The bivariate Sinh-Elliptical distribution with applications to Birnbaum–Saunders distribution and associated regression and measurement error models

Abstract

The bivariate Sinh-Elliptical (BSE) distribution is a generalization of the well-known Rieck’s (1989) Sinh-Normal distribution that is quite useful in Birnbaum–Saunders (BS) regression model. The main aim of this paper is to define the BSE distribution and discuss some of its properties, such as marginal and conditional distributions and moments. In addition, the asymptotic properties of method of moments estimators are studied, extending some existing theoretical results in the literature. These results are obtained by using some known properties of the bivariate elliptical distribution. This development can be viewed as a follow-up to the recent work on bivariate Birnbaum–Saunders distribution by Kundu et al. (2010) towards some applications in the regression setup. The measurement error models are also introduced as part of the application of the results developed here. Finally, numerical examples using both simulated and real data are analyzed, illustrating the usefulness of the proposed methodology.

Introduction

Many univariate distributions have been generalized to form bivariate absolutely continuous distributions. These include several common distributions such as uniform, normal, exponential, beta, Weibull and gamma (see  Balakrishnan and Lai, 2009). The univariate Sinh-Normal (SN) distribution has received considerable attention in lifetime regression models, especially when the lifetime data involve modeling by the distribution of Birnbaum and Saunders, 1969a, Birnbaum and Saunders, 1969b, in which the scale parameter depends on the covariates. The Birnbaum–Saunders (BS) distribution has applications in a wide range of problems and so does the SN distribution due to its close relationship to the BS distribution. Johnson (1949) mentioned that it is natural and also convenient to build non-normal distributions through transformations of a random variable (rv) that follows a normal distribution. He used the translation method to generate statistical distributions that assume a wide variety of shapes via the transformation

$$Z=\nu +\delta g\left(\frac{Y-\mu }{\sigma }\right)\text{,}$$

where $Z\sim N(0,1)$ and $g(\cdot )$ is a monotone function. These non-normal distributions of $Y$ have four parameters $\nu ,\delta ,\mu$ and $\sigma$, where $\nu$ and $\delta$ correspond to shape parameters while $\mu$ and $\sigma$ are location and scale parameters, respectively. The parameters $\delta$ and $\sigma$ are assumed to be positive. Based on (1), with $\delta =2/\alpha$, Rieck (1989, p. 13) defined that, if

$$Z=\nu +\frac{2}{\alpha }sinh\left(\frac{Y-\mu }{\sigma }\right)\text{,}$$

then $Y$ follows a four-parameter Sinh-Normal (SN) distribution, with its probability density function (pdf) given by

$$f_Y\left(y\right)={\sigma }^{-1}\varphi ({\xi }_{2y}+\nu ){\xi }_{1y}\text{,}y\in \mathbb{R}\text{,}$$

where ${\xi }_{1y}={\xi }_{1y}(\alpha ,\mu ,\sigma )=\frac{2}{\alpha }cosh\left(\frac{y-\mu }{\sigma }\right),{\xi }_{2y}={\xi }_{2y}(\alpha ,\mu ,\sigma )=\frac{2}{\alpha }sinh\left(\frac{y-\mu }{\sigma }\right)$, and $\varphi (\cdot )$ denotes the standard normal density function. In this case, the notation $Y\sim SN(\alpha ,\mu ,\sigma ,\nu )$ is used, which is reduced simply to $Y\sim SN(\alpha ,\mu ,\sigma )$ when $\nu =0$ in (3); see Rieck and Nedelman (1991). Moreover, when $\sigma =2$, the r.v. $T=exp\left(Y\right)$ follows the Birnbaum–Saunders (BS) distribution (Birnbaum and Saunders, 1969a) that is usually denoted by $BS(\alpha ,\beta )$, where $\beta =exp\left(\mu \right)$. This is the reason why the Sinh-Normal is called the log-Birnbaum–Saunders distribution. Some extensions of the SN distributions have been considered in the literature by replacing the normal distribution by a symmetric or asymmetric distribution, as can be seen in Barros et al. (2008) and Leiva et al. (2010), respectively. Another extension could be obtained by considering the log-linear regression model for the $\beta$-Birnbaum–Saunders distribution; see Cordeiro and Lemonte (2011).

Even though considerable amount of work has been done on univariate Sinh-Normal and Birnbaum–Saunders distributions and their extensions, bivariate versions of these models have received little attention in the literature. The bivariate BS distribution was presented by Kundu et al. (2010) who also derived several interesting and attractive properties of it by using some known properties of the bivariate normal distribution. Hence, it will be natural to study the log-bivariate BS distribution that could also be called the bivariate Sinh-Normal (BSN) distribution. Only recently, Kundu (2014) introduced a BSN distribution based on the normal distribution, which is obtained through the Gaussian copula property.

Even though normality may be a reasonable model to build the BSN distribution, it may lack robustness in parameter estimation under departures from normality (in the form of heavy tails) and/or outliers (Lange and Sinsheimer, 1993).

In this paper, we derive a bivariate extension of the Sinh-Normal distribution by basing it on the bivariate elliptical distribution in place of bivariate normal distribution, and refer to it as the bivariate Sinh-Elliptical (BSE) distribution.

Before presenting our derivation, let us first recall the definition of the multivariate elliptical distribution (Fang et al., 1990): We say that a $p$-variate random vector $Z$ has an elliptical distribution, denoted by $E{C}_p(\eta ,\Psi ;f)$, where $\eta$ is the location vector and $\Psi$ is a positive definite dispersion matrix, if its pdf is given by

$$f_Z\left(z\right)={\left|\Psi \right|}^{-1/2}f\left({(z-\eta )}^{\top }{\Psi }^{-1}(z-\eta )\right)\text{,}z\in {\mathbb{R}}^p\text{.}$$

The function $f$ is referred to as the density generator. Alternatively, the elliptical distribution is denoted by $E{C}_p(\eta ,\Psi ;\psi )$, where its characteristic function is ${\psi }_Z\left(t\right)=exp\left(i{t}^{\top }\eta \right)\psi \left(t^{\top }\Psi t\right),t\in {\mathbb{R}}^p$. Note that the expectation and the covariance matrix of $Z$ are $E\left(Z\right)=\eta$ and Var$\left(Z\right)=\Sigma =c_f\Psi$, respectively, where $c_f=-2{\psi }^{\prime }\left(0\right)$ with ${\psi }^{\prime }$ being the first derivative of the function $\psi$ associated with the characteristic function of $Z$. For this distribution, the kurtosis measure is $\kappa ={\psi }^{″}\left(0\right)/{\left({\psi }^{\prime }\left(0\right)\right)}^2-1$, where ${\psi }^{″}$ is the second derivative of $\psi$. Elliptical distributions include several special cases, such as the normal, Student-$t$, Logistic, and Laplace. For an elaborate discussion on elliptical distributions, one may refer to Cambanis et al. (1981), Muirhead (1982), and Fang et al. (1990). For $Z\sim E{C}_2(0,\Psi ;f)$, with ${\psi }_{11}={\psi }_{22}=1$, the joint pdf is given by

$$f_Z(z;\rho )=\frac{1}{\sqrt{1-{\rho }^2}}f\left(z^{\top }{\Psi }^{-1}z\right)\text{,}z\in {\mathbb{R}}^2\text{,}$$

where $z^{\top }{\Psi }^{-1}z=\frac{1}{(1-{\rho }^2)}(z_1^2+z_2^2-2\rho z_1{z}_2)$ and $\rho$ is the correlation coefficient.

The main aim of this paper is to present the BSE distribution, which has its marginal distributions and conditional distributions as univariate Sinh-Elliptical distributions that include Sinh-Normal (Rieck, 1989) and Sinh-Student-$t$ (Barros et al., 2008), as particular cases. Also, the special case $(\lambda =0)$ has been discussed in Leiva et al. (2010). The asymptotic behavior of the method of moments estimators is also discussed, and the relationship to the bivariate BS distribution introduced by Kundu et al. (2010) is pointed out. In addition, the associated regression and linear measurement error models are analyzed.

The rest of the paper is organized as follows. In Section  2, we define the BSE distribution and discuss some of its properties. Section  3 discusses the asymptotic properties of the method of moments estimators of the model parameters. Section  4 discusses the relationship between the bivariate Sinh-Elliptical distribution and the bivariate Birnbaum–Saunders distribution of Kundu et al. (2010). In addition, the associated regression model and linear measurement error model are also discussed as an application of the results developed here. In Section  5, numerical examples using both simulated and real data are presented to illustrate the proposed methodology. Finally, some concluding remarks are made in Section  6.