sigma }_1,{\sigma }_2)=(1,1,0.5,0.5)$, $\rho \in \{0,0.25,0.5,0.75,0.95\}$ (negative values of ρ produce the same results and so are omitted). To study the performance of the ML estimators, we computed the bias, root mean square error (RMSE), and coverage probability (CP) as

$$\begin{array}{rl}\widehat{\mathrm{Bias}}(\hat{\theta })& =\frac{1}{N}\sum _{i=1}^N{\hat{\theta }}^{(i)}-\theta ,\\ \widehat{\mathrm{RMSE}}(\hat{\theta })& =\sqrt{\frac{1}{N}\sum _{i=1}^N({\hat{\theta }}^{(i)}-\theta {)}^2},\\ \widehat{\mathrm{CP}}(\hat{\theta })& =\frac{1}{N}\sum _{i=1}^N\mathcal{I}(\theta \in [L_{\hat{\theta }}^{(i)},U_{\hat{\theta }}^{(i)}]),\end{array}$$

where θ and ${\hat{\theta }}^{(i)}$ are the true parameter value and its i-th ML estimate, N is the number of Monte Carlo replications, $\mathcal{I}$ is an indicator function taking the value 1 if $\theta \in [L_{\hat{\theta }}^{(i)},U_{\hat{\theta }}^{(i)}]$, and 0 otherwise, where $L_{\hat{\theta }}^{(i)}$ and $U_{\hat{\theta }}^{(i)}$ are the i-th upper and lower limit estimates of the 95% confidence interval. We expect that, as the sample size increases, the bias and RMSE would decrease, and the CP would approach the 95% nominal level.

The obtained simulation results are presented in Figure 1. We observe that the results obtained for the chosen bivariate unit-log-normal distribution are as expected in that as the sample size increases, the bias and RMSE both decrease and that the CP approaches the 95% nominal level. Finally, in general, the results do not seem to depend on the parameter ρ.

Figure 1.

Monte Carlo simulation results for the bivariate unit-log-normal model.

6. Application to soccer data

In this section, two real soccer data sets, corresponding to times elapsed until scored goals of UEFA Champions League and pass completions of 2022 FIFA World Cup, are analyzed. The UEFA Champions League data set was extracted from Meintanis [17], while the 2022 FIFA World Cup data set is new and is analyzed for the first time here.

6.1. UEFA champions league

We consider a bivariate data set on the group stage of the UEFA Champions League for the seasons 2004/05 and 2005/06. Only matches with at least one goal scored directly from a kick by any team, and with at least one goal scored by the home team, are considered here; see Meintanis [17]. The first variable ($W_1$) is the time (in minutes) elapsed until a first kick goal is scored by any team, and the second one $W_2$ is the time (in minutes) elapsed until a first goal of any type is scored by the home team. The times are divided by 90 minutes (full game time) to obtain data on the unit square $(0,1)\times (0,1)$; see Table A1.

Table 4 provides descriptive statistics for the variables $W_1$ and $W_2$, including minimum, median, mean, maximum, standard deviation (SD), coefficient of variation (CV), coefficient of skewness (CS), and coefficient of kurtosis (CK). We observe in the variable $W_1$, the mean and median to be, respectively, 0.454 and 0.456, i.e. the mean is almost equal to the median, which indicates symmetry in the data. The CV is $49.274\mathrm{\%}$, which means a moderate level of dispersion is present around the mean. Furthermore, the CS value also confirms the symmetry nature. The variable $W_2$ has mean 0.365 and median 0.311, which indicates a small positively skewed feature in the distribution of the data. Moreover, the CV value is $69.475\mathrm{\%}$, showing a moderate level of dispersion around the mean. The CS confirms the small skewed nature and the CK value indicates the small kurtosis feature in the data.

Table 4.

Summary statistics for the UEFA Champions League data set.

Variables	n	Minimum	Median	Mean	Maximum	SD	CV	CS	CK
$W_1$	37	0.022	0.456	0.454	0.911	0.224	49.274	0.164	−0.930
$W_2$	37	0.022	0.311	0.365	0.944	0.254	69.475	0.522	−0.839

The ML estimates and the standard errors (in parentheses) for the bivariate unit-log-symmetric model parameters are presented in Table 5. The extra parameters, associated with the log-Student-t, log-hyperbolic and log-slash models, were estimated by using the profile log-likelihood; see Saulo et al. [22]. Table 5 also presents the log-likelihood value, and the values of the Akaike (AIC) and Bayesian (BIC) information criteria. We observe that the log-hyperbolic model provides better fit than all other models based on the values of log-likelihood, AIC and BIC.

Table 5.

ML estimates (with standard errors in parentheses), and the log-likelihood, AIC and BIC values for the indicated bivariate unit-log-symmetric models.

Distribuiton	${\hat{\eta }}_1$	${\hat{\eta }}_2$	${\hat{\sigma }}_1$	${\hat{\sigma }}_2$	$\hat{\rho }$	$\hat{\nu }$	Log-likelihood	AIC	BIC
Log-normal	0.5288*	0.3414*	0.8865*	1.1355*	0.4956*	–	−36.693	83.386	91.441
	(0.0771)	(0.0637)	(0.1031)	(0.1320)	(0.1240)
Log-Student-t	0.5541*	0.3783*	0.7431*	0.9734*	0.4723*	7	−35.487	80.974	89.029
	(0.0751)	(0.0672)	(0.1033)	(0.1308)	(0.1463)
Log-hyperbolic	0.5458*	0.3816*	0.8456*	1.0950*	0.4893*	2	−35.470	80.940	88.996
	(0.0752)	(0.0677)	(0.1162)	(0.1462)	(0.1428)
Log-Laplace	0.5680*	0.5679*	0.9928*	1.3231*	0.5281*	–	−36.009	82.019	90.073
	(0.0020)	(0.0021)	(0.1692)	(0.2164)	(0.1639)
Log-slash	0.5629*	0.3715*	0.6203*	0.8302*	0.4472*	5	−35.560	81.120	89.174
	(0.0749)	(0.0666)	(0.0847)	(0.1096)	(0.1472)

Note: ^* significant at 5% level.

Figure 2 shows the QQ plots of $\mathrm{Resid}(\mathit{W})$, defined in (23), for the bivariate unit-log-symmetric models considered in Table 5. The QQ plot is a plot of the empirical quantiles of $\mathrm{Resid}(\mathit{W})$ against the theoretical quantiles of the respective reference distribution (see Section 4.1). Therefore, points falling along a straight line would indicate a good fit. From Figure 2, we see clearly that, with the exception of log-Student-t case, the empirical distributions of $\mathrm{Resid}(\mathit{W})$ in the considered models conform relatively well with their reference distributions. We also see that, in all the cases, there is a point away from the reference line, which may be an outlier.

Figure 2.

QQ plot of $\mathrm{Resid}(\mathit{W})$ for the indicated models. (a) Log-normal (b) Log-Student-t (c) Log-hyperbolic (d) Log-Laplace (e) Log-slash

We can use the Kolmogorov-Smirnov test [9] to verify the assumption of the theoretical distribution of $\mathrm{Resid}(\mathit{W})$. As the log-hyperbolic model provided the best fit in terms of log-likelihood, AIC and BIC, we only report the result for this case. The p-value is 0.9998, and therefore we can not reject the null hypothesis that the sample is drawn from the reference distribution.

6.2. 2022 FIFA world cup

We now use the data on the 2022 FIFA World Cup to illustrate the model developed in the preceding sections. These data are available at https://www.kaggle.com/. The first variable ($W_1$) is the medium pass completion proportion, that is, successful passes between 14 and 18 meters. The second variable ($W_2$) is the long pass completion proportion, namely, passes longer than 37 meters; see Table A1.

Table 6 provides descriptive statistics for the variables $W_1$ and $W_2$. We observe that the variable $W_1$ has mean equal to the median, which indicates symmetry in the data. The CV is $4.376\mathrm{\%}$, which means a low level of dispersion around the mean. Furthermore, the CS value is relatively small, which also confirms the symmetric nature. The variable $W_2$ has mean equal to 0.550 and median equal to 0.556, which indicates a symmetric feature in the distribution of the data. Moreover, the CV value is $13.713\mathrm{\%}$, showing a low level of dispersion around the mean. The CS confirms the symmetric nature and the CK value indicates the small kurtosis feature in the data.

Table 6.

Summary statistics for the 2022 FIFA World Cup data set.

Variables	n	Minimum	Median	Mean	Maximum	SD	CV	CS	CK
$W_1$	32	0.769	0.860	0.860	0.931	0.038	4.376	−0.373	−0.194
$W_2$	32	0.427	0.556	0.550	0.751	0.075	13.713	0.308	−0.425

Table 7 presents the estimation results for the bivariate unit-log-symmetric models, and these reveal that the log-normal model provides better fit than all other models based on the values of log-likelihood, AIC and BIC.

Table 7.

ML estimates (with standard errors in parentheses), and the log-likelihood, AIC and BIC values for the indicated bivariate unit-log-symmetric models.

Distribuiton	${\hat{\eta }}_1$	${\hat{\eta }}_2$	${\hat{\sigma }}_1$	${\hat{\sigma }}_2$	$\hat{\rho }$	$\hat{\nu }$	Log-likelihood	AIC	BIC
Log-normal	1.9872*	0.7953*	0.1364*	0.2089*	0.7343*	–	20.791	−31.581	−24.252
	(0.0479)	(0.0294)	(0.0171)	(0.0261)	(0.0815)
Log-Student-t	1.9954*	0.7936*	−0.1257*	−0.1949*	0.7423*	9	20.130	−30.260	−22.931
	(0.0485)	(0.0299)	(0.0178)	(0.0271)	(0.0868)
Log-hyperbolic	1.9908*	0.7942*	0.3956*	0.6088*	0.7378*	10	20.618	−31.236	−23.907
	(0.0482)	(0.0296)	(0.0523)	(0.0800)	(0.0841)
Log-Laplace	1.9908*	0.8089*	0.1563*	0.2425*	0.7471*	–	16.915	−23.830	−16.501
	(0.0023)	(0.0021)	(0.0278)	(0.0415)	(0.0938)
Log-slash	1.9897*	0.7935*	0.1173*	0.1802*	0.7392*	8	20.613	−28.919	−23.898
	(0.0482)	(0.0295)	(0.0154)	(0.0237)	(0.0844)

Note: ^* significant at 5% level.

Figure 3 shows the QQ plots of $\mathrm{Resid}(\mathit{W})$ (see Section 4.1) for the bivariate unit-log-symmetric models considered in Table 7. We see clearly that the log-normal model provides better fit than all other bivariate unit-log-symmetric models. By applying the Kolmogorov-Smirnov test for the log-normal case, the corresponding p-value is found to be 0.9993, and therefore, the null hypothesis is not rejected.

Figure 3.

QQ plot of $\mathrm{Resid}(\mathit{W})$ for the indicated models. (a) Log-normal (b) Log-Student-t (c) Log-hyperbolic (d) Log-Laplace (e) Log-slash.

7. Conclusions

In this paper, we have proposed a family of bivariate distributions over the unit square. By suitably defining the density generator, we can transform any distribution over the real line into a bivariate distribution over the region $(0,1)\times (0,1)$. Such a model has several potential applications, since the simultaneous modeling of quantities like proportions, rates or indices frequently arises in applied sciences like economics, medicine, engineering and social sciences. We have discussed several theoretical properties such as stochastic representation, quantiles, conditional distributions, independence and moments. We have also carried out a Monte Carlo simulation study and have demonstrated some applications in the analyses soccer data. The present research can be extended in several possible directions. By changing the density generator, numerous special forms of the BULS distribution can be constructed. Furthermore, generalizations to higher dimensions can be studied. We are currently working in these directions and hope to report the findings in a future paper.

Appendices.

Appendix 1. Some proofs and additional results

For the convenience of readers, we present here some complementary results relating to Section 3.3 and the proofs of Propositions 3.13 and 3.14.

Definition A.1

We say that a random variable X follows a univariate generalized hyperbolic (GH) distribution, denoted by $X\sim \mathrm{GH}(\lambda ,\alpha ,\delta )$, if its PDF is given by

$$f_{\mathrm{GH}}(x;\lambda ,\alpha ,\delta )={\displaystyle \frac{(\sqrt{{\alpha }^2}/\delta {)}^{\lambda }}{\sqrt{2\pi }{K}_{\lambda }(\delta \sqrt{{\alpha }^2})}}{\displaystyle \frac{K_{\lambda -1/2}(\alpha \sqrt{{\delta }^2+x^2})}{(\sqrt{{\delta }^2+x^2}/\alpha {)}^{1/2-\lambda }}},-\mathrm{\infty }<x<\mathrm{\infty }.$$

Here, $K_r$ is the modified Bessel function of the third kind with index r, $\lambda \in \mathbb{R},\alpha \in \mathbb{R}$ and $\delta >0$ is a scale parameter.

The following result has appeared in a multivariate version in Proposition 3 of Deng and Yao [7].

Proposition A.2 Hyperbolic generator —

Let $\mathit{Z}=(Z_1,Z_2{)}^{\mathrm{\top }}$ be a random vector as in Proposition 3.1. If $g_c(x)=\exp (-\nu \sqrt{1+x})$, then the conditional distribution of $Z_2$, given $Z_1=x$, is $\mathrm{GH}(1,\nu ,\sqrt{1+x^2})$ and both of its unconditional distributions are $\mathrm{GH}(3/2,\nu ,1)$.

Proof.

By using (8), the joint PDF of $Z_1$ and $Z_2$ is (see Table 1)

$$f_{Z_1,Z_2}(x,y)=\frac{{\nu }^2\exp (\nu )}{2\pi (\nu +1)}\exp (-\nu \sqrt{1+x^2+y^2}).$$ (A1)

So, the marginal PDF of $Z_1$ is given by

$$\begin{array}{rl}{f}_{Z_1}(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{Z_1,Z_2}(x,y)\mathrm{d}y& =\frac{{\nu }^2\exp (\nu )}{2\pi (\nu +1)}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp (-\nu \sqrt{1+x^2+y^2})\mathrm{d}y\\ & =\frac{2{\nu }^2\exp (\nu )}{2\pi (\nu +1)}{\int }_0^{\mathrm{\infty }}\exp (-\nu \sqrt{1+x^2+y^2})\mathrm{d}y.\end{array}$$

By using Formula 6 of Section 3.46–3.48 of (see p. 364 of Gradshteyn and Ryzhik [10]) that ${\int }_0^{\mathrm{\infty }}\exp (-a\sqrt{b^2+x^2})\mathrm{d}x=b{K}_1(\mathrm{ab})$, the above integral becomes

$$\frac{2{\nu }^2\exp (\nu )}{2\pi (\nu +1)}\sqrt{1+x^2}{K}_1(\nu \sqrt{1+x^2}).$$ (A2)

Now, as $K_{3/2}(\nu )=2\sqrt{\pi }\exp (-\nu )(\nu +1)(\nu /2{)}^{3/2}/{\nu }^3$, the above espression becomes

$$\frac{{\nu }^{3/2}}{\sqrt{2\pi }{K}_{3/2}(\nu )}\frac{K_1(\nu \sqrt{1+x^2})}{(\sqrt{1+x^2}/\nu {)}^{-1}}=f_{\mathrm{GH}}(x;3/2,\nu ,1),$$

which proves that $Z_1\sim \mathrm{GH}(3/2,\nu ,1)$. Similarly, we can show that $Z_2\sim \mathrm{GH}(3/2,\nu ,1)$, as well.

On the other hand, from (A1) and (A2), and by using the well-known identity $K_{1/2}(z)=\sqrt{\pi /(2z)}\exp (-z)$, the conditional PDF of $Z_2$, given $Z_1=x$, is obtained as

$$\begin{array}{rl}{f}_{Z_2|Z_1}(y|x)={\displaystyle \frac{\exp (-\nu \sqrt{1+x^2+y^2})}{2\sqrt{1+x^2}{K}_1(\nu \sqrt{1+x^2})}}& ={\displaystyle \frac{\nu /\sqrt{1+x^2}}{\sqrt{2\pi }{K}_1(\nu \sqrt{1+x^2})}}{\displaystyle \frac{K_{1/2}(\nu \sqrt{1+x^2+y^2})}{(\sqrt{1+x^2+y^2}/\nu {)}^{-1/2}}}\\ & =f_{\mathrm{GH}}(x;1,\nu ,\sqrt{1+x^2}),\end{array}$$

which completes the proof.

The following result has also appeared in a multivariate version in Theorem 6.7.1 of Kotz et al. [14].

Proposition A.3 Laplace generator —

Let $\mathit{Z}=(Z_1,Z_2{)}^{\mathrm{\top }}$ be a random vector as in Proposition 3.1. If $g_c(x)=K_0(\sqrt{2x})$, then the conditional distribution of $Z_2$, given $Z_1=x$, is $\mathrm{GH}(1/2,\sqrt{2},|x|)$ and both of its unconditional distributions are $\mathrm{Laplace}(0,1/\sqrt{2})$.

Proof.

By (8) and using the definitions of $K_0$ and $Z_{g_c}$ in Table 1, we have

$$f_{Z_1,Z_2}(x,y)=\frac{1}{\pi }{K}_0\left(\sqrt{2(x^2+y^2)}\right)=\frac{1}{2\pi }{\int }_0^{\mathrm{\infty }}\frac{1}{t}\exp (-t-\frac{x^2+y^2}{2t})\mathrm{d}t.$$ (A3)

We then find the marginal density of $Z_1$ to be

$$f_{Z_1}(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{Z_1,Z_2}(x,y)\mathrm{d}y=\frac{1}{\sqrt{2}}\exp (-\sqrt{2}|x|)=f_{\mathrm{L}}(x),$$ (A4)

where $f_{\mathrm{L}}(x)=\exp (-\sqrt{2}|x|)/\sqrt{2}$ is the Laplace PDF with scale parameter $1/\sqrt{2}$; that is, $Z_1\sim \mathrm{Laplace}(0,1/\sqrt{2})$. Similarly, we can show that $Z_2\sim \mathrm{Laplace}(0,1/\sqrt{2})$, as well.

On the other hand, by using (A3), (A4) and the well-known identity $K_{1/2}(z)=\sqrt{\pi /(2z)}\exp (-z)$, the conditional PDF of $Z_2$, given $Z_1=x$, is obtained as

$$\begin{array}{rl}{f}_{Z_2|Z_1}(y|x)={\displaystyle \frac{{\displaystyle \frac{1}{\pi }{K}_0\left(\sqrt{2(x^2+y^2)}\right)}}{{\displaystyle \frac{1}{\sqrt{2}}\exp (-\sqrt{2}|x|)}}}& ={\displaystyle \frac{(\sqrt{2}/|x|{)}^{1/2}}{\sqrt{2\pi }{K}_{1/2}(\sqrt{2}|x|)}}{K}_0\left(\sqrt{2}\sqrt{x^2+y^2}\right)\\ & =f_{\mathrm{GH}}(x;1/2,\sqrt{2},|x|).\end{array}$$

Then, from Definition A.1, we simply have $Z_2|(Z_1=x)\sim \mathrm{GH}(1/2,\sqrt{2},|x|)$, as required.

Definition A.4

We say that a random variable X follows a univariate extended slash (ESL) distribution, denoted by $X\sim \mathrm{ESL}(a,q)$, if its PDF is given by

$$f_{\mathrm{ESL}}(x;a,q)={\displaystyle \frac{{\int }_0^1{t}^q\varphi (\mathrm{ta})\varphi (\mathrm{tx})\mathrm{d}t}{{\int }_0^1{u}^{q-1}\varphi (\mathrm{ua})\mathrm{d}u}},-\mathrm{\infty }<x<\mathrm{\infty },$$

where ϕ denotes the PDF of the standard normal distribution.

If we now choose a=0, the classical slash (SL) PDF is obtained, given by

$$\begin{array}{rl}{f}_{\mathrm{SL}}(x;q)& =q{\int }_0^1{t}^q\varphi (\mathrm{tx})\mathrm{d}t,-\mathrm{\infty }<x<\mathrm{\infty }\\ & =\frac{q{2}^{\frac{q}{2}-1}}{\sqrt{\pi }}|x{|}^{-(q+1)}\gamma (\frac{q+1}{2},\frac{x^2}{2}).\end{array}$$

In this case, we denote it by $X\sim \mathrm{SL}(q)$.

Proposition A.5 Slash generator —

Let $\mathit{Z}=(Z_1,Z_2{)}^{\mathrm{\top }}$ be a random vector as in Proposition 3.1. If $g_c(x)=x^{-(q+2)/2}\gamma ((q+2)/2,x/2)$, then the conditional distribution of $Z_2$, given $Z_1=x$, is $\mathrm{ESL}(x,q+1)$ and both of its unconditional distributions are $\mathrm{SL}(q)$.

Proof.

By (8) and using the definition of $Z_{g_c}$ in Table 1, the joint PDF of $Z_1$ and $Z_2$ is given by

$$\begin{array}{rl}{f}_{Z_1,Z_2}(x,y)& =\frac{q{2}^{\frac{q}{2}-1}}{\pi }(x^2+y^2{)}^{-\frac{q+2}{2}}\gamma (\frac{q+2}{2},\frac{x^2+y^2}{2})\\ & =q{\int }_0^1{t}^{q+1}\varphi (\mathrm{tx})\varphi (\mathrm{ty})\mathrm{d}t.\end{array}$$ (A5)

So, the marginal PDF of $Z_1$ is

$$\begin{array}{rl}{f}_{Z_1}(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{Z_1,Z_2}(x,y)\mathrm{d}y& =q{\int }_0^1{t}^{q+1}\varphi (\mathrm{tx})[{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\varphi (\mathrm{ty})\mathrm{d}y]\mathrm{d}t\\ & =q{\int }_0^1{t}^q\varphi (\mathrm{tx})\mathrm{d}t=f_{\mathrm{SL}}(x;q),\end{array}$$ (A6)

which proves that (see Definition A.4), $Z_1\sim \mathrm{SL}(q)$. Similarly, we can show that $Z_2\sim \mathrm{SL}(q)$.

On the other hand, by (A5) and (A6), the conditional PDF of $Z_2$, given $Z_1=x$, is obtained as

$$f_{Z_2|Z_1}(y|x)={\displaystyle \frac{{\int }_0^1{t}^{q+1}\varphi (\mathrm{tx})\varphi (\mathrm{ty})\mathrm{d}t}{{\int }_0^1{u}^q\varphi (\mathrm{ux})\mathrm{d}u}}=f_{\mathrm{ESL}}(y;x,q+1).$$

From Definition A.4, we then find that $Z_2|(Z_1=x)\sim \mathrm{ESL}(x,q+1)$, as required.

Proof of Proposition 3.13.

Writing $S_1=[W_1-{\mu }_1(1)]/{\sigma }_{W_1}$ and ${\rho }_{\mathit{W}}{S}_1+\sqrt{1-{\rho }_{\mathit{W}}^2}{S}_2=[W_2-{\mu }_2(1)]/{\sigma }_{W_2}$, a simple algebraic manipulation shows that

$$d^2=S_1^2+S_2^2.$$

Upon using the law of total expectation, the corresponding CDF of $d^2$ is given by (for 0<w<1)

$$\begin{array}{rl}{F}_{d^2}(w)=\mathbb{E}[\mathbb{E}({\mathbb{1}}_{\{S_1^2+S_2^2⩽w\}})|S_1]& =\mathbb{E}[\mathbb{E}({\mathbb{1}}_{\{|S_2|⩽\sqrt{w-S_1^2}\}})|S_1{\mathbb{1}}_{\{|S_1|⩽\sqrt{w}\}}]\\ & ={\int }_{-\sqrt{w}}^{\sqrt{w}}[{\int }_{-\sqrt{w-s_1^2}}^{\sqrt{w-s_1^2}}{f}_{S_2}(s_2|S_1=s_1)\mathrm{d}{s}_2]f_{S_1}(s_1)\mathrm{d}{s}_1.\end{array}$$ (A7)

If $S_1=s_1$, then $W_1={\mu }_1(1)+{\sigma }_{W_1}{s}_1$. So, the conditional distribution of $S_2$, given $S_1=s_1$, is the same as the distribution of

$$\frac{1}{\sqrt{1-{\rho }_{\mathit{W}}^2}}[\frac{W_2-{\mu }_2(1)}{{\sigma }_{W_2}}-{\rho }_{\mathit{W}}{s}_1]|W_1={\sigma }_{W_1}{s}_1+{\mu }_1(1).$$

Hence, the PDF of $S_2$, given $S_1=s_1$, is given by

$$f_{S_2}(s_2|S_1=s_1)={\sigma }_{W_2}\sqrt{1-{\rho }_{\mathit{W}}^2}{f}_{W_2}({\mu }_2(1)+{\sigma }_{W_2}[{\rho }_{\mathit{W}}{s}_1+\sqrt{1-{\rho }_{\mathit{W}}^2}{s}_2]|W_1={\mu }_1(1)+{\sigma }_{W_1}{s}_1).$$

By using Lemma 3.3, the above density can be expressed as

$$\begin{array}{rl}{f}_{S_2}(s_2|S_1=s_1)& ={\sigma }_{W_2}\sqrt{1-{\rho }_{\mathit{W}}^2}{f}_{W_2}(w_{2,s}|W_1=w_{1,s})\\ & =k\frac{1}{(1-w_{1,s})t_{1,s}(1-w_{2,s})t_{2,s}}{f}_{Z_2}(\frac{1}{\sqrt{1-{\rho }^2}}({\tilde{w}}_{2,s}-\rho {\tilde{w}}_{1,s})|Z_1={\tilde{w}}_{1,s}),\end{array}$$ (A8)

where $k\equiv {\sigma }_{W_1}{\sigma }_{W_2}\sqrt{1-{\rho }_{\mathit{W}}^2}/({\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2})$, and $w_{i,s}$, ${\tilde{w}}_{i,s}$ and $t_{i,s}$ are as given in (18).

Replacing (A8) in (A7), we get the following formula for the CDF of the squared Mahalanobis distance:

$$\begin{array}{rl}{F}_{d^2}(w)& =k{\int }_{-\sqrt{w}}^{\sqrt{w}}[{\int }_{-\sqrt{w-s_1^2}}^{\sqrt{w-s_1^2}}\frac{1}{(1-w_{1,s})t_{1,s}(1-w_{2,s})t_{2,s}}\\ & f_{Z_2}(\frac{1}{\sqrt{1-{\rho }^2}}({\tilde{w}}_{2,s}-\rho {\tilde{w}}_{1,s})|Z_1={\tilde{w}}_{1,s})\mathrm{d}{s}_2]f_{Z_1}({\tilde{w}}_{1,s})\mathrm{d}{s}_1.\end{array}$$

Moreover, the vector $(Z_1,Z_2{)}^{\mathrm{\top }}$ has a standard elliptical distribution (spherical) (see Item 2.11 of Saulo et al. [23]), denoted by $(Z_1,Z_2{)}^{\mathrm{\top }}\sim \mathrm{BSY}({\mu }_1=0,{\mu }_2=0,{\sigma }_1=1,{\sigma }_2=1,\rho =0,g_c)$, where $g_c$ is the density generator in (5). Hence, the required result.

Proof of Proposition 3.14.

By differentiating $F_{d^2}(w)$ (in Proposition 3.13) with respect to w and then by using the following well-known Leibniz integral rule

$$\frac{\mathrm{d}}{\mathrm{d}x}{\int }_{a(x)}^{b(x)}h(x,y)\mathrm{d}y=h(x,b(x))b^{\prime }(x)-h(x,a(x))a^{\prime }(x)+{\int }_{a(x)}^{b(x)}\frac{\mathrm{\partial }h(x,y)}{\mathrm{\partial }x}\mathrm{d}y,$$

the required result follows.

Appendix 2. Data sets

Table A1.

UEFA Champions League and 2022 FIFA World Cup data sets.

	UEFA		FIFA
	W1	W2	W1	W2
1	0.289	0.222	0.888	0.541
2	0.700	0.200	0.815	0.474
3	0.211	0.211	0.907	0.624
4	0.733	0.944	0.891	0.606
5	0.444	0.444	0.827	0.517
6	0.544	0.544	0.898	0.557
7	0.089	0.089	0.856	0.462
8	0.767	0.789	0.861	0.618
9	0.433	0.433	0.890	0.603
10	0.911	0.533	0.860	0.477
11	0.800	0.800	0.920	0.646
12	0.733	0.689	0.894	0.587
13	0.278	0.100	0.913	0.648
14	0.456	0.033	0.849	0.471
15	0.178	0.833	0.781	0.427
16	0.200	0.200	0.828	0.442
17	0.244	0.156	0.864	0.581
18	0.467	0.467	0.820	0.527
19	0.022	0.022	0.846	0.526
20	0.400	0.578	0.879	0.601
21	0.378	0.378	0.860	0.481
22	0.589	0.433	0.885	0.616
23	0.600	0.078	0.862	0.592
24	0.567	0.311	0.769	0.463
25	0.844	0.711	0.845	0.495
26	0.711	0.167	0.846	0.489
27	0.289	0.533	0.931	0.751
28	0.178	0.178	0.863	0.555
29	0.489	0.144	0.856	0.447
30	0.278	0.156	0.879	0.569
31	0.611	0.122	0.812	0.613
32	0.544	0.544	0.841	0.594
33	0.267	0.267	–	–
34	0.489	0.333	–	–
35	0.467	0.033	–	–
36	0.300	0.522	–	–
37	0.311	0.311	–	–

Funding Statement

Roberto Vila and Helton Saulo gratefully acknowledge financial support from CNPq, CAPES and FAPDF, Brazil.

Disclosure statement

No potential conflict of interest was reported by the author(s).