# An Extension of the Fréchet Distribution and Applications

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## Abstract

**:**

## 1. Introduction

**Lemma**

**1.**

- (a)
- Let $X\sim Fr\left(\lambda \right)$. Then its cumulative distribution function (cdf) is$${F}_{X}(x;\lambda )=\mathrm{exp}\left(-{x}^{-\lambda}\right),\phantom{\rule{1.em}{0ex}}x>0.$$
- (b)
- Let $V\sim Exp\left(1\right)$. Then ${V}^{-\frac{1}{\lambda}}\sim Fr\left(\lambda \right)$.
- (c)
- If $X\sim Fr\left(\lambda \right)$, then the rth-moment of X exists for $r<\lambda $ and$$\mathbb{E}\left({X}^{r}\right)=\mathsf{\Gamma}\left(1-\frac{r}{\lambda}\right),$$
- (d)
- Let $X\sim Fr\left(\lambda \right)$. Then the pdf of X is unimodal with mode $m={\left(\frac{\lambda}{1+\lambda}\right)}^{1/\lambda}$ or decreasing (for λ close to zero), the hazard function is always unimodal, see [10].

**Remark**

**1.**

**Definition**

**1.**

**Lemma**

**2.**

- (a)
- Let $Y\sim SE\left(\alpha \right)$. Then the cdf is$${F}_{Y}(y;\alpha )=1-\alpha {y}^{-\alpha}\mathsf{\Gamma}\left(\alpha \right)G(y;1+\alpha ,1)-\mathrm{exp}(-y),\phantom{\rule{2.em}{0ex}}y>0.$$
- (b)
- If $Y\sim SE\left(\alpha \right)$ then the survival and hazard rate functions are as follows$${S}_{Y}(y;\alpha )=\alpha {y}^{-\alpha}\mathsf{\Gamma}\left(\alpha \right)G(y;1+\alpha ,1)+\mathrm{exp}(-y),\phantom{\rule{2.em}{0ex}}y>0,$$$${h}_{Y}(y;\alpha )=\frac{{\alpha}^{2}{y}^{-(1+\alpha )}\mathsf{\Gamma}\left(\alpha \right)G(y;1+\alpha ,1)}{\alpha {y}^{-\alpha}\mathsf{\Gamma}\left(\alpha \right)G(y;1+\alpha ,1)+\mathrm{exp}(-y)},\phantom{\rule{2.em}{0ex}}y>0.$$
- (c)
- Let $Y\sim SE\left(\alpha \right)$. Then the rth-moment of Y exists for $\alpha >r$ and$$\mathbb{E}\left({Y}^{r}\right)=\frac{\alpha}{\alpha -r}r!.$$

## 2. Results for the SEFr Distribution

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 2.1. Properties

**Proposition**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

#### 2.2. Moments

**Proposition**

**6.**

**Proof.**

**Corollary**

**2.**

- 1.
- $E\left[X\right]=\sigma \frac{\alpha \lambda}{\alpha \lambda +1}\mathsf{\Gamma}\left(1-\frac{1}{\lambda}\right)\phantom{\rule{0.277778em}{0ex}},$ provided that $\lambda >1$.
- 2.
- $Var\left[X\right]={\sigma}^{2}\alpha \lambda \left[\frac{1}{\alpha \lambda +2}\mathsf{\Gamma}\left(1-\frac{2}{\lambda}\right)-\frac{\alpha \lambda}{{(\alpha \lambda +1)}^{2}}{\mathsf{\Gamma}}^{2}\left(1-\frac{1}{\lambda}\right)\phantom{\rule{0.277778em}{0ex}}\right],$ provided that $\lambda >2$.
- 3.
- Let ${\mu}_{j}=E\left[{X}^{j}\right]$. Then, the skewness, $\sqrt{{\beta}_{1}}$, and kurtosis, ${\beta}_{2}$, coefficients can be obtained by using$$\begin{array}{ccc}\hfill \sqrt{{\beta}_{1}}& =& \frac{{\mu}_{3}-3{\mu}_{1}{\mu}_{2}+2{\mu}_{1}^{3}}{{({\mu}_{2}-{\mu}_{1}^{2})}^{\frac{3}{2}}},\phantom{\rule{1.em}{0ex}}\lambda >3,\hfill \\ \hfill {\beta}_{2}& =& \frac{{\mu}_{4}-4{\mu}_{1}{\mu}_{3}+6{\mu}_{2}{\mu}_{1}^{2}-3{\mu}_{1}^{4}}{{({\mu}_{2}-{\mu}_{1}^{2})}^{2}},\phantom{\rule{1.em}{0ex}}\lambda >4.\hfill \end{array}$$

- Shannon Entropy.

**Lemma**

**3.**

**Proposition**

**7.**

**Proof.**

#### 2.3. Other Properties of Interest in Reliability

- The reverse hazard rate function.

**Corollary**

**3.**

- The stress-strength parameter.

**Proposition**

**8.**

**Corollary**

**4.**

**Proof.**

- Order Statistics.

## 3. Inference

#### 3.1. ML Estimators

`BFGS`” method of the “

`optim`” subroutine in R software [39]. The “

`BFGS`” method is a limited-memory quasi-Newton method for approximating the Hessian matrix of the target distribution. It is worth mentioning that the parameter vector $\mathit{\theta}=(\sigma ,\lambda ,\alpha )$ can be easily obtained, thanks to the properties of the pdf f. The smooth and continuous nature of the function f, along with the existence and finiteness of its first and second derivatives, ensure that the equation $S\left(\mathit{\theta}\right)=\mathbf{0}$ has roots. These roots correspond to the MLEs of the vector $\mathit{\theta}$. By employing relevant calculus techniques, it is possible to verify that the solutions correspond to a maximum.

#### 3.2. Observed Fisher Information Matrix

#### 3.3. Simulation Study

Algorithm 1: To simulate values from the $X\sim SEFr(\sigma ,\lambda ,\alpha )$. |

Step 1: Generate ${X}_{1}\sim Exp\left(1\right)$ and ${X}_{2}\sim Uniform(0,1).$Step 2: Compute $Y=\frac{{X}_{1}}{{X}_{2}^{1/\alpha}}.$Step 3: Compute $X=\sigma {Y}^{-1/\lambda}.$ |

## 4. Applications

- Slashed Quasi-Gamma, $X\sim SQG(\beta ,\theta ,q)$, introduced in [46]. Its pdf is:$${f}_{X}(x;\beta ,\theta ,q)=\frac{q{\beta}^{q}{x}^{-(q+1)}}{\mathsf{\Gamma}\left(\frac{1}{10}\right)}\mathsf{\Gamma}\left(\frac{q}{\theta}+\frac{1}{10}\right)F\left({\left(\frac{x}{\beta}\right)}^{\theta},\frac{q}{\theta}+\frac{1}{10},1\right),\phantom{\rule{1.em}{0ex}}x>0,$$
- Slash Fréchet, $X\sim SFr(\lambda ,q)$, introduced in [20]. Its pdf is:$${f}_{X}(x;\lambda ,q)={\displaystyle \frac{q}{{x}^{q+1}}}\mathsf{\Gamma}\left(1-\frac{q}{\lambda},{x}^{-\lambda}\right),\phantom{\rule{1.em}{0ex}}x>0,$$

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

#### 4.2. Application 2 (Air Conditioning System Failures)

## 5. Conclusions

- The stochastic representation of the new model in terms of the Slash-Exponential is given. In this way, an additional shape parameter is added to Fréchet model.
- Closed expressions for the pdf and cdf are given, therefore also for the survival and hazard rate function.
- It is shown that the new model is unimodal or decreasing. It is proven that if the new shape parameter tends to infinity then the SEFr approaches to Fréchet model.
- Closed expressions are given for the moments, with particular interest on skewness and kurtosis coefficients.
- We highlight that the new model presents less kurtosis than the basal Fréchet distribution. For the best of our understanding, it is the first time in literature that, as result of applying slash methodology, the new model exhibits a lighter right tail and less kurtosis compared to basal model.
- Maximum likelihood method has been proposed to estimate the parameters in the model. Score equations and the observed Fisher information matrix are studied.
- A simulation study has been carried out. There, bias, standard error, RMSE and empirical coverage probability for MLEs have been obtained for increasing sample size. The good asymptotic properties of MLEs can be seen.
- Two real applications are included where the SEFr model is compared to Fr, Slashed Quasi-Gamma and Slash-Fréchet. By using AIC and BIC, it has been seen that the new model provides a better fit compared to others.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**pdf of $SEFr\phantom{\rule{0.166667em}{0ex}}(1,\lambda ,\alpha )$: (

**a**) for $\alpha =0.5$ and $\lambda \in \{2,4,8\}$, (

**b**) for $\lambda =5$ and $\alpha \in \{0.5,1,8\}$.

**Figure 2.**The cdf, survival function and hazard rate function for the SEFr $(1,\lambda ,0.5)$ model can be visualized through their respective plots.

**Figure 3.**Plot and contour plot of the skewness coefficient in the SEFr $(1,\lambda ,\alpha )$ model.

**Figure 4.**Plot and contour plot of the kurtosis coefficient in the SEFr $(1,\lambda ,\alpha )$ model.

**Figure 6.**Fitted pdf for the remission times of patients with bladder cancer dataset in the Fr and SFr, SQG, and SEFr distributions.

**Figure 7.**QQ plots for the remission times of patients with bladder cancer dataset: (

**a**) SEFr Model; (

**b**) SQG model; (

**c**) SFr model; (

**d**) Fr model.

**Figure 9.**Fitted Density for Air Conditioning System Failures dataset in the Fr and SFr, SQG, and SEFr distributions.

**Figure 10.**QQ plots for Air Conditioning System Failures dataset: (

**a**) SEFr Model; (

**b**) SQG model; (

**c**) SFr model; (

**d**) Fr model.

$\mathit{\alpha}$ | $\mathit{P}\phantom{\rule{0.166667em}{0ex}}[\mathit{X}>1]$ | $\mathit{P}\phantom{\rule{0.166667em}{0ex}}[\mathit{X}>1.5]$ | $\mathit{P}\phantom{\rule{0.166667em}{0ex}}[\mathit{X}>2]$ | $\mathit{P}\phantom{\rule{0.166667em}{0ex}}[\mathit{X}>2.5]$ |
---|---|---|---|---|

$SEFr\phantom{\rule{0.166667em}{0ex}}(1,5,0.5)$ | 0.253 | 0.042 | 0.010 | 0.003 |

$SEFr\phantom{\rule{0.166667em}{0ex}}(1,5,1)$ | 0.368 | 0.063 | 0.016 | 0.005 |

$SEFr\phantom{\rule{0.166667em}{0ex}}(1,5,8)$ | 0.587 | 0.110 | 0.027 | 0.009 |

$Fr\phantom{\rule{0.166667em}{0ex}}\left(5\right)$ | 0.632 | 0.123 | 0.031 | 0.010 |

True Value | n = 50 | n = 100 | n = 200 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\sigma}$ | $\mathit{\lambda}$ | $\mathit{\alpha}$ | Estimator | Bias | SE | RMSE | CP | Bias | SE | RMSE | CP | Bias | SE | RMSE | CP |

1.5 | 2 | 0.5 | $\widehat{\sigma}$ | 0.029 | 0.305 | 0.347 | 90.5 | 0.009 | 0.208 | 0.230 | 90.8 | 0.002 | 0.149 | 0.153 | 93.8 |

$\widehat{\lambda}$ | 0.316 | 0.682 | 2.067 | 94.0 | 0.099 | 0.356 | 0.407 | 95.1 | 0.037 | 0.241 | 0.262 | 93.3 | |||

$\widehat{\alpha}$ | 0.073 | 0.374 | 0.726 | 90.5 | 0.015 | 0.133 | 0.144 | 93.7 | 0.007 | 0.091 | 0.096 | 93.5 | |||

1.2 | $\widehat{\sigma}$ | −0.037 | 0.253 | 0.291 | 90.6 | −0.025 | 0.177 | 0.196 | 93.8 | −0.006 | 0.122 | 0.124 | 94.9 | ||

$\widehat{\lambda}$ | 0.124 | 0.404 | 0.598 | 92.8 | 0.027 | 0.259 | 0.278 | 94.1 | 0.024 | 0.181 | 0.185 | 95.1 | |||

$\widehat{\alpha}$ | 1.523 | 7.629 | 4.558 | 91.2 | 0.427 | 1.460 | 1.795 | 94.8 | 0.077 | 0.328 | 0.436 | 93.8 | |||

4 | 0.5 | $\widehat{\sigma}$ | −0.010 | 0.149 | 0.176 | 89.1 | −0.004 | 0.106 | 0.109 | 93.3 | −0.003 | 0.074 | 0.077 | 93.9 | |

$\widehat{\lambda}$ | 0.512 | 1.222 | 3.100 | 92.9 | 0.187 | 0.722 | 0.854 | 94.8 | 0.084 | 0.483 | 0.521 | 94.7 | |||

$\widehat{\alpha}$ | 0.104 | 0.443 | 0.715 | 89.9 | 0.019 | 0.137 | 0.155 | 93.5 | 0.008 | 0.092 | 0.094 | 94.1 | |||

1.2 | $\widehat{\sigma}$ | −0.019 | 0.132 | 0.140 | 92.4 | −0.014 | 0.089 | 0.097 | 95.3 | −0.004 | 0.061 | 0.062 | 94.8 | ||

$\widehat{\lambda}$ | 0.256 | 0.810 | 1.023 | 93.4 | 0.096 | 0.527 | 0.583 | 93.1 | 0.049 | 0.362 | 0.368 | 95.2 | |||

$\widehat{\alpha}$ | 1.227 | 6.490 | 4.114 | 91.1 | 0.397 | 1.452 | 2.013 | 92.4 | 0.066 | 0.317 | 0.410 | 94.5 | |||

2 | 3 | 0.7 | $\widehat{\sigma}$ | −0.018 | 0.242 | 0.262 | 90.9 | −0.005 | 0.170 | 0.183 | 92.9 | −0.002 | 0.120 | 0.125 | 94.3 |

$\widehat{\lambda}$ | 0.259 | 0.733 | 0.946 | 94.6 | 0.108 | 0.472 | 0.518 | 95.3 | 0.065 | 0.323 | 0.343 | 95.6 | |||

$\widehat{\alpha}$ | 0.180 | 0.842 | 1.153 | 92.0 | 0.052 | 0.252 | 0.434 | 94.0 | 0.013 | 0.133 | 0.144 | 93.4 | |||

1 | $\widehat{\sigma}$ | −0.028 | 0.234 | 0.272 | 91.2 | −0.009 | 0.161 | 0.172 | 95.2 | −0.002 | 0.111 | 0.111 | 95.7 | ||

$\widehat{\lambda}$ | 0.180 | 0.631 | 0.820 | 92.2 | 0.068 | 0.416 | 0.439 | 95.3 | 0.040 | 0.286 | 0.297 | 94.8 | |||

$\widehat{\alpha}$ | 0.889 | 4.261 | 3.233 | 91.3 | 0.164 | 0.595 | 0.912 | 93.5 | 0.036 | 0.222 | 0.266 | 94.6 | |||

5 | 0.7 | $\widehat{\sigma}$ | −0.016 | 0.147 | 0.166 | 91.7 | 0.003 | 0.103 | 0.109 | 94.4 | 0.001 | 0.072 | 0.074 | 94.5 | |

$\widehat{\lambda}$ | 0.442 | 1.238 | 1.679 | 93.5 | 0.218 | 0.797 | 0.885 | 95.7 | 0.105 | 0.538 | 0.561 | 96.1 | |||

$\widehat{\alpha}$ | 0.197 | 0.898 | 1.110 | 91.8 | 0.024 | 0.204 | 0.234 | 93.1 | 0.009 | 0.132 | 0.145 | 93.8 | |||

1 | $\widehat{\sigma}$ | −0.025 | 0.147 | 0.165 | 92.9 | −0.008 | 0.096 | 0.101 | 93.6 | −0.005 | 0.067 | 0.067 | 94.7 | ||

$\widehat{\lambda}$ | 0.267 | 1.045 | 1.305 | 91.7 | 0.117 | 0.690 | 0.737 | 94.6 | 0.081 | 0.480 | 0.513 | 94.0 | |||

$\widehat{\alpha}$ | 0.974 | 6.529 | 3.370 | 92.4 | 0.152 | 0.561 | 0.858 | 93.9 | 0.046 | 0.222 | 0.254 | 95.2 | |||

3 | 1.5 | 0.3 | $\widehat{\sigma}$ | 0.170 | 0.957 | 1.184 | 88.1 | 0.054 | 0.684 | 0.733 | 92.1 | 0.020 | 0.475 | 0.495 | 94.5 |

$\widehat{\lambda}$ | 0.535 | 0.966 | 3.147 | 93.9 | 0.112 | 0.350 | 0.423 | 94.4 | 0.055 | 0.227 | 0.251 | 95.0 | |||

$\widehat{\alpha}$ | 0.013 | 0.123 | 0.152 | 88.9 | 0.005 | 0.083 | 0.098 | 92.1 | 0.002 | 0.057 | 0.059 | 94.6 | |||

0.9 | $\widehat{\sigma}$ | −0.033 | 0.689 | 0.766 | 90.6 | −0.014 | 0.487 | 0.526 | 92.8 | −0.006 | 0.340 | 0.338 | 95.0 | ||

$\widehat{\lambda}$ | 0.104 | 0.327 | 0.390 | 94.4 | 0.039 | 0.215 | 0.249 | 93.9 | 0.023 | 0.148 | 0.155 | 94.7 | |||

$\widehat{\alpha}$ | 0.622 | 2.755 | 2.598 | 91.7 | 0.142 | 0.544 | 0.807 | 93.4 | 0.034 | 0.189 | 0.209 | 95.2 | |||

3.5 | 0.3 | $\widehat{\sigma}$ | 0.022 | 0.394 | 0.461 | 89.9 | 0.007 | 0.287 | 0.306 | 92.5 | 0.001 | 0.204 | 0.202 | 95.3 | |

$\widehat{\lambda}$ | 0.923 | 1.649 | 4.378 | 93.2 | 0.289 | 0.832 | 1.182 | 95.2 | 0.093 | 0.524 | 0.547 | 95.5 | |||

$\widehat{\alpha}$ | 0.015 | 0.123 | 0.164 | 88.3 | 0.004 | 0.081 | 0.089 | 92.1 | 0.003 | 0.057 | 0.057 | 94.5 | |||

0.9 | $\widehat{\sigma}$ | −0.021 | 0.301 | 0.352 | 89.7 | −0.005 | 0.209 | 0.232 | 93.2 | −0.003 | 0.145 | 0.150 | 93.3 | ||

$\widehat{\lambda}$ | 0.280 | 0.794 | 1.344 | 93.7 | 0.123 | 0.509 | 0.549 | 95.5 | 0.064 | 0.346 | 0.370 | 95.1 | |||

$\widehat{\alpha}$ | 0.681 | 3.211 | 2.885 | 90.6 | 0.100 | 0.438 | 0.728 | 94.9 | 0.022 | 0.184 | 0.202 | 93.8 |

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

128 | 9.366 | 10.508 | 3.287 | 18.483 |

**Table 4.**Estimates, SE in parentheses, log-likelihood, AIC, and BIC values for the remission times of patients with bladder cancer dataset.

Parameters | Fr (SE) | SFr (SE) | SQG (SE) | SEFr (SE) |
---|---|---|---|---|

$\widehat{\sigma}$ | - | - | - | 9.9436 (1.3592) |

$\widehat{\lambda}$ | 0.6726 (0.0479) | 0.9242 (0.0688) | - | 1.8586 (0.2660) |

$\widehat{\alpha}$ | - | - | - | 0.6329 (0.1558) |

$\widehat{\beta}$ | - | - | 7.7993 (0.9893) | - |

$\widehat{\mathit{\theta}}$ | - | - | 10.8627 (1.3211) | - |

$\widehat{q}$ | - | 0.9623 (0.1302) | 1.5211 (0.2282) | - |

log-likelihood | −481.0559 | −448.1104 | −411.7342 | −410.0634 |

AIC | 964.1118 | 900.2208 | 829.4683 | 826.1268 |

BIC | 966.9638 | 905.9249 | 838.0244 | 834.6829 |

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

30 | 59.6333 | 71.8996 | 1.6914 | 4.9595 |

**Table 6.**Estimates of parameters, SE in parentheses, log-likelihood, AIC and BIC values for Air Conditioning System Failures.

Parameters | Fr (SE) | SFr (SE) | SQG (SE) | SEFr (SE) |
---|---|---|---|---|

$\widehat{\sigma}$ | - | - | - | 38.5732 (13.9807) |

$\widehat{\lambda}$ | 0.3924 (0.0601) | 0.9508 (0.3089) | - | 1.0968 (0.2273) |

$\widehat{\alpha}$ | - | - | - | 1.1344 (0.5565) |

$\widehat{\beta}$ | - | - | 14.7397 (3.1532) | - |

$\widehat{\mathit{\theta}}$ | - | - | 14.3648 (4.2492) | - |

$\widehat{q}$ | - | 0.4067 (0.0960) | 0.7165 (0.1559) | - |

log-likelihood | −177.5930 | −163.9272 | −153.0741 | −152.3953 |

AIC | 357.1859 | 331.8543 | 312.1481 | 310.7905 |

BIC | 358.5871 | 334.6567 | 316.3517 | 314.9941 |

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## Share and Cite

**MDPI and ACS Style**

Gómez, Y.M.; Barranco-Chamorro, I.; Castillo, J.S.; Gómez, H.W.
An Extension of the Fréchet Distribution and Applications. *Axioms* **2024**, *13*, 253.
https://doi.org/10.3390/axioms13040253

**AMA Style**

Gómez YM, Barranco-Chamorro I, Castillo JS, Gómez HW.
An Extension of the Fréchet Distribution and Applications. *Axioms*. 2024; 13(4):253.
https://doi.org/10.3390/axioms13040253

**Chicago/Turabian Style**

Gómez, Yolanda M., Inmaculada Barranco-Chamorro, Jaime S. Castillo, and Héctor W. Gómez.
2024. "An Extension of the Fréchet Distribution and Applications" *Axioms* 13, no. 4: 253.
https://doi.org/10.3390/axioms13040253