Bivariate Log-Symmetric Regression Models Applied to Newborn Data

Abstract

This paper introduces bivariate log-symmetric models for analyzing the relationship between two variables, assuming a family of log-symmetric distributions. These models offer greater flexibility than the bivariate lognormal distribution, allowing for better representation of diverse distribution shapes and behaviors in the data. The log-symmetric distribution family is widely used in various scientific fields and includes distributions such as log-normal, log-Student-t, and log-Laplace, among others, providing several options for modeling different data types. However, there are few approaches to jointly model continuous positive and explanatory variables in regression analysis. Therefore, we propose a class of generalized linear model (GLM) regression models based on bivariate log-symmetric distributions, aiming to fill this gap. Furthermore, in the proposed model, covariates are used to describe its dispersion and correlation parameters. This study uses a dataset of anthropometric measurements of newborns to correlate them with various biological factors, proposing bivariate regression models to account for the relationships observed in the data. Such models are crucial for preventing and controlling public health issues.

1. Introduction

Bivariate log-symmetric models are used to analyze the relationship between two variables, assuming that both follow a log-symmetric distribution. This distribution is characterized by the fact that the original random variable and its reciprocal have the same distribution; see, [1]. This property is fundamental to understanding the behavior of log-symmetric distributions in relation to their transformations. Additionally, bivariate log-symmetric distributions are a generalization of the bivariate log-normal distribution and offer greater flexibility in modeling bivariate data, allowing for better representation of a wider range of distribution shapes and behaviors observed in the data. As a result, this flexibility can lead to more accurate modeling and robust interpretations. For more details about its properties, see [2].

The family of log-symmetric distributions is widely used in various areas of science, especially when the data are continuous, strictly positive, asymmetric, bimodal, and/or have a light/heavy tail. These distributions include examples such as log-normal, log-Student-t, log-power-exponential, log-hyperbolic, log-logistic, log-Laplace, and log-slash, among others. This diversity of distributions offers a wide range of options for modeling different types of data. References for more information include [1,3].

Analyzing data for correlated variables separately with marginal distributions can be problematic as important information is overlooked. Furthermore, the inclusion of explanatory variables (covariates) can improve the precision in estimating the mean/median of the response variable and its prediction. Ignoring the correlation between responses and incorporating covariates can lead to inaccurate decisions, as reported by [4]. Furthermore, standard regression modeling is often not suitable for describing the behavior of bimodal and/or light- and heavy-tailed data. Therefore, bivariate versions of statistical distributions, along with associated regression models, are needed to address these issues. There are few approaches in bivariate and/or multivariate regression analysis to model the relationship between a vector of continuous positive response variables and a set of explanatory variables. Some recent works on bivariate regression models can be found in [5,6,7]. In Ref. [5], the authors considered bivariate logistic regression diagnostics to study characteristics of cancer patients with comorbid diabetes and hypertension in Malawi. Ref. [6] explored the phenomenon of regression to the mean, offering methods for its estimation and adjustment under the bivariate normal distribution. Finally, Ref. [7] developed methods for accounting for regression to the mean under the bivariate t distribution. The proposed model is applied to patient data related to schizophrenia, and the observed conditional mean difference is decomposed into treatment effects and regression to the mean.

The main objective of this work is to introduce a GLM-type regression model based on the class of bivariate log-symmetric distributions. Similar to what was proposed by [8], in the implementation of the GLM-type regression model based on a bivariate Birnbaum–Saunders RBS (BRBS) distribution; see [9]. Then, we have covariates explaining possible dispersion (heteroscedasticity) and correlation, respectively. The remainder of this paper is as follows: In Section 2, we provide a motivating example for our study. In Section 3, we describe preliminary aspects regarding the family of log-symmetric distributions. Section 4 presents the bivariate log-symmetric regression model, as well as the maximum likelihood estimators of the unknown parameters, and their corresponding asymptotic results. Furthermore, we also consider residual analysis for the proposed model. In Section 5, a Monte Carlo simulation study is performed to evaluate the performance of the maximum likelihood estimation method. For illustrative purposes, we only present the results for the bivariate log-normal model. In Section 6, the proposed bivariate log-symmetric regression models are applied to a dataset related to the analysis of anthropometric data of newborns to demonstrate the practical utility of the proposed models. Finally, in Section 7, we outline some final considerations and also point out some problems that deserve further investigation. Some mathematical derivations are given in Appendix A.

2. Motivating Example

In this section, we present an example that motivates our study, highlighting the importance of the analysis of anthropometric data of newborns for the diagnosis and prognosis of neonatal health. Analyzing these data can provide valuable insights into growth patterns, identification of anomalies, and risk factors for health conditions. In addition to assisting in the planning of neonatal care, the statistical analysis of these data has the potential to contribute significantly to the advancement of neonatal medicine, promoting the health and well-being of newborns.

2.1. Anthropometric Measurements

Anthropometry is an important tool in assessing human health in various phases of life, especially in newborns. It involves the evaluation of physical measurements, such as weight, compression, head circumference, abdominal circumference, head circumference, chest circumference, and other parameters. These measures are frequently recorded at the time of birth and during the first months of life to monitor the patterns of growth and development of newborns. When interpreted in an epidemiological and clinical context, we can provide fundamental information for the diagnosis and prognosis of newborns, revealing crucial syndromic symptoms.

In this context, analysis of anthropometric data from newborns is important for screening and identifying possible genetic, infectious, nutritional, and endocrinological health problems, among others. Two relevant measurements are the weight and length of newborns, which are generally associated with sex and gestational age at birth; see [10]. Several reference percentile curves have been established for these measurements depending on sex and gestational age at birth, which have been used to record and evaluate the growth conditions of the newborn until childhood. Most of these curves were obtained using parametric and non-parametric regression models for each anthropometric measurement separately, and generally ignoring the association between them and the presence of outliers; see [11,12,13].

2.2. The Dataset

This study uses a dataset studied by [14], composed of information from newborns collected at the Manuel Uribe Ángel Hospital, in Envigado, Antioquia, Colombia, between January 2018 and September 2019. The newborn dataset contains records collected from 6714 recently born children. This study seeks to correlate anthropometric data, such as weight and crown-heel length, of newborns with biological factors of the babies and their parents, which sets up a bivariate response problem. The objective is to model the anthropometric data of newborns in relation to these factors. The dependent variables to be considered are as follows:

• $T_1$: weight (in hectograms);
• $T_2$: length (crown-heel length, in centimeters),

• and the corresponding covariates are:

• $x_1$: sex (0 for female, 1 for male);
• $x_2$: gestational age at birth (in weeks);
• $x_3$: mother’s age (in years);
• $x_4$: father’s age (in years).

A correlation analysis between the variables $T_1$, $T_2$, $x_1$, $x_2$, $x_3$, and $x_4$ revealed non-linear relationships between the responses $T_1$, $T_2$ and the covariates $x_1$, $x_2$, $x_3$, and $x_4$. Figure 1 provides the scatterplots and correlation coefficients for all responses and covariates. The analysis reveals that (i) $x_1$, $x_2$, and $x_3$ are not correlated; (ii) $x_1$ and $x_2$ exhibit moderate correlation, but the values of the variance inflation factors (which measure how much the variance of an estimated regression coefficient increases if your explanatory variables are correlated) do not exceed 2, thereby eliminating concerns about multicollinearity; (iii) $T_1$ and $T_2$ have a high positive correlation; and (iv) there are different correlations between $T_1$, $T_2$, and the covariates. This analysis suggests support for a GLM-type bivariate regression model.

Table 1. Summary of the statistics for the newborn dataset.

Variables	n	Minimum	Median	Mean	Maximum	SD	CV	CS	CK
$T_1$	6714	5.4	31.1	30.791	50.9	4.931	16.015	−0.747	2.168
$T_2$	6714	28	49	48.72	58	2.634	5.406	−1.609	6.816

provides descriptive statistics for the weight ($T_1$) and length ($T_2$) variables, such as the minimum, median, mean, maximum, standard deviation (SD), coefficient of variation, coefficient of skewness (CS), and coefficient of (excess) kurtosis (CK). For weight, the mean and median are, respectively, $30.791$ and $31.1$, i.e., the mean is almost equal to the median, which indicates symmetry in the data. The CV is $16.015\%$, which means a low level of dispersion around the mean. Furthermore, the CS value suggests slight negative skewness, while the CK indicates a high degree of kurtosis. For length, the mean of $48.72$ and a median of 49, also suggest symmetry in the distribution of data. Moreover, the CV is $5.406\%$, showing a low level of dispersion around the mean. The CS indicates a skewed nature and the CK value indicates the high kurtosis feature in the data; see Figure 2 for the plots of histograms for these data. In general, the results support the use of bivariate log-symmetric models as they have flexibility in terms of distributional assumptions and flexibility in describing the behavior of light- and heavy-tailed data.

3. Preliminaries

We say that a continuous random vector $\mathit{T}={(T_1,T_2)}^{\top }$, with support ${\mathbb{R}}^{2+}$, follows a bivariate log-symmetric (BLS) distribution with parameters, as follows:

• ${\eta }_i>0$ (medians);
• ${\sigma }_i>0$ (dispersion); and
• $\rho \in (-1,1)$ (correlation)

• for $i=1,2$, and density generator $g_c$ [15], denoted by $\mathit{T}\sim \mathrm{BLS}({\eta }_1,{\eta }_2,{\sigma }_1,{\sigma }_2,\rho ,g_c)$, if its probability density function (PDF) is given by the following:

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

  (1)

  where

  $${\tilde{t}}_i=\log \left[{\left({\displaystyle \frac{t_i}{{\eta }_i}}\right)}^{1/{\sigma }_i}\right],i=1,2,$$

  and $Z_{g_c}>0$ is a partition function given by the following:

  $$Z_{g_c}={\int }_0^{\infty }{\int }_0^{\infty }{\displaystyle \frac{1}{t_1{t}_2{\sigma }_1{\sigma }_2\sqrt{1-{\rho }^2}}}{g}_c\left({\displaystyle \frac{{\tilde{t}}_1^2-2\rho {\tilde{t}}_1{\tilde{t}}_2+{\tilde{t}}_2^2}{1-{\rho }^2}}\right)\mathrm{d}{t}_1\mathrm{d}{t}_2.$$

  (2)

  The variance–covariance matrix, when it exists, of a random vector $\mathit{T}\sim \mathrm{BLS}({\eta }_1,{\eta }_2,{\sigma }_1,{\sigma }_2,\rho ,g_c)$ is given by $K_{\mathit{T}}=\psi (\mathbf{\Sigma })$ for some matrix function $\psi :{\mathcal{M}}_{2,2}⟼{\mathcal{M}}_{2,2}$, where ${\mathcal{M}}_{2,2}$ denotes the set of all $2\times 2$ real matrices, and

  $$\mathbf{\Sigma }=\left(\begin{array}{cc}{\sigma }_1^2& \rho {\sigma }_1{\sigma }_2\\ \rho {\sigma }_1{\sigma }_2& {\sigma }_2^2\end{array}\right).$$

  (3)

Table 2. Partition functions $\left(Z_{g_c}\right)$, density generators $\left(g_c\right)$, and extra parameters for some BLS distributions.

Bivariate Distribution	${\mathit{Z}}_{{\mathit{g}}_{\mathit{c}}}$	${\mathit{g}}_{\mathit{c}}$	Extra Parameter
Log-normal	$2\pi$	$exp(-x/2)$	−
Log-Student-t	${\displaystyle \frac{\mathrm{\Gamma }\left(\nu /2\right)\nu \pi }{\mathrm{\Gamma }\left((\nu +2)/2\right)}}$	${(1+{\displaystyle \frac{x}{\nu }})}^{-(\nu +2)/2}$	$\nu >0$
Log-hyperbolic	${\displaystyle \frac{2\pi (\nu +1)exp(-\nu )}{{\nu }^2}}$	$exp(-\nu \sqrt{1+x})$	$\nu >0$
Log-Laplace	$\pi$	$K_0\left(\sqrt{2x}\right)$	−
Log-slash	${\displaystyle \frac{\pi }{\nu -1}}{2}^{\frac{3-\nu }{2}}$	$x^{-\frac{\nu +1}{2}}\gamma (\frac{\nu +1}{2},\frac{x}{2})$	$\nu >1$
Log-power-exponential	$2^{\xi +1}(1+\xi )\mathrm{\Gamma }(1+\xi )\pi$	$exp\left(-\frac{1}{2}{x}^{1/(1+\xi )}\right)$	$-1<\xi ⩽1$
Log-logistic	$\pi /2$	$\frac{exp(-x)}{{(1+exp(-x))}^2}$	−

lists some examples of bivariate log-symmetric distributions, such as the log-normal, log-Student-t, log-hyperbolic, log-Laplace, log-slash, log-power-exponential, and log-logistic. In Table 2, $\mathrm{\Gamma }\left(t\right)={\int }_0^{\infty }{x}^{t-1}exp(-x)\mathrm{d}x$, where $t>0$, denotes the complete gamma function, $K_0\left(u\right)={\int }_0^{\infty }{t}^{-1}exp(-t-{\displaystyle \frac{u^2}{4t}})\mathrm{d}t/2$, $u>0$, denotes the Bessel function of the third kind [16], and $\gamma (s,x)={\int }_0^x{t}^{s-1}exp(-t)\mathrm{d}t$ denotes the lower incomplete gamma function.

4. Bivariate Log-Symmetric Regression Models

In this section, we shall introduce the bivariate log-symmetric regression model and discuss parameter estimation based on the maximum likelihood method. Moreover, we also consider residual analysis for the proposed model. The proposed bivariate log-symmetric regression model can be viewed as a generalization of the log-normal/independent linear regression model (in the bivariate case) studied by [14]. Its advantage over the log-normal/independent counterpart lies in its flexibility in describing heteroscedasticity and variable dispersion, as covariates are employed to describe its dispersion and correlation parameters.

4.1. Bivariate Regression

Let ${\mathit{T}}_i={(T_{1i},T_{2i})}^{\top }\sim \mathrm{BLS}({\eta }_{1i},{\eta }_{2i},{\sigma }_{1i},{\sigma }_{2i},{\rho }_i,g_c)$, for $i=1,\dots ,n$, be a set of n independent bivariate log-symmetric random variables. We propose a regression model based on the bivariate log-symmetric distribution as follows:

$$\begin{array}{cc}\hfill g_{{\eta }_1}\left({\eta }_{1i}\right)& ={\mathit{x}}_i^{\top }\mathit{\beta }={\beta }_0+{\beta }_1{x}_{i1}+\cdots +{\beta }_k{x}_{ik},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill g_{{\eta }_2}\left({\eta }_{2i}\right)& ={\mathit{w}}_i^{\top }\mathit{\tau }={\tau }_0+{\tau }_1{w}_{i1}+\cdots +{\tau }_l{x}_{il},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill g_{{\sigma }_1}\left({\sigma }_{1i}\right)& ={\mathit{v}}_i^{\top }\mathit{\kappa }={\kappa }_0+{\kappa }_1{v}_{i1}+\cdots +{\kappa }_p{v}_{ip},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill g_{{\sigma }_2}\left({\sigma }_{2i}\right)& ={\mathit{z}}_i^{\top }\mathit{\omega }={\omega }_0+{\omega }_1{z}_{i1}+\cdots +{\omega }_q{z}_{iq},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill g_{\rho }\left({\rho }_i\right)& ={\mathit{s}}_i^{\top }\mathit{\psi }={\psi }_0+{\psi }_1{s}_{i1}+\cdots +{\psi }_r{s}_{ir},\hfill \end{array}$$

(8)

where $\mathit{\beta }={({\beta }_0,{\beta }_1,\dots ,{\beta }_k)}^{\top }$, $\mathit{\tau }={({\tau }_0,{\tau }_1,\dots ,{\tau }_l)}^{\top }$, $\mathit{\kappa }={({\kappa }_0,{\kappa }_1,\dots ,{\kappa }_p)}^{\top }$, $\mathit{\omega }={({\omega }_0,{\omega }_1,\dots ,{\omega }_q)}^{\top }$, and $\mathit{\psi }={({\psi }_0,{\psi }_1,\dots ,{\psi }_r)}^{\top }$ are the vectors of unknown parameters to be estimated, whereas ${\mathit{x}}_i^{\top }={(1,x_{i1},\dots ,x_{ik})}^{\top }$, ${\mathit{w}}_i^{\top }={(1,w_{i1},\dots ,w_{il})}^{\top }$, ${\mathit{v}}_i^{\top }={(1,v_{i1},\dots ,v_{ip})}^{\top }$, ${\mathit{z}}_i^{\top }={(1,z_{i1},\dots ,z_{iq})}^{\top }$, and ${\mathit{s}}_i^{\top }={(1,s_{i1},\dots ,s_{ir})}^{\top }$ are vectors that contain the values of k, l, p, q, and l covariates, respectively. The link functions $g_{{\eta }_1}:{\mathbb{R}}^{+}\to \mathbb{R}$, $g_{{\eta }_2}:{\mathbb{R}}^{+}\to \mathbb{R}$, $g_{{\sigma }_1}:{\mathbb{R}}^{+}\to \mathbb{R}$, $g_{{\sigma }_2}:{\mathbb{R}}^{+}\to \mathbb{R}$ and $g_{\rho }:(-1,1)\to \mathbb{R}$ in (4)–(8) must be strictly monotone, positive, and at least twice differentiable, with $g_{{\eta }_1}^{-1}(·)$, $g_{{\eta }_2}^{-1}(·)$, $g_{{\sigma }_1}^{-1}(·)$, $g_{{\sigma }_2}^{-1}(·)$ and $g_{\rho }^{-1}(·)$ being the inverse functions of $g_{{\eta }_1}(·)$, $g_{{\eta }_2}(·)$, $g_{{\sigma }_1}(·)$, $g_{{\sigma }_2}(·)$ and $g_{\rho }(·)$, respectively. The most common choice for the first four functions is the log link (it does not require to impose non-negativity restrictions on the parameters), whereas for the latter, the most common choice is the ArcTanh link (it guarantees that $g_{\rho }^{-1}\left({\mathit{s}}_i^{\top }\mathit{\psi }\right)$ lies in the interval [−1, 1]). Other common choices for the first four functions are the identity and inverse links, which are often used in generalized linear models. The choice of the appropriate link can be made by using information criteria. A simulation study can be carried out to assess the choice of the link functions. However, it is beyond the scope of this paper and we will address this topic in future work.

4.2. Maximum Likelihood Estimation

Let $({\mathit{T}}_1,\dots ,{\mathit{T}}_n)$ be a bivariate (not identically distributed) random sample of size n from the BLS distribution (1), with ${\mathit{T}}_i\sim \mathrm{BLS}({\eta }_{1i},{\eta }_{2i},{\sigma }_{1i},{\sigma }_{2i},{\rho }_i,g_c)$, $i=1,\dots ,n$, and let $({\mathit{t}}_1,\dots ,{\mathit{t}}_n)$ denote its respective observed value. The log-likelihood function (without the additive constant) for $\mathit{\theta }={(\mathit{\beta },\mathit{\tau },\mathit{\kappa },\mathit{\omega },\mathit{\psi })}^{\top }$ is given by the following:

$$\ell \left(\mathit{\theta }\right)=-\sum _{i=1}^n\log \left({\sigma }_{1i}\right)-\sum _{i=1}^n\log \left({\sigma }_{2i}\right)-{\displaystyle \frac{1}{2}}\sum _{i=1}^n\log (1-{\rho }_i^2)+\sum _{i=1}^n\log g_c\left({\displaystyle \frac{{\tilde{t}}_{1i}^2-2{\rho }_i{\tilde{t}}_{1i}{\tilde{t}}_{2i}+{\tilde{t}}_{2i}^2}{1-{\rho }_i^2}}\right),$$

where

$${\tilde{t}}_{1i}=\log \left[{\left({\displaystyle \frac{t_{1i}}{{\eta }_{1i}}}\right)}^{1/{\sigma }_{1i}}\right],{\tilde{t}}_{2i}=\log \left[{\left({\displaystyle \frac{t_{2i}}{{\eta }_{2i}}}\right)}^{1/{\sigma }_{2i}}\right],$$

and the vector parameters $\mathit{\theta }$ and $({\eta }_{1i},{\eta }_{2i},{\sigma }_{1i},{\sigma }_{2i},{\rho }_i)$, $i=1,\dots ,n$, are connected through equations in (4)–(8). As $\ell \left(\mathit{\theta }\right)$ is differentiable in $\mathit{\theta }$, the necessary conditions for the occurrence of a maximum (or a possible minimum) are as follows:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial \ell \left(\mathit{\theta }\right)}{\partial {\beta }_j}}=0,j=0,1,\dots ,k,\hfill \\ \hfill & {\displaystyle \frac{\partial \ell \left(\mathit{\theta }\right)}{\partial {\tau }_j}}=0,j=0,1,\dots ,l,\hfill \\ \hfill & {\displaystyle \frac{\partial \ell \left(\mathit{\theta }\right)}{\partial {\kappa }_j}}=0,j=0,1,\dots ,p,\hfill \\ \hfill & {\displaystyle \frac{\partial \ell \left(\mathit{\theta }\right)}{\partial {\omega }_j}}=0,j=0,1,\dots ,q,\hfill \\ \hfill & {\displaystyle \frac{\partial \ell \left(\mathit{\theta }\right)}{\partial {\psi }_j}}=0,j=0,\dots ,r,\hfill \end{array}$$

(9)

known as the likelihood equations. It is a routine task to check the following:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial \ell \left(\mathit{\theta }\right)}{\partial {\beta }_j}}=2\sum _{i=1}^n{G}_c\left(t_i^{*}\right){\displaystyle \frac{\partial t_i^{*}}{\partial {\eta }_{1i}}}{\displaystyle \frac{\partial {\eta }_{1i}}{\partial {\beta }_j}},j=0,1,\dots ,k,\hfill \\ \hfill & {\displaystyle \frac{\partial \ell \left(\mathit{\theta }\right)}{\partial {\tau }_j}}=2\sum _{i=1}^n{G}_c\left(t_i^{*}\right){\displaystyle \frac{\partial t_i^{*}}{\partial {\eta }_{2i}}}{\displaystyle \frac{\partial {\eta }_{2i}}{\partial {\tau }_j}},j=0,1,\dots ,l,\hfill \\ \hfill & {\displaystyle \frac{\partial \ell \left(\mathit{\theta }\right)}{\partial {\kappa }_j}}=\left[-\sum _{i=1}^n{\displaystyle \frac{1}{{\sigma }_{1i}}}+2\sum _{i=1}^n{G}_c\left(t_i^{*}\right){\displaystyle \frac{\partial t_i^{*}}{\partial {\sigma }_{1i}}}\right]{\displaystyle \frac{\partial {\sigma }_{1i}}{\partial {\kappa }_j}},j=0,1,\dots ,p,\hfill \\ \hfill & {\displaystyle \frac{\partial \ell \left(\mathit{\theta }\right)}{\partial {\omega }_j}}=\left[-\sum _{i=1}^n{\displaystyle \frac{1}{{\sigma }_{2i}}}+2\sum _{i=1}^n{G}_c\left(t_i^{*}\right){\displaystyle \frac{\partial t_i^{*}}{\partial {\sigma }_{2i}}}\right]{\displaystyle \frac{\partial {\sigma }_{2i}}{\partial {\omega }_j}},j=0,1,\dots ,q,\hfill \\ \hfill & {\displaystyle \frac{\partial \ell \left(\mathit{\theta }\right)}{\partial {\psi }_j}}=\left[-\sum _{i=1}^n{\displaystyle \frac{{\rho }_i}{1-{\rho }_i^2}}+2\sum _{i=1}^n{G}_c\left(t_i^{*}\right){\displaystyle \frac{\partial t_i^{*}}{\partial {\rho }_i}}\right]{\displaystyle \frac{\partial {\rho }_i}{\partial {\psi }_j}},j=0,1,\dots ,r,\hfill \end{array}$$

where $G_c\left(x\right)=g_c^{\prime }\left(x\right)/g_c\left(x\right)$ and

$$t_i^{*}={\displaystyle \frac{{\tilde{t}}_{1i}^2-2{\rho }_i{\tilde{t}}_{1i}{\tilde{t}}_{2i}+{\tilde{t}}_{2i}^2}{1-{\rho }_i^2}}.$$

(10)

In the above, note that (for $i=1,\dots ,n$), we have the following:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial t_i^{*}}{\partial {\eta }_{1i}}}=2{\displaystyle \frac{{\rho }_i{\tilde{t}}_{2i}-{\tilde{t}}_{1i}}{(1-{\rho }_i^2){\eta }_{1i}{\sigma }_{1i}}},\hfill \\ \hfill & {\displaystyle \frac{\partial t_i^{*}}{\partial {\eta }_{2i}}}=2{\displaystyle \frac{{\rho }_i{\tilde{t}}_{1i}-{\tilde{t}}_{2i}}{(1-{\rho }_i^2){\eta }_{2i}{\sigma }_{2i}}},\hfill \\ \hfill & {\displaystyle \frac{\partial t_i^{*}}{\partial {\sigma }_{1i}}}=2{\displaystyle \frac{{\rho }_i{\tilde{t}}_{1i}{\tilde{t}}_{2i}-{\tilde{t}}_{1i}^2}{(1-{\rho }_i^2){\sigma }_{1i}}},\hfill \\ \hfill & {\displaystyle \frac{\partial t_i^{*}}{\partial {\sigma }_{2i}}}=2{\displaystyle \frac{{\rho }_i{\tilde{t}}_{1i}{\tilde{t}}_{2i}-{\tilde{t}}_{2i}^2}{(1-{\rho }_i^2){\sigma }_{2i}}},\hfill \\ \hfill & {\displaystyle \frac{\partial t_i^{*}}{\partial {\rho }_i}}=2{\displaystyle \frac{{\rho }_i{t}_i^{*}-{\tilde{t}}_{1i}{\tilde{t}}_{2i}}{1-{\rho }_i^2}},\hfill \end{array}$$

(11)

and

$$\begin{array}{ccc}\hfill & {\displaystyle \frac{\partial {\eta }_{1i}}{\partial {\beta }_0}}={\displaystyle \frac{1}{g_{{\eta }_1}^{\prime }\left({\eta }_{1i}\right)}},\hfill & {\displaystyle \frac{\partial {\eta }_{1i}}{\partial {\beta }_j}}={\displaystyle \frac{x_{ij}}{g_{{\eta }_1}^{\prime }\left({\eta }_{1i}\right)}},j=1,\dots ,k,\hfill \\ \hfill & {\displaystyle \frac{\partial {\eta }_{2i}}{\partial {\tau }_0}}={\displaystyle \frac{1}{g_{{\eta }_2}^{\prime }\left({\eta }_{2i}\right)}},\hfill & {\displaystyle \frac{\partial {\eta }_{2i}}{\partial {\tau }_j}}={\displaystyle \frac{w_{ij}}{g_{{\eta }_2}^{\prime }\left({\eta }_{2i}\right)}},j=1,\dots ,l,\hfill \\ \hfill & {\displaystyle \frac{\partial {\sigma }_{1i}}{\partial {\kappa }_0}}={\displaystyle \frac{1}{g_{{\sigma }_1}^{\prime }\left({\sigma }_{1i}\right)}},\hfill & {\displaystyle \frac{\partial {\sigma }_{1i}}{\partial {\kappa }_j}}={\displaystyle \frac{v_{ij}}{g_{{\sigma }_1}^{\prime }\left({\sigma }_{1i}\right)}},j=1,\dots ,p,\hfill \\ \hfill & {\displaystyle \frac{\partial {\sigma }_{2i}}{\partial {\omega }_0}}={\displaystyle \frac{1}{g_{{\sigma }_2}^{\prime }\left({\sigma }_{2i}\right)}},\hfill & {\displaystyle \frac{\partial {\sigma }_{2i}}{\partial {\omega }_j}}={\displaystyle \frac{z_{ij}}{g_{{\sigma }_2}^{\prime }\left({\sigma }_{2i}\right)}},j=1,\dots ,q,\hfill \\ \hfill & {\displaystyle \frac{\partial {\rho }_i}{\partial {\psi }_0}}={\displaystyle \frac{1}{g_{\rho }^{\prime }\left({\rho }_i\right)}},\hfill & {\displaystyle \frac{\partial {\rho }_i}{\partial {\psi }_j}}={\displaystyle \frac{s_{ij}}{g_{\rho }^{\prime }\left({\rho }_i\right)}},j=1,\dots ,r.\hfill \end{array}$$

(12)

Any nontrivial root $\mathit{\theta }$ of the likelihood equations in (9) is called a maximum likelihood estimator in the loose sense. When the parameter value gives the absolute maximum of the log-likelihood function, it is called a maximum likelihood estimator in the strict sense.

No closed-form solution to the maximization problem is available for the BLS regression model, and a maximum likelihood estimator can only be found via numerical optimization. The extra parameters, associated with the bivariate log-Student-t, log-hyperbolic, log-slash, and log-power-exponential regression models, were estimated by using the profile log-likelihood; see [2]. Under some regularity conditions, Ref. [17] formalized the existence and uniqueness of the maximum likelihood estimator $\widehat{\mathit{\theta }}$ for $\mathit{\theta }$.

Under some restrictions, Ref. [18] verified that the asymptotic behavior of the maximum likelihood estimator is Gaussian, in other words:

$$\sqrt{n}(\widehat{\mathit{\theta }}-\mathit{\theta })\stackrel{d}{⟶}N(\mathbf{0},{\mathit{I}}^{-1}\left(\mathit{\theta }\right)),$$

where “$\stackrel{d}{⟶}$” denotes convergence in law, $\mathbf{0}$ is the zero mean vector, and ${\mathit{I}}^{-1}\left(\mathit{\theta }\right)$ is the inverse expected Fisher information matrix. It is preferable to use the observed Fisher information, denoted by $\mathit{J}\left(\mathit{\theta }\right)$, instead of $\mathit{I}\left(\mathit{\theta }\right)$ [19], because the diagonal elements of the inverse observed Fisher information matrix, denoted by ${\mathit{J}}^{-1}\left(\mathit{\theta }\right)$, can be used to approximate the standard errors of the maximum likelihood estimates. The elements of matrix $\mathit{J}\left(\mathit{\theta }\right)$ can be found in Appendix A.

4.3. Residual Analysis

Consider $({\mathit{T}}_1,\dots ,{\mathit{T}}_n)$ a bivariate (not identically distributed) random sample of size n from the BLS distribution (1), with ${\mathit{T}}_i\sim \mathrm{BLS}({\eta }_{1i},{\eta }_{2i},{\sigma }_{1i},{\sigma }_{2i},{\rho }_i,g_c)$, where ${\eta }_{1i}$, ${\eta }_{2i}$, ${\sigma }_{1i}$, ${\sigma }_{2i}$, and ${\rho }_i$ are defined in (4)–(8), for $i=1,\dots ,n$. By deriving the distribution of the following stochastic relation (see the argument of the function $g_c$ in (1)), we have the following:

$${\displaystyle \frac{{\tilde{T}}_{1i}^2-2\rho {\tilde{T}}_{1i}{\tilde{T}}_{2i}+{\tilde{T}}_{2i}^2}{1-{\rho }^2}},$$

(13)

where

$${\tilde{T}}_{ji}=\log \left[{\left({\displaystyle \frac{T_{ji}}{{\eta }_{ji}}}\right)}^{1/{\sigma }_{ji}}\right],j=1,2,i=1,\dots ,n,$$

we propose a residual to assess the goodness of fit and departures from the assumed model. Note that from (13), we have the following:

$${\mathrm{Residual}}_i={\displaystyle \frac{{\left\{\log \left[{\left(\frac{T_{1i}}{{\eta }_{1i}}\right)}^{1/{\sigma }_{1i}}\right]\right\}}^2-2\rho \left\{\log \left[{\left(\frac{T_{1i}}{{\eta }_{1i}}\right)}^{1/{\sigma }_{1i}}\right]\right\}\left\{\log \left[{\left(\frac{T_{2i}}{{\eta }_{2i}}\right)}^{1/{\sigma }_{2i}}\right]\right\}+{\left\{\log \left[{\left(\frac{T_{2i}}{{\eta }_{2i}}\right)}^{1/{\sigma }_{2i}}\right]\right\}}^2}{(1-{\rho }_i^2)}},$$

(14)

for $i=1,\dots ,n$, and the corresponding cumulative distribution function (CDF) and PDF of (14) are given, respectively, by the following:

$$\begin{array}{ccc}F\left(x\right)\hfill & =& {\displaystyle \frac{4}{Z_{g_c}}}{\int }_0^{\sqrt{x}}\left[{\int }_0^{\sqrt{x-z_1^2}}{g}_c(z_1^2+z_2^2)\mathrm{d}{z}_2\right]\mathrm{d}{z}_1,x>0,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}f\left(x\right)\hfill & =& {\displaystyle \frac{\pi }{Z_{g_c}}}{g}_c\left(x\right),x>0,\hfill \end{array}$$

(16)

where $Z_{g_c}$ and $g_c$ are as in (1). We can then use the relation (14) to check the goodness of fit. This is done by contrasting the empirical distribution with the theoretical one through quantile-quantile (QQ) plots.

5. Monte Carlo Simulation

In this section, we present the results of a Monte Carlo simulation study for the BLS regression model. The purpose of this simulation study is to evaluate the performance of the maximum likelihood estimators. For illustrative purposes, we only present the results for the bivariate log-normal model, with simulated data generated according to the following:

$$\begin{array}{cc}\hfill \log \left({\eta }_{1i}\right)& =\log \left({\mathit{x}}_i^{\top }\mathit{\beta }\right)={\beta }_0+{\beta }_1{x}_{i1}+{\beta }_2{x}_{i2}+{\beta }_3{x}_{i3},\hfill \\ \hfill \log \left({\eta }_{2i}\right)& =\log \left({\mathit{w}}_i^{\top }\mathit{\tau }\right)={\tau }_0+{\tau }_1{x}_{i1}+{\tau }_2{x}_{i2}+{\tau }_3{x}_{i3},\hfill \\ \hfill \log \left({\sigma }_{1i}\right)& =\log \left({\mathit{v}}_i^{\top }\mathit{\kappa }\right)={\kappa }_0+{\kappa }_1{x}_{i1},\hfill \\ \hfill \log \left({\sigma }_{2i}\right)& =\log \left({\mathit{z}}_i^{\top }\mathit{\omega }\right)={\omega }_0+{\omega }_1{x}_{i1},\hfill \\ \hfill \mathrm{ArcTanh}\left({\rho }_i\right)& =\mathrm{ArcTanh}\left({\mathit{s}}_i^{\top }\mathit{\psi }\right)={\psi }_0+{\psi }_1{x}_{i1},\hfill \end{array}$$

for $i\in \{1,\cdots ,n\}$, where $x_{1i}$, $x_{2i}$, and $x_{3i}$ are covariates obtained from a standard normal distribution. We consider three different sets of true parameter values, leading to scenarios that cover low, moderate, and high correlations, as follows:

• Scenario 1: $\mathit{\beta }={(1.1,0.7,1.5,0.5)}^{\top }$, $\mathit{\tau }={(0.9,1.5,1.1,0.6)}^{\top }$, $\mathit{\kappa }={(-0.4,0.7)}^{\top }$, $\mathit{\omega }={(-0.4,0.7)}^{\top }$ and $\mathit{\psi }={(0.3,0.5)}^{\top }$ –low correlation–.
• Scenario 2: $\mathit{\beta }={(1.1,0.7,1.5,0.5)}^{\top }$, $\mathit{\tau }={(0.9,1.5,1.1,0.6)}^{\top }$, $\mathit{\kappa }={(-0.2,0.5)}^{\top }$, $\mathit{\omega }={(-0.2,0.5)}^{\top }$ and $\mathit{\psi }={(2.0,2.5)}^{\top }$ –moderate correlation–.
• Scenario 3: $\mathit{\beta }={(1.1,0.7,1.5,0.5)}^{\top }$, $\mathit{\tau }={(0.9,1.5,1.1,0.6)}^{\top }$, $\mathit{\kappa }={(0.1,0.5)}^{\top }$, $\mathit{\omega }={(0.1,0.5)}^{\top }$ and $\mathit{\psi }={(1.0,3.0)}^{\top }$ –high correlation–.

• The sample sizes for all three simulation scenarios are $n\in \{100,500,1000\}$, with 500 Monte Carlo replications for each combination of above given parameters.

Table 3. Empirical biases from simulated bivariate log-normal regression data.

	Scenario 1
$\mathit{n}$	${\mathit{\beta }}_{\mathbf{0}}$	${\mathit{\beta }}_{\mathbf{1}}$	${\mathit{\beta }}_{\mathbf{2}}$	${\mathit{\beta }}_{\mathbf{3}}$	${\mathit{\tau }}_{\mathbf{0}}$	${\mathit{\tau }}_{\mathbf{1}}$	${\mathit{\tau }}_{\mathbf{2}}$	${\mathit{\tau }}_{\mathbf{3}}$	${\mathit{\kappa }}_{\mathbf{0}}$	${\mathit{\kappa }}_{\mathbf{1}}$	${\mathit{\omega }}_{\mathbf{0}}$	${\mathit{\omega }}_{\mathbf{1}}$	${\mathit{\psi }}_{\mathbf{0}}$	${\mathit{\psi }}_{\mathbf{1}}$
100	0.0505	0.0548	0.0226	0.0686	0.0605	0.0241	0.0313	0.0572	0.1552	0.0923	0.1538	0.0975	0.2758	0.2204
500	0.0203	0.0188	0.0086	0.0294	0.0265	0.0094	0.0126	0.0219	0.0627	0.0337	0.0618	0.0368	0.1188	0.0701
1000	0.0153	0.0129	0.0067	0.0197	0.0171	0.0061	0.0082	0.0163	0.0420	0.0221	0.0449	0.0221	0.0814	0.0508
	Scenario 2
100	0.0394	0.0314	0.0078	0.0229	0.0485	0.0146	0.0108	0.0189	0.2779	0.1108	0.2775	0.1114	0.0414	0.0654
500	0.0077	0.0048	0.0010	0.0029	0.0095	0.0022	0.0013	0.0025	0.1122	0.0433	0.1125	0.0433	0.0176	0.0167
1000	0.0040	0.0023	0.0004	0.0013	0.0049	0.0011	0.0006	0.0011	0.0795	0.0297	0.0795	0.0298	0.0121	0.0101
	Scenario 3
100	0.0166	0.0136	0.0028	0.0080	0.0201	0.0065	0.0036	0.0070	0.5120	0.1078	0.5101	0.1084	0.0797	0.0633
500	0.0023	0.0016	0.0003	0.0009	0.0030	0.0007	0.0004	0.0008	0.2097	0.0441	0.2097	0.0442	0.0318	0.0145
1000	0.0012	0.0007	0.0001	0.0004	0.0014	0.0003	0.0002	0.0003	0.1591	0.0315	0.1587	0.0315	0.0216	0.0091

and

Table 4. Empirical RMSEs from simulated bivariate log-normal regression data.

	Scenario 1
$\mathit{n}$	${\mathit{\beta }}_{\mathbf{0}}$	${\mathit{\beta }}_{\mathbf{1}}$	${\mathit{\beta }}_{\mathbf{2}}$	${\mathit{\beta }}_{\mathbf{3}}$	${\mathit{\tau }}_{\mathbf{0}}$	${\mathit{\tau }}_{\mathbf{1}}$	${\mathit{\tau }}_{\mathbf{2}}$	${\mathit{\tau }}_{\mathbf{3}}$	${\mathit{\kappa }}_{\mathbf{0}}$	${\mathit{\kappa }}_{\mathbf{1}}$	${\mathit{\omega }}_{\mathbf{0}}$	${\mathit{\omega }}_{\mathbf{1}}$	${\mathit{\psi }}_{\mathbf{0}}$	${\mathit{\psi }}_{\mathbf{1}}$
100	0.0692	0.0496	0.0432	0.0433	0.0695	0.0493	0.0436	0.0438	0.0787	0.0813	0.0768	0.0867	0.1045	0.1397
500	0.0282	0.0167	0.0163	0.0187	0.0304	0.0181	0.0171	0.0166	0.0311	0.0303	0.0314	0.0324	0.0449	0.0442
1000	0.0211	0.0114	0.0123	0.0121	0.0195	0.0112	0.0114	0.0124	0.0211	0.0195	0.0226	0.0193	0.0304	0.0315
	Scenario 2
100	0.0574	0.0298	0.0173	0.0166	0.0575	0.0295	0.0174	0.0166	0.0692	0.0699	0.0688	0.0699	0.1041	0.1964
500	0.0118	0.0048	0.0021	0.0022	0.0117	0.0048	0.0020	0.0022	0.0282	0.0269	0.0283	0.0269	0.0436	0.0515
1000	0.0057	0.0021	0.0009	0.0009	0.0058	0.0021	0.0009	0.0009	0.0200	0.0184	0.0199	0.0184	0.0303	0.0322
	Scenario 3
100	0.0256	0.0137	0.0060	0.0057	0.0252	0.0140	0.0057	0.0058	0.0638	0.0679	0.0636	0.0682	0.0992	0.2213
500	0.0034	0.0016	0.0006	0.0006	0.0037	0.0015	0.0006	0.0006	0.0264	0.0274	0.0264	0.0274	0.0401	0.0535
1000	0.0017	0.0006	0.0003	0.0003	0.0016	0.0006	0.0003	0.0003	0.0204	0.0199	0.0203	0.0199	0.0275	0.0339

present the results of the Monte Carlo simulation study to assess the performance of the maximum likelihood estimators. In particular, we report the empirical biases and root mean squared errors (RMSEs), which are computed as follows:

$$\begin{array}{c}\hfill \widehat{\mathrm{RB}}\left(\widehat{\theta }\right)=\left|{\displaystyle \frac{\frac{1}{N}{\sum }_{i=1}^N{\widehat{\theta }}^{\left(i\right)}-\theta }{\theta }}\right|,\\ \hfill \widehat{\mathrm{RMSE}}\left(\widehat{\theta }\right)=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{({\widehat{\theta }}^{\left(i\right)}-\theta )}^2},\end{array}$$

where $\theta$ and ${\widehat{\theta }}^{\left(i\right)}$ are the true parameter value and its i-th maximum likelihood estimate, and N is the number of Monte Carlo replications. In terms of parameter recovery, from Table 3 and Table 4, we observe that both biases and RMSEs approach zero as the sample size (n) increases, as expected. We also observe that the bias (RMSE) tends to be larger (smaller) for smaller correlation values.

6. Application to Newborn Data

We illustrate the proposed methodology with a dataset related to the analysis of anthropometric data of newborns for the diagnosis and prognosis of neonatal health presented in [14] and described in Section 2. We consider the regression structure defined in (4)–(8) for analyzing the newborn data using the bivariate log-symmetric regression model, expressed as follows:

$$\begin{array}{cc}\hfill \log \left({\eta }_{1i}\right)& =\log \left({\mathit{x}}_i^{\top }\mathit{\beta }\right)={\beta }_0+{\beta }_1{x}_{i1}+{\beta }_2{x}_{i2}+{\beta }_3{x}_{i3}+{\beta }_4{x}_{i4},\hfill \\ \hfill \log \left({\eta }_{2i}\right)& =\log \left({\mathit{w}}_i^{\top }\mathit{\tau }\right)={\tau }_0+{\tau }_1{x}_{i1}+{\tau }_2{x}_{i2}+{\tau }_3{x}_{i3}+{\tau }_4{x}_{i4},\hfill \\ \hfill \log \left({\sigma }_{1i}\right)& =\log \left({\mathit{v}}_i^{\top }\mathit{\kappa }\right)={\kappa }_0+{\kappa }_1{x}_{i1}+{\kappa }_2{x}_{i2}+{\kappa }_3{x}_{i3}+{\kappa }_4{x}_{i4},\hfill \\ \hfill \log \left({\sigma }_{2i}\right)& =\log \left({\mathit{z}}_i^{\top }\mathit{\omega }\right)={\omega }_0+{\omega }_1{x}_{i1}+{\omega }_2{x}_{i2}+{\omega }_3{x}_{i3}+{\omega }_4{x}_{i4},\hfill \\ \hfill \mathrm{ArcTanh}\left({\rho }_i\right)& =\mathrm{ArcTanh}\left({\mathit{s}}_i^{\top }\mathit{\psi }\right)={\psi }_0+{\psi }_1{x}_{i1}+{\psi }_2{x}_{i2}+{\psi }_3{x}_{i3}+{\psi }_4{x}_{i4},\hfill \end{array}$$

recalling that ${\eta }_{1i}=\mathrm{Median}\left(Y_{1i}\right)$, with $Y_{1i}$ being the weight (in hectograms), ${\eta }_{2i}=\mathrm{Median}\left(T_{2i}\right)$, with $T_{2i}$ being the length (crown-heel length, in centimeters), $x_{i1}$ is the sex (0 for female, 1 for male), $x_{i2}$ is the gestational age at birth (in weeks), $x_{i3}$ is the mother’s age (in years) and $x_{i4}$ is the father’s age (in years).

The choice of an appropriate member of the log-symmetric family can be conducted by using information criteria.

Table 5. Model selection and goodness-of-fit measures for the newborn data.

Distribution	Log-Likelihood	AIC	BIC
Log-normal	−30,300.5199	60,651.0398	60,821.3385
Log-Student-t	−30,236.8269	60,523.6538	60,693.9526
Log-hyperbolic	−30,230.5438	60,511.0876	60,681.3864
Log-Laplace	−30,814.1601	61,678.3202	61,848.6189
Log-slash	−30,220.9040	60,491.8080	60,662.1068
Log-power-exponential	−30,249.1342	60,548.2684	60,718.5672
Log-logistic	−30,606.7447	61,263.4893	61,433.7881

provides the values of the log-likelihood, as well as the Akaike (AIC) and Bayesian information (BIC) criteria for the adjusted bivariate log-symmetric regression models. In general, the bivariate regression model based on the log-slash distribution provides the best fit for the data. The estimated extra parameter for the log-slash model was $\widehat{\nu }=7$. For comparison, the values of the log-likelihood, AIC, and BIC for the bivariate gamma regression model [20], a well-accepted bivariate regression approach, are −34,397.03, 68,826.07, and 68,935.06, respectively. Therefore, all the bivariate log-symmetric regression models outperform the gamma one.

Figure 3a–g present the QQ plots of the proposed residuals (see Section 4.3) for the models considered in Table 5. By comparing the quantiles of a dataset to the quantiles of a theoretical distribution, the QQ plot can be used to determine whether the dataset follows the reference distribution. From the QQ plots, we observe that the bivariate log-slash model provides the best fit. Therefore, the results of these plots corroborate the ones obtained in Table 5.

Table 6. Maximum likelihood estimates with SEs, t-values, and p-values for the bivariate log-slash regression model for the newborn data.

Coefficients for ${\mathit{\eta }}_1$
Parameter	Estimate	SE	$\mathit{t}$-Value	$\mathit{p}$-Value
${\beta }_0$	1.0757653	0.0436945	24.620	<$2\times {10}^{-16}$ ***
${\beta }_1$	0.0303725	0.0028133	10.796	<$2\times {10}^{-16}$ ***
${\beta }_2$	0.0592572	0.0011096	53.405	<$2\times {10}^{-16}$ ***
${\beta }_3$	0.0018078	0.0002887	6.263	$4.02\times {10}^{-10}$ ***
${\beta }_4$	0.0001755	0.0002369	0.741	0.459
Coefficients for ${\mathit{\eta }}_2$
Parameter	Estimate	SE	$\mathit{t}$-Value	$\mathit{p}$-Value
${\tau }_0$	$3.245$	$1.409\times {10}^{-2}$	$230.249$	$<2\times {10}^{-16}$ ***
${\tau }_1$	$1.301\times {10}^{-2}$	$9.115\times {10}^{-4}$	$14.269$	$<2\times {10}^{-16}$ ***
${\tau }_2$	$1.635\times {10}^{-2}$	$3.571\times {10}^{-4}$	$45.772$	$<2\times {10}^{-16}$ ***
${\tau }_3$	$2.593\times {10}^{-4}$	$9.251\times {10}^{-5}$	$2.803$	$0.00508$ ***
${\tau }_4$	$3.102\times {10}^{-5}$	$7.507\times {10}^{-5}$	$0.413$	$0.67942$
Coefficients for ${\mathit{\sigma }}_1$
Parameter	Estimate	SE	t-Value	p-Value
${\kappa }_0$	$1.243$	$1.742\times {10}^{-1}$	$7.139$	$1.04\times {10}^{-12}$ ***
${\kappa }_1$	$-7.563\times {10}^{-3}$	$1.866\times {10}^{-2}$	$-0.405$	$0.6852$
${\kappa }_2$	$-9.594\times {10}^{-2}$	$4.420\times {10}^{-3}$	$-21.704$	$<2\times {10}^{-16}$ ***
${\kappa }_3$	$4.902\times {10}^{-3}$	$1.871\times {10}^{-3}$	$2.620$	$0.0088$ ***
${\kappa }_4$	$-9.401\times {10}^{-5}$	$1.563\times {10}^{-3}$	$-0.060$	$0.9520$
Coefficients for ${\mathit{\sigma }}_2$
Parameter	Estimate	SE	t-Value	p-Value
${\omega }_0$	$0.320772$	$0.173395$	$1.850$	$0.0644$ **
${\omega }_1$	$-0.006767$	$0.018696$	$-0.362$	$0.7174$
${\omega }_2$	$-0.098184$	$0.004396$	$-22.335$	$<2\times {10}^{-16}$ ***
${\omega }_3$	$0.003571$	$0.001881$	$1.898$	$0.0577$ **
${\omega }_4$	$-0.002913$	$0.001567$	$-1.860$	$0.0630$ **
Coefficients for $\mathit{\rho }$
Parameter	Estimate	SE	t-Value	p-Value
${\psi }_0$	$3.385383$	$0.249032$	$13.594$	$<2\times {10}^{-16}$ ***
${\psi }_1$	$0.011318$	$0.025801$	$0.439$	$0.6609$
${\psi }_2$	$-0.069482$	$0.006295$	$-11.037$	$<2\times {10}^{-16}$ ***
${\psi }_3$	$0.005343$	$0.002594$	$2.060$	$0.0395$ ***
${\psi }_4$	$-0.003870$	$0.002153$	$-1.798$	$0.0723$ **

*** significant at 5% level. ** significant at 10% level.

reports the maximum likelihood estimates with SEs, t-values, and p-values for the bivariate log-slash regression model. From this table, we observe the following results: the covariates $x_{i1}$, $x_{i2}$, and $x_{i3}$ are significant for ${\eta }_{1i}$ and ${\eta }_{2i}$. The covariates $x_{i2}$ and $x_{i3}$, which impact the dispersion ${\sigma }_1$, are significant, and the covariates $x_{i2}$, $x_{i3}$, and $x_{i4}$, which are associated with the dispersion ${\sigma }_1$, are significant. Therefore, there is evidence of heteroscedasticity. For the covariates related to the correlation parameter, we observe that $x_{i2}$, $x_{i3}$, and $x_{i4}$ are significant.

7. Concluding Remarks

In this paper, we propose a class of regression models based on bivariate log-symmetric distributions. These models offer greater flexibility compared to the bivariate lognormal distribution. In the proposed bivariate regression models, covariates are used to describe the dispersion and correlation parameters, which is an important aspect given the possible presence of heteroscedasticity. We discussed the maximum likelihood estimation of the model parameters and introduced a type of residual. We conducted a Monte Carlo simulation study and demonstrated its application in analyzing anthropometric measurements of newborns. The application to newborn data favored the use of the bivariate log-slash model over other bivariate log-symmetric regression models. Notably, we detected the presence of heteroscedasticity and variable dispersion, features not considered by [14].

Despite the strengths of this study, limitations must be acknowledged. The proposed models have limitations with respect to the presence of nonlinearity. Therefore, addressing non-linear structures that incorporate both parametric and nonparametric components remains an open problem. Moreover, it will be of interest to study bivariate tobit models; see [21]. In addition, influence diagnostic tools can be applied to the proposed models. Furthermore, the behaviors of the Wald, score, likelihood ratio, and gradient tests could also be investigated; see [22]. Finally, comparisons with other approaches such as copula regression can be performed [23]. Work on these issues is currently in progress, and we hope to report the findings in a future paper.