Multivariate skew distributions with mode-invariance through the transformation of scale

Abstract

The skew-symmetric distribution is often-used as a skew distribution, but it is not always unimodal even when the underlying distribution is unimodal. Recently, another type of skew distribution was proposed using the transformation of scale (ToS). It is always unimodal and shows the monotonicity of skewness. In this paper, a multivariate skew distribution is considered using the ToS. The skewness for the multivariate skew distribution is proposed and the monotonicity of skewness is shown. The proposed multivariate skew dist ribution is more flexible than the conventional multivariate skew-symmetric distributions. This is illustrated in numerical examples. Additional properties are also presented, including random number generation, half distribution, parameter orthogonality, non-degenerated Fisher information, entropy maximization distribution.

Derivation of \(\partial \xi /\partial \lambda _j\) and \(\partial \xi /\partial a\)

The differential of a with respect to $\lambda _j$ is given by

$$\begin{aligned} \frac{\partial a}{\partial \lambda _j} = - \frac{\partial \xi }{\partial \lambda _j} \Bigg / \frac{\partial \xi }{\partial a} . \end{aligned}$$

The numerator of $\partial a/\partial \lambda _j$ is

$$\begin{aligned} \frac{\partial \xi }{\partial \lambda _j}=\frac{\partial }{\partial \lambda _j} f(\varvec{r}(a \varvec{z}_0;{\varvec{\lambda }})) = \frac{\partial f}{\partial \varvec{y}'} \frac{\partial \varvec{r}}{\partial \lambda _j}. \end{aligned}$$

Let $\varvec{r}(\varvec{x};{\varvec{\lambda }})=\varvec{y}({\varvec{\lambda }})$. We have

$$\begin{aligned} \varvec{x}=\varvec{s}(\varvec{y}({\varvec{\lambda }});{\varvec{\lambda }})=\varvec{y}({\varvec{\lambda }}) + \varvec{H}(\varvec{y}({\varvec{\lambda }});{\varvec{\lambda }}). \end{aligned}$$

The differential of the above with respect to $\lambda _j$ is

$$\begin{aligned} \varvec{0}= \frac{\partial \varvec{y}}{\partial \lambda _j} + \frac{\partial \varvec{H}}{\partial \varvec{y}'} \frac{\partial \varvec{y}}{\partial \lambda _j} + \frac{\partial \varvec{H}}{\partial \lambda _j}. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{\partial \varvec{y}}{\partial \lambda _j} = - \left( I + \frac{\partial \varvec{H}}{\partial \varvec{y}'}\right) ^{-1} \frac{\partial \varvec{H}}{\partial \lambda _j}. \end{aligned}$$

Therefore, the numerator of $\partial a_+/\partial \lambda _j$ is given by

$$\begin{aligned} \frac{\partial \xi }{\partial \lambda _j} = - \frac{\partial f}{\partial \varvec{y}'} \left( I + \frac{\partial \varvec{H}}{\partial \varvec{y}'}\right) ^{-1} \frac{\partial \varvec{H}}{\partial \lambda _j}. \end{aligned}$$

The denominator of $\partial a/\partial \lambda _j$ is

$$\begin{aligned} \frac{\partial \xi }{\partial a} = \frac{\partial f}{\partial \varvec{y}'} \frac{\partial \varvec{r}}{\partial \varvec{x}'} \varvec{z}_0. \end{aligned}$$

Let $\varvec{r}(\varvec{x};{\varvec{\lambda }})=\varvec{y}(\varvec{x})$. We have

$$\begin{aligned} \varvec{x}=\varvec{s}(\varvec{y}(\varvec{x});{\varvec{\lambda }})=\varvec{y}(\varvec{x}) + \varvec{H}(\varvec{y}(\varvec{x});{\varvec{\lambda }}). \end{aligned}$$

The differential of the above with respect to $\varvec{x}$ is

$$\begin{aligned} I= \frac{\partial \varvec{y}}{\partial \varvec{x}'} + \frac{\partial \varvec{H}}{\partial \varvec{y}'} \frac{\partial \varvec{y}}{\partial \varvec{x}'}. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{\partial \varvec{y}}{\partial \varvec{x}'}= \left( I + \frac{\partial \varvec{H}}{\partial \varvec{y}'} \right) ^{-1}. \end{aligned}$$

Therefore, the denominator of $\partial a_+/\partial \lambda _j$ is given by

$$\begin{aligned} \frac{\partial \xi }{\partial \lambda _j} = \frac{\partial f}{\partial \varvec{y}'} \left( I + \frac{\partial \varvec{H}}{\partial \varvec{y}'}\right) ^{-1} \varvec{z}_0. \end{aligned}$$