# A Note on Pareto-Type Distributions Parameterized by Its Mean and Precision Parameters

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Pareto-Type Distributions with Alternative Parameterizations

#### 2.1. Pareto Distribution

#### 2.2. Power Function Distribution

#### 2.3. Lomax Distribution

#### 2.4. Generalized Pareto Distribution

#### 2.5. Other Models Parameterized in Terms of the Mean and Precision Parameters

## 3. Modelling and Inference

- $exp\left({\beta}_{0}\right)$ represents the mean of the response variable when all the covariates are equal to 0. Of course, this interpretation is valid as long as it makes sense.
- $exp\left({\beta}_{j}\right)$, $j=1,\dots ,p$ represents the increment (in percentage terms) when the j-th covariates increased in 1 unit and the others are fixed.

#### A Simulation Study

## 4. Real-World Data Analysis

#### 4.1. Lomax Regression Model

#### 4.2. Pareto Regression Model

## 5. More Concluding Remarks and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Pareto, V. Cours d’economie Politique; F. Rouge: Lausanne, Switzerland, 1987; Volume II. [Google Scholar]
- Wang, X.; Li, X. Generalized Confidence Intervals for Zero-Inflated Pareto Distribution. Mathematics
**2021**, 9, 3272. [Google Scholar] [CrossRef] - Shrahili, M.; Al-Omari, A.I.; Alotaibi, N. Acceptance Sampling Plans from Life Tests Based on Percentiles of New Weibull–Pareto Distribution with Application to Breaking Stress of Carbon Fibers Data. Processes
**2021**, 9, 2041. [Google Scholar] [CrossRef] - Sharpe, J.; Juárez, M.A. Estimation of the Pareto and related distributions—A reference-intrinsic approach. Commun. Stat.-Theory Methods
**2021**. [Google Scholar] [CrossRef] - Arnold, B.C. Pareto Distribution; International Cooperative Publishing House: Burtonsville, MD, USA, 1983. [Google Scholar]
- Lomax, K.S. Business Failures: Another Example of the Analysis of Failure Data. J. Am. Assoc.
**1954**, 49, 847–852. [Google Scholar] [CrossRef] - Pickands, J. Statistical inference using extreme order statistics. Ann. Stat.
**1975**, 3, 119–131. [Google Scholar] - Bourguignon, M.; do Nascimento, F.F. Regression models for exceedance data: A new approach. Stat. Methods Appl.
**2021**, 30, 157–173. [Google Scholar] [CrossRef] - Stasinopoulos, M.; Rigby, R. gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape. 2021 R Package Version 5.3-2. Available online: https://CRAN.R-project.org/package=gamlss.dist (accessed on 25 December 2021).
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2021; Available online: https://www.R-project.org/ (accessed on 25 December 2021).
- Santos-Neto, M.; Cysneiros, F.J.A.; Leiva, V. and Barros, M. On a reparameterized Birnbaum–Saunders distribution and its moments, estimation and applications. REVSTAT–Stat. J.
**2021**, 12, 247–272. [Google Scholar] - Gómez, Y.M.; Gallardo, D.I.; De Castro, M. A regression model for positive data based on the slashed half-normal distribution. REVSTAT–Stat. J.
**2021**, 19, 553–573. [Google Scholar] - Akaike, H. A new look at the statistical model identification. IEEE Trans. Auto Contr.
**1974**, 19, 716–723. [Google Scholar] [CrossRef] - Schwarz, G. Estimating the dimension of a model. Ann. Stat.
**1978**, 6, 461–464. [Google Scholar] [CrossRef] - Dunn, P.K.; Smyth, G.K. Randomized quantile residuals. J. Comput. Graph. Stat.
**1996**, 5, 236–244. [Google Scholar]

**Table 1.**Estimated bias, standard errors and coverage probabilities for the estimators in the reparametrized Lomax model.

$\mathit{n}=50$ | $\mathit{n}=100$ | $\mathit{n}=200$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Scenario | Estimator | bias | se | cp | bias | se | cp | bias | se | cp |

1 | ${\beta}_{0}$ | −0.0604 | 0.4442 | 0.909 | −0.0481 | 0.2941 | 0.941 | −0.0254 | 0.2046 | 0.947 |

${\beta}_{1}$ | −0.0326 | 0.6318 | 0.929 | −0.0125 | 0.4417 | 0.935 | −0.0052 | 0.3351 | 0.941 | |

${\nu}_{0}$ | 0.0669 | 0.6213 | 0.968 | 0.0407 | 0.5665 | 0.961 | 0.0316 | 0.4049 | 0.955 | |

${\nu}_{1}$ | 0.0934 | 0.5640 | 0.960 | 0.0689 | 0.4717 | 0.958 | 0.0283 | 0.3449 | 0.953 | |

2 | ${\beta}_{0}$ | −0.0992 | 0.4245 | 0.909 | −0.0616 | 0.2895 | 0.930 | −0.0427 | 0.1885 | 0.949 |

${\beta}_{1}$ | 0.0562 | 0.6192 | 0.920 | 0.0487 | 0.4412 | 0.932 | 0.0308 | 0.3064 | 0.946 | |

${\nu}_{0}$ | 0.0634 | 0.9408 | 0.938 | 0.0417 | 0.7089 | 0.941 | 0.0307 | 0.6292 | 0.945 | |

${\nu}_{1}$ | 0.0764 | 0.6649 | 0.936 | 0.0658 | 0.4683 | 0.943 | 0.0357 | 0.3787 | 0.946 |

RGa | RLo | |||
---|---|---|---|---|

Estimate | se | Estimate | se | |

${\beta}_{0}$ | 4.7518 | 0.1198 | 4.7114 | 0.1572 |

${\beta}_{1}$ | −0.0288 | 0.0100 | −0.0260 | 0.0108 |

${\nu}_{0}$ | −0.1212 | 0.0915 | 1.2420 | 0.9530 |

${\nu}_{1}$ | 0.0041 | 0.0079 | 0.0187 | 0.0732 |

AIC | 958.01 | 953.79 | ||

BIC | 967.83 | 963.61 |

RWe | RPa | |||
---|---|---|---|---|

Estimate | se | Estimate | se | |

${\beta}_{0}$ | 3.7712 | 0.3870 | 3.7549 | 0.8921 |

${\beta}_{1}$ | −0.0239 | 0.0080 | −0.0228 | 0.0175 |

${\nu}_{0}$ | −0.9357 | 1.1873 | −8.5153 | 1.3215 |

${\nu}_{1}$ | 0.0487 | 0.0255 | 0.1982 | 0.0132 |

AIC | 345.11 | 326.47 | ||

BIC | 353.48 | 334.84 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bourguignon, M.; Gallardo, D.I.; Gómez, H.J.
A Note on Pareto-Type Distributions Parameterized by Its Mean and Precision Parameters. *Mathematics* **2022**, *10*, 528.
https://doi.org/10.3390/math10030528

**AMA Style**

Bourguignon M, Gallardo DI, Gómez HJ.
A Note on Pareto-Type Distributions Parameterized by Its Mean and Precision Parameters. *Mathematics*. 2022; 10(3):528.
https://doi.org/10.3390/math10030528

**Chicago/Turabian Style**

Bourguignon, Marcelo, Diego I. Gallardo, and Héctor J. Gómez.
2022. "A Note on Pareto-Type Distributions Parameterized by Its Mean and Precision Parameters" *Mathematics* 10, no. 3: 528.
https://doi.org/10.3390/math10030528