# Multivariate Classes of GB2 Distributions with Applications

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## Abstract

**:**

## 1. Introduction

## 2. The GB2 Distribution

#### 2.1. Representations of the GB2 Distribution

#### 2.2. Previous Work about Multivariate GB2 Distributions

## 3. Multivariate Distributions with GB2 Marginals

#### 3.1. Multivariate GB2 Income Distributions with p Fixed

#### 3.2. Multivariate GB2 Distributions with q Fixed

#### 3.3. More Multivariate Distributions

## 4. A General Multivariate GB2 Distribution

#### 4.1. The Model

#### 4.2. The Bivariate Case

**Theorem**

**1.**

**Proof of Theorem**

**1.**

**Theorem**

**2.**

**Proof of Theorem**

**2.**

**Lemma**

**1.**

**Proof of Lemma**

**1.**

**Theorem**

**3.**

**Proof of Theorem**

**3.**

**Theorem**

**4.**

**Proof of Theorem**

**4.**

## 5. Multivariate GB2 Distribution with Support above the Diagonal

#### Marginal and Conditional Distributions

## 6. Applications

#### 6.1. Modeling Bivariate Income Distributions

#### 6.2. Modeling Compound Precipitation and Wind Events

#### 6.2.1. Data

#### 6.2.2. Methods

`R`software function

`optimx`, with the limited memory quasi-Newton L-BFGS-B algorithm, considering the $pr$ variable expressed in ${10}^{-4}$ kg·m${}^{-2}$·s${}^{-1}$ and the $sfcWind$ variable expressed in m·s${}^{-1}$. We also calculated the value of the Akaike information criterion ($AIC$), given by following expression [64]

#### 6.2.3. Results

## 7. Conclusions and Future Research

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Some Results about Whittaker’s and Hypergeometric Functions

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**Figure 1.**Joint pdf and contour plots of the Bivariate GB2 distributions with support above than diagonal, with ${\sigma}_{i}=1$, $i=1,2$ and ${p}_{i}=2$, $i=1,2,3$.

**Figure 2.**Joint pdf and contour plots of the bivariate income Singh–Maddala distribution for the years: $(1960,1966)$ (

**top**); and $(1966,1972)$ (

**bottom**).

**Figure 3.**Daily time series of precipitation ($pr$) and near-surface wind speed ($sfcWind$) over four selected grid points (see Table 1), using EWEMBI data, from 1 January 2007 to 31 December 2016.

**Figure 4.**Contour plots and joint pdf of the bivariate GB2 distribution with q fixed, corresponding to the four grid points indicated in Table 1.

**Table 1.**Selected locations, their coordinates, the coordinates of the grid points chosen in our dataset and their probability of compound event (simultaneous precipitation and wind events, over the thresholds of 0.1 mm·d${}^{-1}$ and 0.01 m·s${}^{-1}$, in the period 2007–2016).

Location | Coordinates | Grid Point | Probability of |
---|---|---|---|

Coordinates | Compound Event | ||

Alhambra, Granada, Spain | ${37.17627}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{3.58810}^{\circ}$ W | ${37.25}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{3.75}^{\circ}$ W | 0.2935 |

Sagrada Familia, Barcelona, Spain | ${41.40404}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{2.17443}^{\circ}$ E | ${41.75}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{1.75}^{\circ}$ E | 0.3452 |

Statue of Liberty, New York City, USA | ${40.68944}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{74.04453}^{\circ}$ W | ${40.75}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{74.25}^{\circ}$ W | 0.4610 |

Taipei World Financial Center, Taiwan | ${25.03428}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{121.56450}^{\circ}$ E | ${24.75}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{121.75}^{\circ}$ E | 0.8555 |

**Table 2.**Parameter estimates and $AIC$ statistics, from the bivariate GB2 model with q fixed (Equation (11)), to the four bivariate truncated datasets by maximum likelihood, with standard errors in parenthesis ($pr$ expressed in ${10}^{-4}$ kg·m${}^{-2}$·s${}^{-1}$ and $sfcWind$ expressed in m·s${}^{-1}$).

Location | ${\widehat{\mathbf{q}}}_{0}$ | ${\widehat{\mathbf{p}}}_{1}$ | ${\widehat{\mathbf{p}}}_{2}$ | ${\widehat{\mathbf{a}}}_{1}$ | ${\widehat{\mathbf{a}}}_{2}$ | ${\widehat{\mathit{\sigma}}}_{1}$ | ${\widehat{\mathit{\sigma}}}_{2}$ | $\mathit{AIC}$ |
---|---|---|---|---|---|---|---|---|

Granada | 16.3187 | 3.0369 | 7.9799 | 0.4702 | 1.1355 | 12.1607 | 5.3893 | −3992.44 |

(${37.25}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{3.75}^{\circ}$ W) | (2.8526) | (1.0185) | (2.4878) | (0.0783) | (0.1419) | (4.0787) | (1.2403) | |

Barcelona | 16.6511 | 2.0810 | 11.7500 | 0.5627 | 0.9280 | 16.4814 | 3.9937 | −5004.23 |

(${41.75}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{1.75}^{\circ}$ E) | (2.1999) | (0.4537) | (3.1215) | (0.0626) | (0.0878) | (4.0430) | (0.9927) | |

New York | 13.8062 | 2.5009 | 6.7216 | 0.4862 | 1.3179 | 16.5038 | 5.6226 | −7793.07 |

(${40.75}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{74.25}^{\circ}$ W) | (1.8616) | (0.5979) | (1.3626) | (0.0587) | (0.1096) | (4.0350) | (0.8037) | |

Taiwan | 11.0710 | 2.1498 | 2.3624 | 0.6120 | 2.0889 | 11.1861 | 5.4784 | −14,855.81 |

(${24.75}^{\circ}$ N $\phantom{\rule{0.277778em}{0ex}}{121.75}^{\circ}$ E) | (2.3232) | (0.3306) | (0.4214) | (0.0542) | (0.2086) | (3.9155) | (0.5138) |

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**MDPI and ACS Style**

Sarabia, J.M.; Jordá, V.; Prieto, F.; Guillén, M.
Multivariate Classes of GB2 Distributions with Applications. *Mathematics* **2021**, *9*, 72.
https://doi.org/10.3390/math9010072

**AMA Style**

Sarabia JM, Jordá V, Prieto F, Guillén M.
Multivariate Classes of GB2 Distributions with Applications. *Mathematics*. 2021; 9(1):72.
https://doi.org/10.3390/math9010072

**Chicago/Turabian Style**

Sarabia, José María, Vanesa Jordá, Faustino Prieto, and Montserrat Guillén.
2021. "Multivariate Classes of GB2 Distributions with Applications" *Mathematics* 9, no. 1: 72.
https://doi.org/10.3390/math9010072