# A Selective Overview of Skew-Elliptical and Related Distributions and of Their Applications

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## Abstract

**:**

## 1. Context and Motivation

#### 1.1. The Wider Perspective

#### 1.2. The Specific Target

## 2. Background Concepts

#### 2.1. Basics of Elliptical Distributions

#### 2.2. The Multivariate Skew-Normal Distribution

#### 2.3. A General Result

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

#### 2.4. Symmetry-Modulated Distributions

**Proposition**

**3.**

## 3. SEC and Other Modulations of EC Distributions

#### 3.1. SEC Distributions with a Linear-Combination Argument

#### 3.2. SEC Distributions via Conditioning

#### 3.3. Constructions Based on Several Latent Variables

#### 3.4. Other Mixtures of SN Variables

#### 3.5. Miscellanea

#### 3.6. Cave Nomina

## 4. Applications as Methodological Developments

#### 4.1. Regression Models and Variants

#### 4.2. Finite Mixtures and Model-Based Clustering

#### 4.3. Spatial and Spatio-Temporal Models

#### 4.4. Methods for Biostatistics and Medical Statistics

#### 4.5. Methods for Observational Studies and the Social Sciences

#### 4.6. A Popular Benchmark

`R`package

`sn`which provides computational tools for SN and ST model fitting; see [128]. Given the purely numerical nature of these exercises, they cannot be described as ‘applied work’, but they provide illustrative examples for a wide range of methods. Notable references which make use of the AIS data include [10,11,19,32,53,55,59,68], but this list could be far longer. There are seldom cross-paper comparisons of the various methods that are applied to the AIS data.

`R`computing environment, it is appropriate to mention also similar tools for the Stata environment, presented in [129].

## 5. Applications to Real Problems

#### 5.1. Economics and Applied Financial Economics

#### 5.2. Quantitative Finance

#### 5.3. Risk

#### 5.4. Biology and Life Sciences

#### 5.5. Environmental Issues

#### 5.6. Industrial and Technological Applications

## 6. Two Illustrations in Applied Domains

#### 6.1. A Finance Application

#### 6.2. Stochastic Frontier Analysis: An Simple Case

`milkProd`available within the

`R`package

`Benchmarking`; see [228]. The data set refers to the diary industry; the production units are Danish milk producers. For each of $n=108$ units, four variables are available:

`milk`(the amount produced, in kg),

`energy`(energy expenses),

`vet`(veterinary expenses),

`cows`(number of cows).

`R`package

`sn`[128], after suitable conversion of the parameters. The fitted model is then

## 7. Final Comments

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Contour level plots of two bivariate skew-normal densities given by (8); in both cases, $\overline{\mathrm{\Omega}}$ is equal to the identity matrix, $\alpha $ is $(2,4)$ in the first plot, it is $(2,-6)$ in the second plot.

**Figure 2.**Examples of symmetry-modulated density generated from a baseline bivariate standard normal density with independent components.

**Figure 3.**Diagram representing the relative position of some selected classes of distributions. SUN = unified skew-normal; SEC = skew-elliptically contoured; SUEC = unified skew-elliptical; SN = skew-normal; MEC = modulated elliptical contoured.

**Figure 4.**Finance application.

**Left panel**: mean-variance-skewness efficient surface;

**right panel**: number of non-zero weights.

**Figure 5.**Fitting SFA models to milk production data.

**Left panel**: scatter plot of fitted and observed data;

**right panel**: fitted SN density (continuous blue line) and ST density (dashed magenta line).

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Adcock, C.; Azzalini, A.
A Selective Overview of Skew-Elliptical and Related Distributions and of Their Applications. *Symmetry* **2020**, *12*, 118.
https://doi.org/10.3390/sym12010118

**AMA Style**

Adcock C, Azzalini A.
A Selective Overview of Skew-Elliptical and Related Distributions and of Their Applications. *Symmetry*. 2020; 12(1):118.
https://doi.org/10.3390/sym12010118

**Chicago/Turabian Style**

Adcock, Chris, and Adelchi Azzalini.
2020. "A Selective Overview of Skew-Elliptical and Related Distributions and of Their Applications" *Symmetry* 12, no. 1: 118.
https://doi.org/10.3390/sym12010118