The Kendall and Spearman rank correlations of the bivariate skew normal distribution

Abstract

We derive the Kendall and Spearman rank correlation coefficients of the bivariate skew normal (SN) distribution. For a given correlation parameter, we provide conditions on the shape parameters, under which the SN is more dependent than the normal in terms of each of the two-rank correlations. We further show how our results can be used for rank-based estimation procedures of the correlation parameter and the equal shape parameter of the SN, whose consistency and asymptotic normality we establish.

Higher‐order functional connectivity analysis of resting‐state functional magnetic resonance imaging data using multivariate cumulants

“…where Z 1 and Z 2 are independent and standard-normally distributed and σ U ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi π= π À 2δ 2 À Á q (Henze, 1986).…”

A Bayesian approach for mixed effects state‐space models under skewness and heavy tails

“…A univariate random variable X is said to follow an SN distribution, $$X\sim \mathcal {SN}(\zeta ,\omega ^2,\lambda )$$, with location, scale, and shape parameters given by ζ, ω 2 , and λ, respectively, if the density of this distribution has the form, $$$$\begin{eqnarray*} p{\left(x \mid \zeta ,\omega ^2,\lambda \right)}&=& \frac{2}{\omega }\phi {\left(\frac{x-\zeta }{\omega }\right)}\Phi {\left(\frac{\lambda }{\omega }(x-\zeta ) \right)}, \end{eqnarray*}$$$$where $$\phi (\cdot )$$ and $$\Phi (\cdot )$$ are, respectively, the probability density function (pdf) and the cumulative distribution function (cdf) of the standard normal distribution (Azzalini, 1985, 2005). Azzalini (1986) and Henze (1986) proposed the following stochastic representation of the SN distribution, which is helpful for states estimation, $$$$\begin{eqnarray*} X&\stackrel{d}{=}&\zeta +\omega \delta W+\omega \sqrt {1-\delta ^2}\varepsilon , \end{eqnarray*}$$$$where $$\delta =\lambda / \sqrt {1+\lambda ^2}$$, …”

“…where 𝜙(⋅) and Φ(⋅) are, respectively, the probability density function (pdf) and the cumulative distribution function (cdf) of the standard normal distribution (Azzalini, 1985(Azzalini, , 2005. Azzalini (1986) and Henze (1986) proposed the following stochastic representation of the SN distribution, which is helpful for states estimation,…”

A semiparametric isotonic regression model for skewed distributions with application to DNA–RNA–protein analysis

“…where 𝜙(⋅) and Φ(⋅) are the standard normal density and distribution functions, respectively, and 𝜂 ∈ ℝ is the shape parameter that accounts for the skewness. Since its introduction, the SN distribution has been extensively investigated with respect to its statistical properties and extended (e.g., into the generalized skew-elliptical distributions) (Henze, 1986;Azzalini and Capitanio, 1999;Azzalini, 2005). In general, SN likelihood equations do not have explicit solutions for obtaining the MLEs of the model parameters and numeric methods such as the EM algorithm are needed (e.g., see Lin et al, 2007).…”

“…The standard SN density function has the following form $$$$\begin{equation*} f(w) = 2 \phi (w) \Phi (\eta w), \end{equation*}$$$$where $$\phi (\cdot )$$ and $$\Phi (\cdot )$$ are the standard normal density and distribution functions, respectively, and $$\eta \in \mathbb {R}$$ is the shape parameter that accounts for the skewness. Since its introduction, the SN distribution has been extensively investigated with respect to its statistical properties and extended (e.g., into the generalized skew‐elliptical distributions) (Henze, 1986; Azzalini and Capitanio, 1999; Azzalini, 2005). In general, SN likelihood equations do not have explicit solutions for obtaining the MLEs of the model parameters and numeric methods such as the EM algorithm are needed (e.g., see Lin et al.…”