**1. Introduction**

Mudholkar and Hutson (2000) [1] studied an asymmetric normal distribution that they called the epsilon–skew–normal {*ESN*(*ε*); |*ε*| < <sup>1</sup>}, with asymmetry or skewness parameter *ε*. When the parameter *ε* assumes the value 0, the distribution simplifies to become a standard normal distribution. The family thus consists of a parameterized set of usually asymmetric distributions that includes the symmetric standard normal density as a special case. Specifically, we say that *X* ∼ *ESN*(*ε*) if its density is of the form:

$$\lg\left(\mathbf{x};\varepsilon\right) = \phi\left(\frac{\mathbf{x}}{1 - s\mathfrak{g}n\left(\mathbf{x}\right)\varepsilon}\right),$$

where *x* ∈ R, *φ* is the standard normal density and *sgn*(·) is the sign function.

Arellano-Valle et al. (2005) [2] discuss extension of this model, together with associated inference procedures. They consider a class of Epsilon-skew-symmetric distributions associated with a particular symmetric density *f* (·) that is indexed by an asymmetry parameter *ε* with densities given by

$$h(\mathbf{x}; \boldsymbol{\varepsilon}) = f\left(\frac{\mathbf{x}}{1 - \operatorname{sgn}\left(\mathbf{x}\right)\boldsymbol{\varepsilon}}\right) \tag{1}$$

where *x* ∈ R and |*ε*| < 1.

If *X* has density of the form (1), then we say that *X* is an epsilon-skew-symmetric random variable and we write *X* ∼ *ESf*(*ε*). Arellano-Valle et al. (2005) [2] extend this family to the model epsilon-skew-exponential-power, a model that has major and minor asymmetry and kurtosis that the ESN model. On the other hand Gómez et al. (2007) [3] study the Fisher information matrix for epsilon-skew-t model, which was used before in the study a financial series by Hansen (1994) [4]; see also Gómez et al. (2008) [5]. Note that if in (1) we set *f*(*t*) = 1/ *π* 1 + *t* <sup>2</sup>, we obtain the epsilon–skew–Cauchy model.

We will write *X* ∼ *N*(0, 1) to indicate that *X* has a standard normal distribution, and we will write *Y* ∼ *HN*(0, 1) to indicate that *Y* has a standard half-normal distribution, i.e., that *Y* = |*X*| where *X* ∼ *N*(0, <sup>1</sup>).

The distribution of the ratio *X*/*Y* of two random variables is of interest in problems in biological and physical sciences, econometrics, and ranking and selection. It is well known that if *X* ∼ *N*(0, <sup>1</sup>), *Y*1 ∼ *N*(0, 1) and *Y*2 ∼ *HN*(0, 1) are independent, then the random variables *X*/*Y*1 and *X*/*Y*2 both have Cauchy distributions; see Johnson et al. (1994, [6] Chapter 16). Behboodian, et al. (2006) [7] and Huang and Chen (2007) [8] study the distribution of such quotients when the component random variables are skew-normal (of the form studied in Azzalini (1985) [9]). The principle objective of the present paper is to study the behavior of such quotients when the component random variables have epsilon-skew-normal distributions.

The paper is organized in the following manner. In Section 2, we describe a representation of the epsilon–skew–Cauchy model. In Section 3, we consider the distribution of the ratio of two independent random variables, one of which has an *ESN* (*ε*) distribution and the other a standard normal distribution. In addition, an extension of the epsilon–skew–Cauchy (ESC) distribution is introduced. Bivariate versions of these distributions are discussed in Section 4. Extensions to higher dimensions can be readily envisioned, but are not discussed here. In Section 5, some of the bivariate distributions introduced in this paper are considered as possible models for a particular real-world data set.

### **2. Representation of the ESC (Epsilon–Skew–Cauchy) Model**

**Proposition 1.** *If X* ∼ *ESN* (*ε*) *and Y* ∼ *HN*(0, 1) *are independent random variables, then Z*1 *d*= *XY has an epsilon–skew–Cauchy distribution with asymmetry parameter ε and density given by*

$$f\text{z}\_1\left(z;\varepsilon\right) = \frac{1}{\pi \left[1 + \left(\frac{z}{1 - s\chi n(z)\varepsilon}\right)^2\right]},$$

*where z* ∈ R*,* |*ε*| < 1 *and we write Z*1 ∼ *ESC*(*ε*)*.*

**Proof.** With the transformation *Z*1 = *X*/*Y* and *W* = *Y*, whose Jacobian |*J*| = *w*, we obtain

$$f\_{Z\_1, \mathcal{W}}(z, w) = \frac{w}{\pi} \exp\left\{ -\frac{1}{2} \left[ \frac{z^2}{\left(1 - \operatorname{sgn}\left(z\right)\varepsilon\right)^2} + 1 \right] w^2 \right\},$$

where *z* ∈ R, *w* > 0.

> It follows directly that

$$f\_{Z\_1}(z; \mathfrak{e}) = \int\_0^\infty f\_{Z\_1, W}(z, w) \, \mathrm{d}w = \frac{1}{\pi \left[1 + \left(\frac{z}{1 - \mathrm{sgn}(z)\mathfrak{e}}\right)^2\right]}$$

.

Figure 1 depicts the behavior of the ESC density for a variety of parameter values.

**Figure 1.** Examples of the *ESC*(*ε*) density for : *ε* = 0 (green line), *ε* = −0.5 (blue line), *ε* = −0.8 (black line), *ε* = 0.5 (red line) and *ε* = 0.8 (pink line).
