# Comparison and Classification of Flexible Distributions for Multivariate Skew and Heavy-Tailed Data

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## Abstract

**:**

## 1. Introduction

## 2. Spherically and Elliptically Symmetric Distributions

#### 2.1. Definition, Examples and Properties

#### 2.2. Modelling Limitations

## 3. Skew-Elliptical Distributions

#### 3.1. Multivariate Skew-t Distribution

## 4. (Multiple) Scale and Location-Scale Mixtures of Multinormal Distributions

#### 4.1. Scale Mixtures of Multinormal Distributions

#### 4.2. Location-Scale Mixtures of Multinormal Distributions

#### 4.3. Multiple Scale Mixtures of Multinormal Distributions

#### 4.4. Multiple Location-Scale Mixtures of Multinormal Distributions

## 5. Multivariate Distributions Obtained via the Transformation Approach

#### 5.1. Tukey’s Transformation

#### 5.2. SAS Transformation

## 6. Copula-Based Multivariate Distributions

- -
- A d-variate Archimedean copula is the function$$C({u}_{1},{u}_{2},\cdots ,{u}_{d})={\varphi}^{-1}(\varphi \left({u}_{1}\right)+\cdots +\varphi \left({u}_{d}\right)),\phantom{\rule{1.em}{0ex}}{({u}_{1},\dots ,{u}_{d})}^{\prime}\in {[0,1]}^{d},$$
- -
- As the terminology suggests, elliptical copulas are the copulas of elliptically symmetric distributions. From (9) one readily sees that an elliptical copula is expressed as $F({F}_{g}^{-1}\left({x}_{1}\right),\dots ,{F}_{g}^{-1}\left({x}_{d}\right))$ where F is the elliptical cdf and ${F}_{g}$ is the same symmetric marginal cdf for every component (see Section 2.1). They naturally inherit good properties as well as drawbacks from elliptical distributions. More detailed information about elliptical copulas is provided in the next section on Meta-Elliptical distributions as these are, de facto, nothing else but multivariate distributions obtained via elliptical copulas.

#### 6.1. Meta-Elliptical Distributions

#### 6.2. Our Proposal: t-Copula with SAS Marginals

## 7. Classification and Comparison of the Families

- Simulation: In order to be able to simulate data from every distribution, we need to have a stochastic representation that underpins a data generating mechanism. All families but copulas enjoy a stochastic representation, and all families can be simulated.
- Asymmetry: It is natural to require from a flexible distribution to be able to capture asymmetry, all the more as the symmetric situation is then contained as a special case. Spherically and elliptically symmetric distributions are, by nature, spherically and elliptically symmetric, respectively. Scale mixtures of multinormal distributions are also elliptically symmetric, while multiple scale mixtures of multinormal distributions and meta-elliptical distributions are centrally symmetric. All other families are able to model skew phenomena via some skewness parameter and hence fully satisfy this criterion.
- Tail: Like skewness, tail weight ought to be governed by at least a d-dimensional vector allowing for one parameter per dimension. This criterion is not satisfied by the spherically and elliptically symmetric distributions as they only contain a one-dimensional tail weight parameter, and consequently also not by the skew-elliptical distributions as they are directly derived from them. The same holds true for scale and location–scale mixtures of multinormal distributions. The remaining distributions can all be termed multi-tail.
- Tractable Density: It is desirable for a distribution to possess a tractable density, be it for the sake of understanding the roles of the parameters, the development of stochastic properties or for parameter estimation purposes. By tractable we mean a density that can be written out explicitly without involving complicated functions such as integrals, for example. The families of spherically and elliptically symmetric distributions can be considered as very tractable (although they do possess complicated special cases such as the scale mixtures of multinormal distributions and the elliptical $\alpha -$stable distributions). The same holds then true for the skew-elliptical distributions. The multivariate SAS distribution also possesses a tractable density. All other distributions do not have tractable densities.
- Tractable Marginal Distributions: Multivariate distributions should possess tractable marginal laws, as often one is interested not only in the combined behavior of various components, but also in their individual behavior. Similarly, tractable conditional distributions are an asset. However, we did not emphasize the latter as special requirement since a tractable density combined with tractable marginals inevitably leads to tractable conditional distributions, even if the latter are not necessarily of a well-known form. We have seen that spherically and elliptically symmetric distributions are closed under linear transformations, hence they do satisfy this requirement. The multivariate SAS distribution has SAS marginals and the meta-elliptical distributions, being based on copulas, allow by construction to have full control over the marginals. Skew-elliptical distributions also enjoy tractable marginals [30]. The only families of distributions not completely satisfying this criterion are the (multiple) scale and location–scale mixtures of multinormal distributions, where it is not difficult to sample from the marginals but computing their density involves numerical integration.
- Parameter Estimation: It is essential to correctly estimate the parameters of each distribution, in particular by means of maximum likelihood estimation (for the sake of brevity, we do not consider here other estimation methods such as moment-based or Bayesian methods). This ensures that the distributions can serve their purpose in other inferential endeavours such as hypothesis testing. All considered families satisfy this criterion, albeit some require more special care, such as for instance the multiple scale or multiple location–scale mixture of multinormal distributions where an EM algorithm is required, or the $\alpha -$stable distribution that requires the use of the fast Fourier transform. In case of meta-elliptical distributions, recently more efficient maximum likelihood estimation methods have been proposed [58] that decompose the estimation into marginal estimation and dependence structure estimation.

- spherically symmetric distributions (ST, SS)
- elliptically symmetric distributions and scale mixtures of multinormals (both ST, ES)
- skew-elliptical distributions and location–scale mixtures of multinormals (both ST, AS)
- multiple scale mixtures of multinormals and meta-elliptical distributions (both MT, CS)
- multiple location–scale mixtures of multinormals and SAS transformation of multinormal (both MT, AS)

## 8. Finite Sample Performance Comparison

- The distributions based on the multivariate t-copula yield in general the best fit.
- For the dimension $d=2$, the t-copula combined with SAS marginals outperforms its competitors, especially for large sample sizes, while for $d=5$ the meta-elliptical asymmetric t provides the best fit.
- For data generated under the $t-$ SAS distribution, the skew-t exhibits the best performance, which is in line with the fact that it can capture skewness as opposed to the meta-elliptical asymmetric t that is centrally symmetric. In dimension 2, the multiple scaled t-distribution even outperforms the meta-elliptical asymmetric t in this case.
- In most situations where the meta-elliptical asymmetric t is the best, the t-copula combined with SAS marginals has a nearly as good BIC value.
- In no setting does the multivariate t-distribution provide the best fit, nor does the multiple scaled t-distribution.
- The meta-elliptical asymmetric t and t-SAS are copula-based models and they are computationally demanding for large sample sizes and dimensions. Hence, they have large computational times. The same holds for the multiple scaled-t, as the parameters are estimated using the EM algorithm; it has the largest computational times in dimension 2, as mentioned earlier.

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Panel (

**a**) shows the contour plot of the bivariate standard normal distribution. Panel (

**b**) shows the contour plot of the bivariate t-distribution with $\mathit{\mu}=\mathbf{0},\mathsf{\Sigma}={\mathbf{I}}_{2}$ and $\nu =1$ degree of freedom.

**Figure 2.**Panel (

**a**) shows the contour plot of the bivariate standard skew-normal distribution with the skewness parameter $\mathit{\delta}={(1,2)}^{\prime}$. Panel (

**b**) shows the contour plot of the bivariate skew-t distribution with $\mathit{\mu}=\mathbf{0},\mathsf{\Sigma}={\mathbf{I}}_{2},\nu =1$ degree of freedom and with the skewness parameter $\mathit{\delta}={(1,2)}^{\prime}$.

**Figure 3.**Contour plots of bivariate $t-$distributions resulting from multiple scale mixtures of multinormal distributions with $\mathit{\mu}={(0,0)}^{\prime}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathbf{A}=\mathrm{diag}(4,4)$. For $\mathbf{D}$ a parameterization via an angle $\xi $ is used, so ${D}_{11}={D}_{22}=cos\left(\xi \right)\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{D}_{21}=-{D}_{12}=sin\left(\xi \right)$, where ${D}_{md}$ denotes the $(m,d)$ entry of the matrix $\mathbf{D}.$ Panel (

**a**) shows the contour plot for degrees of freedom $\nu =(0.2,0.5)$ and panel (

**b**) for degrees of freedom $\nu =(2,10)$, with in both cases $\xi =\pi /8$.

**Figure 4.**Panel (

**a**) shows the contour plot for the bivariate SAS with the skewness parameter $\mathbf{g}={(0,0)}^{\prime}$ and panel (

**b**) the contour plot for the bivariate SAS with $\mathbf{g}={(1,1)}^{\prime}$. For both cases the tail weight parameter is $\mathbf{h}={(0.5,0.5)}^{\prime}$ and the correlation coefficient is $\rho =0.7.$

**Figure 5.**Panel (

**a**) shows the contour plot for the bivariate asymmetric $t-$distribution with equal marginal degrees of freedom (${m}_{1}={m}_{2}=10$) and panel (

**b**) shows the contour plot for the bivariate asymmetric $t-$distribution with different marginal degrees of freedom (${m}_{1}=10$ and ${m}_{2}=0.5$). For both cases the degrees of freedom for the copula is $m=2$.

**Figure 6.**Contour plots for a bivariate t-copula with SAS marginals and for four combinations of the values of marginal SAS distributions. For all cases the degrees of freedom for the t-copula is $m=2.$

Family of Distributions | Simulation | Asymmetry | Tail | TD | TMD | PE |
---|---|---|---|---|---|---|

Spherical | ✓ | SS | ST | ✓ | ✓ | ✓ |

Elliptical | ✓ | ES | ST | ✓ | ✓ | ✓ |

Skew-elliptical | ✓ | AS | ST | ✓ | ✓ | ✓ |

Scale mixtures | ✓ | ES | ST | ✗ | ✗ | (✓) |

Location-scale mixtures | ✓ | AS | ST | ✗ | ✗ | (✓) |

Multiple scale mixtures | ✓ | CS | MT | ✗ | ✗ | (✓) |

Multiple location–scale mixtures | ✓ | AS | MT | ✗ | ✗ | (✓) |

SAS transformation of multinormal | ✓ | AS | MT | ✓ | ✓ | ✓ |

Meta-elliptical distributions | ✓ | CS | MT | ✗ | ✓ | (✓) |

Fitted Distribution | Data Generator | ||||
---|---|---|---|---|---|

t | Skew-t | Multiple Scaled-t | Meta Elliptical-t | t-SAS | |

n = 100 | |||||

t | 787.0882 | 1274.422 | 731.9593 | 918.0995 | |

(20.50 s) | ( 59.89 s) | (10.49 s) | (35.65 s) | ||

skew-t | 623.4148 | 1080.851 | 668.383 | 736.6848 | |

(3.43 s) | (3.86 s) | (3.09 s) | (4.20 s) | ||

multiple scaled-t | 620.7204 | 698.6299 | 667.6790 | 746.9414 | |

(4.07 min) | (3.54 min) | (3.63 min) | (4.12 min) | ||

meta elliptical-t | 617.5156 | 692.7963 | 1112.939 | 776.8259 | |

(12.10 s) | (1.02 min) | (4.29 min) | (3.10 min) | ||

t-SAS | 634.0599 | 694.6652 | 1075.957 | 667.4588 | |

(1.31 min) | (2.03 min) | (9.73 min) | (2.72 min) | ||

n = 1000 | |||||

t | 7346.343 | 11891.12 | 7849.712 | 9847.058 | |

(16.62 s) | (23.56 s) | (17.45 s) | (55.87 s) | ||

skew-t | 6066.006 | 10513.17 | 6620.216 | 7361.205 | |

(13.74 s) | (14.54 s) | (14.07 s) | (15.58 s) | ||

multiple scaled-t | 6091.425 | 6706.059 | 6585.36 | 7490.495 | |

(46.58 min) | (33.01 min) | (37.33 min) | (45.10 min) | ||

meta elliptical-t | 6054.726 | 6609.865 | 10,945.42 | 7855.342 | |

(7.31 min) | (1.44 min) | (1.34 min) | (10.26 min) | ||

t-SAS | 6095.659 | 6553.907 | 10,434.28 | 6539.835 | |

(19.46 min) | (11.83 min) | (32.67 min) | (11.94 min) | ||

n = 10,000 | |||||

t | 72,318.55 | 118,699.1 | 77,715.79 | 96,475.54 | |

(1.38 min) | (38.73 s) | (1.49 min) | (1.59 min) | ||

skew-t | 59,862.13 | 104,062.7 | 65,793.52 | 72,670.13 | |

(2.48 min) | (2.56 min) | ( 2.79 min) | (2.87 min) | ||

multiple scaled-t | 60,211.49 | 66,097.17 | 65,624.49 | 74,064.37 | |

(6.65 h) | (7.001 h) | (7.92 h) | (8.90 h) | ||

meta elliptical-t | 59,845.96 | 65,078.95 | 108,362.4 | 77,732.89 | |

(11.59 min) | (12.92 min) | (29.52 min) | (1.44 h) | ||

t-SAS | 60,115.14 | 64,506.98 | 103,324.6 | 64,959.53 | |

(2.12 h) | (1.96 h) | (10.17 h) | (1.81 h) |

Fitted Distribution | Data Generator | ||||
---|---|---|---|---|---|

t | Skew-t | Multiple Scaled-t | Meta Elliptical-t | t-SAS | |

n = 100 | |||||

t | 2139.904 | 2910.327 | 2431.283 | 2568.123 | |

(19.13 s) | (8.71 s) | (12.81 s) | (12.35 s) | ||

skew-t | 1541.473 | 3067.713 | 2367.529 | 2050.565 | |

( 10.19 s) | (10.67 s) | (11.59 s) | (13.90 s) | ||

meta elliptical-t | 1530.399 | 1591.337 | 2487.038 | 2275.882 | |

(4.05 min) | (3.49 min) | (3.97 min) | (4.09 min) | ||

t-SAS | 1581.714 | 1638.517 | 2509.191 | 2157.801 | |

(6.12 min) | (7.14 min) | (6.74 min) | (6.92 min) | ||

n = 1000 | |||||

t | 20,440.26 | 28,311.18 | 23,663.64 | 24,483.51 | |

(16.54 s) | (14.02 s) | (20.48 s) | (13.20 s) | ||

skew-t | 14,053.25 | 27,661.15 | 40,215.95 | 19,902.44 | |

(39.59 s) | (41.76 s) | (48.51 s) | (47.05 s) | ||

meta elliptical-t | 14,026.45 | 15,169.81 | 23,467.53 | 22,220.04 | |

(23.87 min) | (23.95 min) | (26.24 min) | (35.05 min) | ||

t-SAS | 14,161.15 | 15,367.83 | 23,475.92 | 21,409.75 | |

(53.33 min) | (55.97 min) | (56.59 min) | (1.01 h) | ||

n = 10,000 | |||||

t | 201,314.9 | 280,266.2 | 238,979.6 | 245,145.6 | |

(1.57 min) | (1.67 min) | (1.96 min) | (1.62 min) | ||

skew-t | 139,361.6 | 298,194.7 | 338,688.8 | 197,373.4 | |

(6.38 min) | (6.58 min) | (7.51 min) | (7.94 min) | ||

meta elliptical-t | 139,317.2 | 149,850.3 | 231,530.1 | 222,997.3 | |

(4.53 h) | (4.69 h) | (4.59 h) | (6.03 h) | ||

t-SAS | 140,195.5 | 152,159.8 | 231,812.7 | 212,482.6 | |

( 9.35 h) | (9.15 h) | (8.88 h) | (9.96 h) |

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**MDPI and ACS Style**

Babić, S.; Ley, C.; Veredas, D.
Comparison and Classification of Flexible Distributions for Multivariate Skew and Heavy-Tailed Data. *Symmetry* **2019**, *11*, 1216.
https://doi.org/10.3390/sym11101216

**AMA Style**

Babić S, Ley C, Veredas D.
Comparison and Classification of Flexible Distributions for Multivariate Skew and Heavy-Tailed Data. *Symmetry*. 2019; 11(10):1216.
https://doi.org/10.3390/sym11101216

**Chicago/Turabian Style**

Babić, Slađana, Christophe Ley, and David Veredas.
2019. "Comparison and Classification of Flexible Distributions for Multivariate Skew and Heavy-Tailed Data" *Symmetry* 11, no. 10: 1216.
https://doi.org/10.3390/sym11101216