# A More Flexible Extension of the Fréchet Distribution Based on the Incomplete Gamma Function and Applications

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## Abstract

**:**

## 1. Introduction

- ${F}_{X}(x;\alpha )=exp\left\{-{x}^{-\alpha}\right\}$, where ${F}_{X}(\xb7)$ is the cumulative distribution function of X.
- $Q\left(p\right)={\left(-log\left(p\right)\right)}^{-1/\alpha},\phantom{\rule{1.em}{0ex}}0<p<1.$ where $Q(\xb7)$ is the quantile function of X.
- $E\left({X}^{r}\right)=\mathrm{\Gamma}\left(1-\frac{r}{\alpha}\right)$, $r=1,2,3,\dots ,$ is the r-th moment of X.

## 2. The Slash Fréchet Distribution

#### 2.1. Density Function

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 2.2. Properties

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

#### 2.3. Moments

**Proposition**

**6.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Proof.**

#### 2.4. Some Mathematical Properties

**Proposition**

**9.**

**Proof.**

**Proposition**

**10.**

**Proof.**

**Proposition**

**11.**

**Proof.**

## 3. Estimation

#### 3.1. Moment Estimators

**Proposition**

**12.**

**Proof.**

`“uniroot”`function of the R software, we obtain ${\widehat{\alpha}}_{M}$; replacing ${\widehat{\alpha}}_{M}$ in Equation (6), we obtain ${\widehat{q}}_{M}$. □

#### 3.2. Maximum Likelihood Estimators

#### 3.3. Simulation Study

- Generate $W\sim U(0,1).$
- Compute $X={\left(-log\left(W\right)\right)}^{-1/\alpha}.$
- Generate $U\sim U(0,1).$
- Compute $Y={\displaystyle \frac{X}{{U}^{1/q}}}.$

## 4. Applications

#### 4.1. Application 1 (Patients with Lung Cancer)

`survival`R package [16], labeled as

`veteran`.

#### 4.2. Application 2 (Patients with Peritoneal Dialysis)

#### 4.3. Application 3 (Patients with Breast Cancer)

`survival`” with the database “

`gbsg`”.

## 5. Conclusions

- A new extension of the Fréchet distribution with the density function, cumulative distribution function, survival function, and hazard function is obtained explicitly (closed) in terms of the incomplete gamma function.
- The moments, expectations, and variances of this new distribution were obtained, leading to closed expressions for all of them.
- By observing the skewness and kurtosis coefficients, it can be seen that the SFr model is more flexible than the Fr model. Furthermore, as shown in Table 1, the tails of the distribution become heavier when parameter q is smaller.
- Analyzing the stochastic representation of the SFr model, it is observed that the SFr distribution is a scale mixture of the Fr and $U(0,1)$ distribution.
- In the simulation study, it is observed that as the sample size increases, the maximum likelihood estimators are closer to the parameter values, suggesting consistent and stable estimators.
- In applications with real data, the SFr distribution demonstrates superior fits compared to the Fr model and other slash distributions, because it has lower values in the AIC, BIC, CAIC, and HQIC criteria.
- In future research, we plan to work on a new extension of the Fr distribution that is more flexible in the kurtosis coefficient than the SFr distribution. We will use this distribution in regression problems and survival analyses.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Graphical comparison of the density function of the Fréchet (Fr) and slash Fréchet (SFr) distributions for fixed alpha ($\alpha =1$) and different values of q.

**Figure 2.**Graphical comparison of the CDF between the Fréchet (Fr) and slash Fréchet (SFr) distribution for the fixed alpha ($\alpha =1$) and different values of q.

**Figure 3.**Graphs of the survival function and hazard function for the SFr distribution with $\alpha =2$ and different values of q.

**Figure 4.**Skewness coefficient plot of the SFr model (

**left side**). Comparison of the skewness coefficient between SFr and Fr for different values of q (

**right side**).

**Figure 5.**Plot of the kurtosis coefficient for the SFr model (

**left side**). Comparison of the kurtosis coefficient between the SFr and Fr models for different values of q (

**right side**).

**Figure 8.**Density adjusted for the dataset of patients undergoing lung cancer in the Fr and SFr distributions.

**Figure 11.**Density adjusted to the dataset of patients undergoing peritoneal dialysis in the Fr, SPN, and SFr distributions.

**Figure 12.**QQ plots for the dataset of patients undergoing peritoneal dialysis: (

**a**) Fr model; (

**b**) SPN model; (

**c**) SFr model.

**Figure 13.**Profile log-likelihoods of $\alpha $ and q for the dataset of patients undergoing peritoneal dialysis.

**Figure 15.**Density adjusted for the dataset of patients undergoing breast cancer in the Fr, SHN, and SFr distributions.

**Figure 16.**QQ plots for the dataset of patients undergoing breast cancer: (

**a**) Fr model; (

**b**) SHN model (

**c**) SFr model.

**Table 1.**Comparison of values of the survival function between the SFr and Fr distributions for $\alpha =1$ and q = 1, 3, 5, 10.

$\mathit{P}(\mathit{Y}>10)$ | $\mathit{P}(\mathit{Y}>11)$ | $\mathit{P}(\mathit{Y}>12)$ | $\mathit{P}(\mathit{Y}>13)$ | $\mathit{P}(\mathit{Y}>14)$ | $\mathit{P}(\mathit{Y}>15)$ |
---|---|---|---|---|---|

Fr (1) | 0.0952 | 0.0869 | 0.0800 | 0.0740 | 0.0689 |

SFr (1, 10) | 0.1051 | 0.0960 | 0.0884 | 0.0819 | 0.0763 |

SFr (1, 5) | 0.1171 | 0.1070 | 0.0986 | 0.0914 | 0.0852 |

SFr (1, 3) | 0.1368 | 0.1253 | 0.1157 | 0.1074 | 0.1002 |

SFr (1, 1) | 0.2775 | 0.2605 | 0.2457 | 0.2327 | 0.2212 |

n | $\mathit{\alpha}$ | q | $\widehat{\mathit{\alpha}}$ | $\mathbf{sd}\left(\widehat{\mathit{\alpha}}\right)$ | $\mathit{C}\left(\widehat{\mathit{\alpha}}\right)$ | $\widehat{\mathit{q}}$ | $\mathbf{sd}\left(\widehat{\mathit{q}}\right)$ | $\mathit{C}\left(\widehat{\mathit{q}}\right)$ |
---|---|---|---|---|---|---|---|---|

50 | 0.5 | 0.5 | 0.5544 | 0.1156 | 97.35 | 0.5329 | 0.1574 | 90.45 |

100 | 0.5225 | 0.0676 | 96.05 | 0.5141 | 0.0972 | 92.65 | ||

150 | 0.5148 | 0.0536 | 95.65 | 0.5074 | 0.0775 | 93.20 | ||

200 | 0.5109 | 0.0455 | 95.65 | 0.5061 | 0.0664 | 93.75 | ||

250 | 0.5068 | 0.0401 | 95.15 | 0.5052 | 0.0593 | 94.30 | ||

300 | 0.5068 | 0.0367 | 95.20 | 0.5035 | 0.0539 | 95.05 | ||

50 | 0.7 | 0.4 | 0.8964 | 0.4138 | 97.60 | 0.4106 | 0.0833 | 93.20 |

100 | 0.7438 | 0.1289 | 96.95 | 0.4059 | 0.0570 | 94.80 | ||

150 | 0.7307 | 0.1006 | 95.75 | 0.4032 | 0.0457 | 94.65 | ||

200 | 0.7205 | 0.0845 | 95.25 | 0.4025 | 0.0394 | 94.80 | ||

250 | 0.7160 | 0.0747 | 96.25 | 0.4022 | 0.0351 | 95.10 | ||

300 | 0.7151 | 0.0681 | 95.45 | 0.4007 | 0.0318 | 93.95 | ||

50 | 1 | 1 | 1.1888 | 0.3298 | 96.90 | 1.0482 | 0.2915 | 90.1 |

100 | 1.0416 | 0.1342 | 95.75 | 1.0178 | 0.1910 | 94.1 | ||

150 | 1.0276 | 0.1066 | 95.40 | 1.0159 | 0.1550 | 93.3 | ||

200 | 1.0200 | 0.0909 | 95.00 | 1.0088 | 0.1322 | 92.8 | ||

250 | 1.0147 | 0.0805 | 95.35 | 1.0096 | 0.1187 | 94.4 | ||

300 | 1.0122 | 0.0731 | 95.30 | 1.0073 | 0.1078 | 94.4 | ||

50 | 3 | 2 | 3.5152 | 2.6864 | 97.25 | 2.0399 | 0.4426 | 93.25 |

100 | 3.1753 | 0.5104 | 96.30 | 2.0275 | 0.3054 | 94.80 | ||

150 | 3.1360 | 0.4020 | 96.20 | 2.0132 | 0.2444 | 93.75 | ||

200 | 3.0860 | 0.3375 | 95.45 | 2.0124 | 0.2113 | 94.85 | ||

250 | 3.0645 | 0.2973 | 96.05 | 2.0127 | 0.1885 | 94.60 | ||

300 | 3.0498 | 0.2694 | 96.25 | 2.0065 | 0.1712 | 95.50 | ||

50 | 5 | 3 | 6.0466 | 2.2275 | 96.55 | 3.0638 | 0.6347 | 93.05 |

100 | 5.3670 | 0.9570 | 96.45 | 3.0306 | 0.4333 | 94.10 | ||

150 | 5.1994 | 0.6965 | 95.70 | 3.0305 | 0.3513 | 94.80 | ||

200 | 5.1491 | 0.5910 | 96.35 | 3.0132 | 0.3003 | 95.25 | ||

250 | 5.1065 | 0.5197 | 95.25 | 3.0230 | 0.2699 | 94.65 | ||

300 | 5.0991 | 0.4726 | 95.45 | 3.0158 | 0.2451 | 95.30 | ||

50 | 2.3 | 2 | 2.5672 | 0.5551 | 96.65 | 2.0735 | 0.5295 | 91.60 |

100 | 2.3919 | 0.3308 | 96.00 | 2.0451 | 0.3593 | 93.55 | ||

150 | 2.3612 | 0.2624 | 95.85 | 2.0196 | 0.2835 | 94.75 | ||

200 | 2.3540 | 0.2253 | 95.65 | 2.0190 | 0.2449 | 95.20 | ||

250 | 2.3387 | 0.1994 | 95.05 | 2.0171 | 0.2186 | 95.35 | ||

300 | 2.3378 | 0.1816 | 95.25 | 2.0128 | 0.1983 | 95.15 | ||

50 | 4.5 | 5 | 4.8658 | 0.8912 | 96.75 | 5.2956 | 1.6178 | 91.10 |

100 | 4.6657 | 0.5677 | 96.05 | 5.1339 | 1.0337 | 92.45 | ||

150 | 4.6148 | 0.4520 | 95.50 | 5.1018 | 0.8302 | 92.35 | ||

200 | 4.5990 | 0.3896 | 94.40 | 5.0426 | 0.7024 | 93.25 | ||

250 | 4.5751 | 0.3441 | 95.25 | 5.0352 | 0.6242 | 93.85 | ||

300 | 4.5670 | 0.3126 | 94.20 | 5.0394 | 0.5719 | 93.45 |

n | $\overline{\mathit{x}}$ | S | $\sqrt{{\mathit{b}}_{1}}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

137 | 8.7737 | 10.6121 | 4.1055 | 26.3882 |

**Table 4.**Estimates, SE in parenthesis, log-likelihood, AIC, BIC, CAIC, and HQIC values for the dataset of patients undergoing lung cancer.

Parameters | Fr | SFr |
---|---|---|

$\alpha $ | 0.7452 (0.0540) | 2.0245 (0.3805) |

q | - | 0.7382 (0.0812) |

log-likelihood | −504.6068 | −444.1976 |

AIC | 1011.214 | 892.3952 |

BIC | 1014.134 | 898.2351 |

CAIC | 1015.134 | 900.2351 |

HQIC | 1012.400 | 894.7684 |

n | $\overline{\mathit{x}}$ | S | $\sqrt{{\mathit{b}}_{1}}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

64 | 27.9547 | 24.9442 | 1.5772 | 5.4244 |

**Table 6.**Estimates, SE in parenthesis, log-likelihood, AIC, BIC, CAIC, and HQIC values for the dataset of patients undergoing peritoneal dialysis.

Parameters | Fr | SPN | SFr |
---|---|---|---|

$\alpha $ | 0.4377 (0.0446) | - | 0.6679 (0.0767) |

$\sigma $ | - | 8.6409 (1.9018) | - |

q | - | 0.3900 (0.0509) | 0.5794 (0.1025) |

log-likelihood | −336.0071 | −319.3558 | −315.4611 |

AIC | 674.0141 | 642.7116 | 634.9221 |

BIC | 676.1730 | 647.0294 | 639.2399 |

CAIC | 677.1730 | 649.0294 | 641.2399 |

HQIC | 674.8646 | 644.4126 | 636.6231 |

n | $\overline{\mathit{x}}$ | S | $\sqrt{{\mathit{b}}_{1}}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

686 | 5.0102 | 5.4755 | 2.8784 | 16.2079 |

**Table 8.**Estimates, SE in parenthesis, log-likelihood, AIC, BIC, CAIC, and HQIC values for the dataset of patients undergoing breast cancer.

Parameters | Fr | SHN | SFr |
---|---|---|---|

$\alpha $ | 1.0452 (0.0348) | - | 2.2209 (0.1934) |

$\sigma $ | - | 3.2493 (0.2384) | - |

q | - | 1.9260 (0.2031) | 1.1304 (0.0631) |

log-likelihood | −1905.598 | −1790.6070 | −1712.0270 |

AIC | 3813.196 | 3585.215 | 3428.054 |

BIC | 3817.727 | 3594.277 | 3437.116 |

CAIC | 3818.727 | 3596.277 | 3439.116 |

HQIC | 3814.949 | 3588.721 | 3431.561 |

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## Share and Cite

**MDPI and ACS Style**

Castillo, J.S.; Rojas, M.A.; Reyes, J.
A More Flexible Extension of the Fréchet Distribution Based on the Incomplete Gamma Function and Applications. *Symmetry* **2023**, *15*, 1608.
https://doi.org/10.3390/sym15081608

**AMA Style**

Castillo JS, Rojas MA, Reyes J.
A More Flexible Extension of the Fréchet Distribution Based on the Incomplete Gamma Function and Applications. *Symmetry*. 2023; 15(8):1608.
https://doi.org/10.3390/sym15081608

**Chicago/Turabian Style**

Castillo, Jaime S., Mario A. Rojas, and Jimmy Reyes.
2023. "A More Flexible Extension of the Fréchet Distribution Based on the Incomplete Gamma Function and Applications" *Symmetry* 15, no. 8: 1608.
https://doi.org/10.3390/sym15081608