# A More Flexible Reliability Model Based on the Gompertz Function and the Generalized Integro-Exponential Function

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. The Gompertz Distribution

#### 1.2. The Slash Standard Distribution

## 2. Slash Gompertz Distribution

#### 2.1. Density Function

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 2.2. The Reliability and Hazard Rate Functions

**Proposition**

**3.**

**Proof.**

#### 2.3. Reliability Function Comparison of Gompertz, Slash Birnbaum Saunders and Slash Gompertz Distributions

#### 2.4. The rth Moment of a Gompertz Slash Distributed Random Variable W

**Proposition**

**4.**

**Proof.**

#### 2.5. Expected Value, Variance, Skewness and Kurtosis of a Gompertz Slash Distributed Random Variable W

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

## 3. Some Statistical Properties

#### 3.1. Failure Rate Function

**Proposition**

**7.**

**Proof.**

#### 3.2. Mean Residual Life

**Proposition**

**8.**

**Proof.**

#### 3.3. Incomplete Moments

**Proposition**

**9.**

**Proof.**

#### 3.4. The Lorenz Curve and the Gini Index

**Proposition**

**10.**

## 4. Inference

#### 4.1. Moment Estimators

**Proposition**

**11.**

**Proof.**

#### 4.2. Maximum Likelihood Estimate

#### 4.3. Simulation Study

- Simulate $Y\sim U(0,1).$
- Compute $X=\left(\frac{1}{\beta}\right)ln(1-\left(\frac{\beta}{\alpha}\right)ln(1-Y)).$
- Simulate $V\sim U(0,1).$
- Compute $W=\frac{X}{{V}^{1/q}}.$

## 5. Applications with Real Data

#### 5.1. Application 1

#### 5.2. Application 2

#### 5.3. Comparison of Estimated Reliability

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plot of Gompertz function for $a=2.5$ (solid line), $a=1$ (dotted line), $a=-0.5$ (dashed line) and $b=c=1$.

**Figure 2.**Graphic of comparison of the G distribution (solid line) with $SG$ distribution for $\alpha $ = 0.0001, $\beta $ = 0.01 and $q=1$ (dashed line), $q=3$ (dotted line), $q=6$ (dashed dotted line).

**Figure 3.**Graphic comparison of the $SG$ distribution for $q=1$ and different values of $\alpha $ and $\beta $.

**Figure 4.**Cumulative distribution functions for the $SG$ distributions compared to the G distribution for values of $\alpha $, $\beta $ and q.

**Figure 5.**Reliability functions for the $SG$ distributions compared to the G distribution for values of $\alpha =0.1$, $\beta =1$ and $q=1$.

**Figure 8.**Rupture data histogram with Density $SG$ (solid line), G density (dashed line) and $SBS$ density (dotted line).

**Figure 10.**Empirical cdf with estimated $SBS$ c.d.f. (red color), estimated G c.d.f. (green color) and estimated $SG$ cdf (blue color).

**Figure 11.**Data nickel histogram with density $SG$ (solid line), $GV$ density (dashed line), $SW$ density (dotted line) and G density (dashed-dotted line).

**Figure 13.**Empirical cdf with estimated G c.d.f. (orange color), $SW$ c.d.f. (green color), estimated $GV$ c.d.f. (red color) and estimated $SG$ cdf (blue color).

**Figure 14.**Reliability function $SG$ (Solid line), $GV$ (dashed line), $SW$ (dotted line), G (dashed-dotted line) and Empirical reliability (Solid Line) for nickel data set.

Distribution | $\mathit{P}(\mathit{Y}>4.1)$ | $\mathit{P}(\mathit{Y}>4.2)$ | $\mathit{P}(\mathit{Y}>4.3)$ | $\mathit{P}(\mathit{Y}>4.4)$ | $\mathit{P}(\mathit{Y}>4.5)$ |
---|---|---|---|---|---|

G | 0.0036 | 0.0014 | 0.0006 | 0.0003 | 0.0001 |

$SBS$ | 0.0260 | 0.0255 | 0.0250 | 0.0240 | 0.0241 |

$SG$ | 0.4912 | 0.4796 | 0.4685 | 0.4578 | 0.4476 |

n | $\mathit{\alpha}$ | $\mathit{\beta}$ | q | $\widehat{\mathit{\alpha}}$ | $\mathit{sd}\left(\widehat{\mathit{\alpha}}\right)$ | $\mathit{ali}\left(\widehat{\mathit{\alpha}}\right)$ | $\mathit{c}\left(\widehat{\mathit{\alpha}}\right)$ | $\widehat{\mathit{\beta}}$ | $\mathit{sd}\left(\widehat{\mathit{\beta}}\right)$ | $\mathit{ali}\left(\widehat{\mathit{\beta}}\right)$ | $\mathit{c}\left(\widehat{\mathit{\beta}}\right)$ | $\widehat{\mathit{q}}$ | $\mathit{sd}\left(\widehat{\mathit{q}}\right)$ | $\mathit{ali}\left(\widehat{\mathit{q}}\right)$ | $\mathit{c}\left(\widehat{\mathit{q}}\right)$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

50 | 0.3 | 0.8 | 2 | 0.2856 | 0.1097 | 0.4300 | 95.80 | 0.9213 | 0.5182 | 2.0312 | 94.20 | 2.3408 | 0.9507 | 3.7266 | 94.20 |

100 | 0.3 | 0.8 | 2 | 0.2970 | 0.0814 | 0.3191 | 95.25 | 0.8320 | 0.3327 | 1.3041 | 95.05 | 2.2171 | 0.6490 | 2.5441 | 94.50 |

200 | 0.3 | 0.8 | 2 | 0.2967 | 0.0574 | 0.2252 | 95.35 | 0.8281 | 0.2219 | 0.8700 | 95.10 | 2.0702 | 0.3626 | 1.4215 | 95.05 |

500 | 0.3 | 0.8 | 2 | 0.2987 | 0.0367 | 0.1437 | 95.25 | 0.8068 | 0.1330 | 0.5215 | 94.45 | 2.0334 | 0.1891 | 0.7413 | 94.65 |

50 | 0.5 | 0.8 | 2 | 0.4674 | 0.1669 | 0.6543 | 95.40 | 1.0022 | 0.6540 | 2.5639 | 93.80 | 2.2997 | 0.9467 | 3.7110 | 94.40 |

100 | 0.5 | 0.8 | 2 | 0.4884 | 0.1237 | 0.4849 | 95.25 | 0.8579 | 0.4110 | 1.6110 | 95.30 | 2.2498 | 0.7464 | 2.9260 | 94.95 |

200 | 0.5 | 0.8 | 2 | 0.4931 | 0.0886 | 0.3472 | 95.40 | 0.8361 | 0.2822 | 1.1064 | 95.00 | 2.1004 | 0.4795 | 1.8795 | 96.20 |

500 | 0.5 | 0.8 | 2 | 0.4973 | 0.0560 | 0.2196 | 95.10 | 0.8080 | 0.1658 | 0.6499 | 94.40 | 2.0424 | 0.2122 | 0.8318 | 94.60 |

50 | 1 | 0.8 | 2 | 0.9335 | 0.3183 | 1.2476 | 95.35 | 1.1682 | 1.0244 | 4.0156 | 94.05 | 2.2903 | 1.0496 | 4.1145 | 94.60 |

100 | 1 | 0.8 | 2 | 0.9627 | 0.2253 | 0.8833 | 95.35 | 0.9407 | 0.6059 | 2.3751 | 94.70 | 2.2923 | 0.9649 | 3.7823 | 95.95 |

200 | 1 | 0.8 | 2 | 0.9803 | 0.1607 | 0.6299 | 95.10 | 0.8621 | 0.4150 | 1.6269 | 95.40 | 2.1575 | 0.6500 | 2.5478 | 96.00 |

500 | 1 | 0.8 | 2 | 0.9918 | 0.1014 | 0.3973 | 94.65 | 0.8131 | 0.2407 | 0.9434 | 95.00 | 2.0629 | 0.2691 | 1.0547 | 94.55 |

50 | 0.5 | 0.8 | 1.5 | 0.4698 | 0.1779 | 0.6972 | 95.45 | 0.9858 | 0.7284 | 2.8553 | 93.25 | 1.7562 | 0.7416 | 2.9073 | 95.30 |

100 | 0.5 | 0.8 | 1.5 | 0.4905 | 0.1325 | 0.5196 | 95.05 | 0.8556 | 0.4724 | 1.8519 | 95.70 | 1.6659 | 0.5575 | 2.1852 | 96.40 |

200 | 0.5 | 0.8 | 1.5 | 0.4926 | 0.0941 | 0.3688 | 95.45 | 0.8431 | 0.3125 | 1.2249 | 95.00 | 1.5527 | 0.2592 | 1.0161 | 95.45 |

500 | 0.5 | 0.8 | 1.5 | 0.4979 | 0.0606 | 0.2376 | 95.30 | 0.8084 | 0.1863 | 0.7303 | 94.20 | 1.5237 | 0.1330 | 0.5215 | 94.55 |

50 | 0.5 | 0.2 | 1.2 | 0.4550 | 0.1628 | 0.6382 | 95.90 | 0.3419 | 0.4403 | 1.7260 | 93.60 | 1.4528 | 0.7541 | 2.9559 | 95.70 |

100 | 0.5 | 0.2 | 1.2 | 0.4779 | 0.1177 | 0.4614 | 95.10 | 0.2615 | 0.2816 | 1.1037 | 94.35 | 1.3398 | 0.4652 | 1.8235 | 96.50 |

200 | 0.5 | 0.2 | 1.2 | 0.4870 | 0.0844 | 0.3310 | 95.55 | 0.2398 | 0.1942 | 0.7611 | 94.50 | 1.2495 | 0.2177 | 0.8535 | 94.70 |

500 | 0.5 | 0.2 | 1.2 | 0.4957 | 0.0554 | 0.2172 | 95.45 | 0.2040 | 0.1177 | 0.4615 | 95.70 | 1.2288 | 0.1245 | 0.4880 | 94.35 |

50 | 1.5 | 1.8 | 2 | 1.4001 | 0.4905 | 1.9227 | 95.25 | 2.4105 | 1.7908 | 7.0200 | 93.95 | 2.2794 | 0.9683 | 3.7958 | 94.75 |

100 | 1.5 | 1.8 | 2 | 1.4543 | 0.3550 | 1.3918 | 95.40 | 2.0053 | 1.0838 | 4.2486 | 94.75 | 2.2610 | 0.8222 | 3.2230 | 95.30 |

200 | 1.5 | 1.8 | 2 | 1.4767 | 0.2547 | 0.9985 | 95.30 | 1.8987 | 0.7376 | 2.8913 | 95.10 | 2.1147 | 0.5043 | 1.9769 | 95.70 |

500 | 1.5 | 1.8 | 2 | 1.4899 | 0.1607 | 0.6299 | 95.10 | 1.8223 | 0.4313 | 1.6907 | 94.70 | 2.0497 | 0.2287 | 0.8966 | 94.40 |

50 | 0.2 | 0.5 | 1.2 | 0.1920 | 0.0814 | 0.3192 | 96.25 | 0.5700 | 0.3803 | 1.4909 | 94.35 | 1.3811 | 0.5239 | 2.0538 | 95.65 |

100 | 0.2 | 0.5 | 1.2 | 0.1987 | 0.0610 | 0.2392 | 95.35 | 0.5259 | 0.2618 | 1.0264 | 95.25 | 1.2864 | 0.2698 | 1.0575 | 94.20 |

200 | 0.2 | 0.5 | 1.2 | 0.1996 | 0.0429 | 0.1683 | 94.80 | 0.5078 | 0.1719 | 0.6738 | 94.85 | 1.2382 | 0.1577 | 0.6183 | 94.55 |

500 | 0.2 | 0.5 | 1.2 | 0.2006 | 0.0274 | 0.1074 | 95.15 | 0.4975 | 0.1032 | 0.4044 | 95.50 | 1.2189 | 0.0904 | 0.3544 | 94.65 |

n | $\overline{\mathit{Y}}$ | ${\mathit{S}}_{\mathit{Y}}$ | $\sqrt{{\mathit{\beta}}_{1}}$ | ${\mathit{\beta}}_{2}$ |
---|---|---|---|---|

100 | 221.98 | 144.6181 | 1.3396 | 5.7435 |

**Table 4.**Maximum likelihood estimators for rupture data with their corresponding standard errors in parentheses and AIC, BIC criteria.

Parameters | MLE (G) | MLE ($\mathit{SBS}$) | MLE ($\mathit{SG}$) |
---|---|---|---|

$\alpha $ | 0.0031 (0.0003) | 0.4307 (0.0967) | 0.0024 (0.0003) |

$\beta $ | 0.0022 (0.0004) | 193.4896 (16.0376) | 0.0083 (0.0018) |

q | 2.1425 (0.7495) | 3.1919 (0.8035) | |

AIC | 1266.907 | 1265.789 | 1256.003 |

BIC | 1272.918 | 1273.602 | 1263.818 |

n | $\overline{\mathit{Y}}$ | ${\mathit{S}}_{\mathit{Y}}$ | $\sqrt{{\mathit{\beta}}_{1}}$ | ${\mathit{\beta}}_{2}$ |
---|---|---|---|---|

85 | 21.588 | 16.573 | 2.392 | 8.325 |

**Table 6.**Maximum likelihood estimators for nickel data with their corresponding standard errors in parentheses and AIC, BIC criteria.

Parámetros | MLE (G) | MLE ($\mathit{SW}$) | MLE($\mathit{GV}$) | MLE ($\mathit{SG}$) |
---|---|---|---|---|

$\alpha $ | 0.0378 (0.0057) | 2.0890 (0.3200) | 2.0631 (0.7925) | 0.0118 (0.0046) |

$\beta $ | 0.0112 (0.0049) | 14.4769 (1.7959) | 0.2146 (0.0433) | |

$\lambda $ | 0.0737 (0.0114) | |||

$\sigma $ | 1.0498 (0.1254) | |||

q | 2.4182 (0.6042) | 2.0852 (0.3395) | ||

AIC | 698.112 | 677.936 | 681.128 | 674.276 |

BIC | 703.021 | 685.339 | 688.481 | 681.633 |

$\mathit{SG}$ | $\mathit{SW}$ | $\mathit{GV}$ | G | |
---|---|---|---|---|

Number of Cycles (t) | $\mathit{P}(\mathit{Y}>\mathit{t})$ | $\mathit{P}(\mathit{Y}>\mathit{t})$ | $\mathit{P}(\mathit{Y}>\mathit{t})$ | $\mathit{P}(\mathit{Y}>\mathit{t})$ |

70 | 0.0283 | 0.0125 | 0.0124 | 0.0180 |

80 | 0.0214 | 0.0090 | 0.0059 | 0.0079 |

90 | 0.0167 | 0.0087 | 0.0028 | 0.0028 |

100 | 0.0134 | 0.0052 | 0.0013 | 0.0009 |

110 | 0.0211 | 0.0041 | 0.0006 | 0.0002 |

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**MDPI and ACS Style**

Reyes, J.; Cortés, P.L.; Rojas, M.A.; Arrué, J.
A More Flexible Reliability Model Based on the Gompertz Function and the Generalized Integro-Exponential Function. *Symmetry* **2022**, *14*, 1207.
https://doi.org/10.3390/sym14061207

**AMA Style**

Reyes J, Cortés PL, Rojas MA, Arrué J.
A More Flexible Reliability Model Based on the Gompertz Function and the Generalized Integro-Exponential Function. *Symmetry*. 2022; 14(6):1207.
https://doi.org/10.3390/sym14061207

**Chicago/Turabian Style**

Reyes, Jimmy, Pedro L. Cortés, Mario A. Rojas, and Jaime Arrué.
2022. "A More Flexible Reliability Model Based on the Gompertz Function and the Generalized Integro-Exponential Function" *Symmetry* 14, no. 6: 1207.
https://doi.org/10.3390/sym14061207