# A Gamma-Type Distribution with Applications

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Slashed Quasi-Gamma Distribution

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

- ${lim}_{q\to \infty}{f}_{T}(t;\beta ,\theta ,q)=\frac{\theta}{{\beta}^{\theta /10}\mathsf{\Gamma}\left(1/10\right)}{t}^{(\theta /10)-1}{e}^{-{\left(t/\beta \right)}^{\theta}}$,
- ${F}_{T}(t;\beta ,\theta ,q)=H\left({\left(t/\beta \right)}^{\theta},1/10,1\right)-(t/q){f}_{T}(t;\beta ,\theta ,q),$

**Proposition**

**2.**

**Proof.**

**Remark**

**1.**

**Corollary**

**1.**

**Corollary**

**2.**

**Remark**

**2.**

## 3. Inference

#### 3.1. Moment Estimators

**Proposition**

**3.**

**Proof.**

#### 3.2. Maximum Likelihood Estimators

#### 3.3. Observed Information Matrix

## 4. Simulation

## 5. Applications

#### 5.1. Application 1

#### 5.2. Application 2

## 6. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Meeker, W.; Escobar, L. Statistical Methods for Reliability Data; Wiley: New York, NY, USA, 1998. [Google Scholar]
- Kleiber, C.; Kotz, S. Statistical Size Distributions in Economics and Actuarial Sciences; Wiley-Interscience: Hoboken, NJ, USA, 2003; ISBN 0-471-15064-9. [Google Scholar]
- Gómez, H.W.; Quintana, F.A.; Torres, F.J. A new family of slash-distribution with elliptical contours. Stat. Probab. Lett.
**2008**, 77, 717–725. [Google Scholar] [CrossRef] - Arslan, O. An alternative multivariate skew-slash distribution. Stat. Probab. Lett.
**2008**, 78, 2756–2761. [Google Scholar] [CrossRef] - Gómez, H.W.; Olivares-Pacheco, J.F.; Venegas, O. An extension of the generalized Birnbaum–Saunders distribution. Stat. Probab. Lett.
**2009**, 79, 331–338. [Google Scholar] [CrossRef] - Olivares-Pacheco, J.F.; Cornide, H.; Monasterio, M. An extension of the two-parameter weibull distribution. Colomb. J. Stat.
**2010**, 33, 219–231. [Google Scholar] - Olmos, N.M.; Varela, H.; Gómez, H.W.; Bolfarine, H. An extension of the half-normal distribution. Stat. Pap.
**2012**, 53, 875–886. [Google Scholar] [CrossRef] - Olmos, N.M.; Varela, H.; Bolfarine, H.; Gómez, H.W. An extension of the generalized half-normal distribution. Stat. Pap.
**2014**, 55, 967–981. [Google Scholar] [CrossRef] - Iriarte, Y.A.; Gómez, H.W.; Varela, H.; Bolfarine, H. Slashed Rayleigh distribution. Colomb. J. Stat.
**2015**, 38, 31–44. [Google Scholar] [CrossRef] - Iriarte, Y.A.; Vilca, F.; Varela, H.; Gómez, H.W. Slashed Generalized Rayleigh distribution. Commun. Stat. Theory Methods
**2017**, 46, 4686–4699. [Google Scholar] [CrossRef] - Reyes, J.; Barranco-Chamorro, I.; Gallardo, D.I.; Gómez, H.W. Generalized Modified Slash Birnbaum–Saunders Distribution. Symmetry
**2018**, 10, 724. [Google Scholar] [CrossRef] [Green Version] - Casella, G.; Berger, R.L. Statistical Inference; Duxbury: Pacific Grove, CA, USA, 2002. [Google Scholar]
- R Core Team, R. A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2019; Available online: http://www.R-project.org/ (accessed on 4 March 2020).
- Andrews, D.F.; Herzberg, A.M. Data: A Collection of Problems from Many Fields for the Student and Research Worker; Springer Series in Statistics; Springer: New York, NY, USA, 1985. [Google Scholar]
- Barlow, R.E.; Toland, R.H.; Freeman, T. A Bayesian Analysis of Stress-Rupture Life of Kevlar 49/Epoxy Spherical Pressure Vessels. In Proceedings of the Canadian Conference in Application of Statistics; Marcel Dekker: New York, NY, USA, 1984. [Google Scholar]
- Salinas, H.; Iriarte, Y.; Astorga, J. A transmuted version of the generalized half-normal distribution. Proyecciones (Antofagasta)
**2019**, 38, 567–583. [Google Scholar] [CrossRef] [Green Version] - Reyes, J.; Iriarte, Y.; Jodrá, P.; Gómez, H.W. The Slash Lindley-Weibull Distribution. Methodol. Comput. Appl. Probab.
**2019**, 21, 235–251. [Google Scholar] [CrossRef] - Martínez-Flórez, G.; Bolfarine, H.; Gómez, H.W. An alpha-power extension for the Birnbaum–Saunders distribution. Statistics
**2014**, 48, 896–912. [Google Scholar] [CrossRef] - Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control
**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.; Smyth, G. Randomized Quantile Residual. J. Comput. Graph. Stat.
**1996**, 5, 236–244. [Google Scholar]

**Figure 1.**Curves of the density function of the generalized gamma distribution (GENG) distribution for $\beta =1$, $k=0.1$ and different values of $\theta $.

**Figure 6.**Residual quantile for the fitted models to life of fatigue fracture data set. The p-values for the Anderson–Darling (AD), Cramer–Von Mises (CVM), and Shapiro–Wilk (SW) normality tests are also presented to check if the residual quantile came from the standard normal distribution. At a 5% significance level, for the three normality tests for the residual quantile, results suggest that the SQG model is appropriate for this data set while the rest of the models are not.

**Figure 9.**Residual quantile for the fitted models in the concentration of the $N{O}_{2}$ data set. The p-values for the AD, CVM, and SW normality tests are also presented to check if the residual quantile came from the standard normal distribution. At a 5% significance level, for the three normality tests for the residual quantile, results suggest that the SQG model is appropriate for this data set while the rest of the models are not.

**Table 1.**Maximum likelihood (ML) estimates for parameters $\beta $, $\theta $, and q of the SQG distribution.

True Value | $\mathit{n}=50$ | ||||
---|---|---|---|---|---|

$\mathbf{\beta}$ | $\mathbf{\theta}$ | $\mathit{q}$ | $\widehat{\mathbf{\beta}}\left(\mathbf{SD}\right)$ | $\widehat{\mathbf{\theta}}\left(\mathbf{SD}\right)$ | $\widehat{\mathit{q}}\left(\mathbf{SD}\right)$ |

3 | 5 | 1 | 3.346(1.427) | 5.201(0.939) | 1.191(0.513) |

2 | 3.105(0.898) | 5.263(0.904) | 2.356(1.126) | ||

3 | 2.909(0.712) | 5.248(0.867) | 3.303(1.212) | ||

5 | 7 | 1 | 5.363(1.842) | 7.380(1.572) | 1.136(0.477) |

2 | 5.140(1.204) | 7.246(1.115) | 2.356(1.006) | ||

3 | 4.976(1.026) | 7.271(1.060) | 3.368(1.477) | ||

10 | 10 | 1 | 10.475(2.780) | 10.593(2.430) | 1.083(0.293) |

2 | 10.190(1.965) | 10.626(2.052) | 2.318(0.949) | ||

3 | 10.016(1.637) | 10.539(1.997) | 3.418(1.307) | ||

True Value | $\mathbf{n}=\mathbf{100}$ | ||||

$\mathbf{\beta}$ | $\mathbf{\theta}$ | $\mathit{q}$ | $\widehat{\mathbf{\beta}}\left(\mathbf{SD}\right)$ | $\widehat{\mathbf{\theta}}\left(\mathbf{SD}\right)$ | $\widehat{\mathit{q}}\left(\mathbf{SD}\right)$ |

3 | 5 | 1 | 3.050(0.829) | 5.141(0.660) | 1.053(0.238) |

2 | 3.087(0.654) | 5.087(0.581) | 2.262(0.773) | ||

3 | 3.062(0.573) | 5.110(0.599) | 3.223(1.058) | ||

5 | 7 | 1 | 5.099(1.100) | 7.238(1.038) | 1.034(0.195) |

2 | 5.080(0.837) | 7.169(0.907) | 2.167(0.587) | ||

3 | 5.088(0.761) | 7.166(0.844) | 3.210(1.122) | ||

10 | 10 | 1 | 10.173(1.859) | 10.376(1.697) | 1.041(0.169) |

2 | 10.187(1.422) | 10.165(1.355) | 2.153(0.530) | ||

3 | 10.141(1.201) | 10.200(1.231) | 3.369(1.007) | ||

True Value | $\mathbf{n}=\mathbf{200}$ | ||||

$\mathbf{\beta}$ | $\mathbf{\theta}$ | $\mathit{q}$ | $\widehat{\mathbf{\beta}}\left(\mathbf{SD}\right)$ | $\widehat{\mathbf{\theta}}\left(\mathbf{SD}\right)$ | $\widehat{\mathit{q}}\left(\mathbf{SD}\right)$ |

3 | 5 | 1 | 3.017(0.588) | 5.069(0.469) | 1.038(0.156) |

2 | 3.061(0.449) | 5.038(0.409) | 2.143(0.482) | ||

3 | 3.050(0.401) | 5.035(0.395) | 3.017(1.019) | ||

5 | 7 | 1 | 5.078(0.786) | 7.039(0.677) | 1.024(0.132) |

2 | 5.032(0.557) | 7.065(0.582) | 2.073(0.349) | ||

3 | 5.075(0.516) | 7.014(0.537) | 3.178(0.891) | ||

10 | 10 | 1 | 10.050(1.215) | 10.207(1.081) | 1.017(0.108) |

2 | 10.077(0.928) | 10.057(0.897) | 2.053(0.298) | ||

3 | 10.064(0.794) | 10.124(0.884) | 3.148(0.569) |

Dataset | n | $\overline{\mathit{X}}$ | ${\mathit{S}}^{2}$ | $\sqrt{{\mathit{b}}_{1}}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|---|

Rupture stress times | 101 | $1.025$ | $1.253$ | $3.047$ | $18.475$ |

**Table 3.**ML estimates with their respective standard errors (SE) for fitting models to the data set.

Estimated | TGHN | LW | EXPBSn | SQG |
---|---|---|---|---|

$\beta $ | - | 0.622 (1.021) | 3.420 (0.666) | 1.541 (0.170) |

$\theta $ | - | 0.837 (0.109) | - | 7.063 (0.771) |

q | - | - | - | 2.471 (0.535) |

$\sigma $ | 2.094 (0.330) | - | - | - |

$\alpha $ | 0.811 (0.064) | 1.046 (1.168) | 0.055 (0.025) | - |

$\lambda $ | 0.854 (0.168) | - | 3.661 (0.697) | - |

LLF | −102.277 | −102.597 | −100.692 | −98.669 |

AIC | 210.554 | 211.195 | 207.385 | 203.338 |

BIC | 218.399 | 219.040 | 215.230 | 211.183 |

Dataset | n | $\overline{\mathit{X}}$ | ${\mathit{S}}^{2}$ | $\sqrt{{\mathit{b}}_{1}}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|---|

Logarithm of the $N{O}_{2}$ | 500 | $3.698$ | $0.563$ | $-0.549$ | $3.748$ |

Estimated | TGHN | LW | EXPBSn | SQG |
---|---|---|---|---|

$\beta $ | - | 9.646 (3.752) | 0.060 (0.010) | 4.281 (0.036) |

$\theta $ | - | 5.723 (0.195) | - | 44.664 (2.214) |

q | - | - | - | 15.710 (1.586) |

$\sigma $ | 3.738 (0.053) | - | - | - |

$\alpha $ | 3.120 (0.163) | 157.002 (345.840) | 5.199 (0.110) | - |

$\lambda $ | −0.892 (0.082) | - | 0.033 (0.012) | - |

LLF | −555.785 | −557.436 | −591.954 | −530.462 |

AIC | 1117.571 | 1120.872 | 1189.908 | 1066.924 |

BIC | 1130.215 | 1133.516 | 1202.552 | 1079.568 |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Iriarte, Y.A.; Varela, H.; Gómez, H.J.; Gómez, H.W.
A Gamma-Type Distribution with Applications. *Symmetry* **2020**, *12*, 870.
https://doi.org/10.3390/sym12050870

**AMA Style**

Iriarte YA, Varela H, Gómez HJ, Gómez HW.
A Gamma-Type Distribution with Applications. *Symmetry*. 2020; 12(5):870.
https://doi.org/10.3390/sym12050870

**Chicago/Turabian Style**

Iriarte, Yuri A., Héctor Varela, Héctor J. Gómez, and Héctor W. Gómez.
2020. "A Gamma-Type Distribution with Applications" *Symmetry* 12, no. 5: 870.
https://doi.org/10.3390/sym12050870