# A Reliability Model Based on the Incomplete Generalized Integro-Exponential Function

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## Abstract

**:**

## 1. Introduction

- ${F}_{X}(x;\beta ,\gamma )=1-exp\left(\right)open="\{"\; close="\}">{\left(\right)}^{\frac{x}{\beta}}\gamma +1$,
- $Q(p)=\beta {\left(\right)}^{\phantom{\rule{0.166667em}{0ex}}}1/\gamma $,
- $E({X}^{r})=e{\beta}^{r}\phantom{\rule{0.166667em}{0ex}}\Gamma \left(\right)open="("\; close=")">\frac{r}{\gamma}+1$

## 2. Probability Density Function

#### 2.1. Stochastic Representation

#### 2.2. Density Function

**Proposition**

**1.**

**Proof.**

#### 2.3. Probabilistic Properties

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

#### 2.4. Moments

**Proposition**

**5.**

**Proof.**

**Corollary**

**1.**

- 1.
- ${\mu}_{1}=E(Z)=\frac{e\beta q}{q-1}{M}_{1}(\gamma )\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{1.em}{0ex}}q>1;$
- 2.
- ${\mu}_{2}=E({Z}^{2})=\frac{e{\beta}^{2}q}{q-2}{M}_{2}(\gamma )\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{1.em}{0ex}}q>2;$
- 3.
- ${\mu}_{3}=E({Z}^{3})=\frac{e{\beta}^{3}q}{q-3}{M}_{3}(\gamma )\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{1.em}{0ex}}q>3;$
- 4.
- ${\mu}_{4}=E({Z}^{4})=\frac{e{\beta}^{4}q}{q-4}{M}_{4}(\gamma )\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{1.em}{0ex}}q>4.$

**Corollary**

**2.**

**Remark**

**1.**

## 3. Inference

#### 3.1. Moment Method Estimators

#### 3.2. Ml Estimation

#### 3.3. Simulation Study

- Simulate $U\sim Uniform(0,1).$
- Compute $V=\beta {\left(\right)}^{\phantom{\rule{0.166667em}{0ex}}}1/\gamma $ where W is the Lambert W function.
- Compute $Z=\frac{V}{{U}^{1/q}}.$

## 4. Applications

#### 4.1. Application I (Rubidium Concentration)

#### 4.2. Application II (Dietary Retinol Consumed)

## 5. Discussion

- The density of the SPM model has a closed-form and is expressed in terms of the incomplete generalized integro-exponential function.
- The SPM model presents more flexible coefficients of asymmetry and kurtosis than those of the PM model. Furthermore, as shown in Table 1, the tails become heavier when the parameter q is smaller.
- We discuss two stochastic representations for the SPM model: one is based on the quotient between two independent random variables, a PM in the numerator, and a power of a U in the denominator; the other is obtained as a mixture of the scale of a PM model and a U model.
- The moments and coefficients of asymmetry and kurtosis have closed-form expressions and are expressed in terms of the generalized integro-exponential function.
- In the applications, two criteria (AIC and BIC) were used to compare the models. In both data sets, the coefficient of kurtosis is high, indicating the presence of atypical observations. The criteria indicate that the SPM model provides the best fit to the data.
- The SPM distribution is a good alternative for modeling continuous positive data sets with atypical observations. These situations are common in all areas of knowledge, for example environmental science, economic, geo-chemistry, survival, reliability, etc. In the future, we will use this distribution in problems of regression, reliability, survival analysis, and Bayesian inference.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Distribution | $\mathit{P}(\mathit{Z}>3)$ | $\mathit{P}(\mathit{Z}>4)$ | $\mathit{P}(\mathit{Z}>5)$ |
---|---|---|---|

PM(1,1) | $2.09\xb7{10}^{-6}$ | $1.98\xb7{10}^{-20}$ | $7.35\xb7{10}^{-60}$ |

SPM(1,1,10) | 0.00360 | 0.00015 | $1.31\xb7{10}^{-5}$ |

SPM(1,1,3) | 0.05910 | 0.01870 | 0.00765 |

SPM(1,1,2) | 0.08834 | 0.03727 | 0.01908 |

SPM(1,1,1) | 0.11111 | 0.06250 | 0.04000 |

n | $\mathit{\beta}$ | $\mathit{\gamma}$ | q | $\widehat{\mathit{\beta}}$ | $\mathit{SD}(\widehat{\mathit{\beta}})$ | $\mathit{C}(\widehat{\mathit{\beta}})$ | $\widehat{\mathit{\gamma}}$ | $\mathit{SD}(\widehat{\mathit{\gamma}})$ | $\mathit{C}(\widehat{\mathit{\gamma}})$ | $\widehat{\mathit{q}}$ | $\mathit{SD}(\widehat{\mathit{q}})$ | $\mathit{C}(\widehat{\mathit{q}})$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

50 | 1 | 1 | 2 | 1.0391 | 0.1372 | 91.6 | 1.0535 | 0.2106 | 91.8 | 2.3550 | 0.8728 | 95.4 |

100 | 1 | 1 | 2 | 1.0260 | 0.0934 | 93.0 | 1.0234 | 0.1399 | 91.8 | 2.2321 | 0.4495 | 95.3 |

150 | 1 | 1 | 2 | 1.0163 | 0.0752 | 92.5 | 1.0089 | 0.1112 | 93.5 | 2.1538 | 0.3294 | 95.0 |

200 | 1 | 1 | 2 | 1.0182 | 0.0651 | 92.0 | 0.9981 | 0.0943 | 91.1 | 2.1515 | 0.2798 | 92.9 |

50 | 1 | 1 | 3 | 1.0078 | 0.1227 | 92.9 | 1.0701 | 0.1963 | 93.5 | 3.3892 | 1.5270 | 93.1 |

100 | 1 | 1 | 3 | 1.0125 | 0.0880 | 93.9 | 1.0214 | 0.1290 | 94.0 | 3.3267 | 0.9484 | 95.0 |

150 | 1 | 1 | 3 | 1.0019 | 0.0694 | 93.3 | 1.0107 | 0.1028 | 94.3 | 3.1859 | 0.6382 | 94.6 |

200 | 1 | 1 | 3 | 1.0076 | 0.0609 | 94.6 | 1.0039 | 0.0880 | 94.8 | 3.1639 | 0.5385 | 95.4 |

50 | 1 | 1 | 4 | 0.9766 | 0.1179 | 93.4 | 1.0465 | 0.1862 | 95.0 | 4.1195 | 2.1138 | 89.4 |

100 | 1 | 1 | 4 | 1.0065 | 0.0852 | 94.2 | 1.0174 | 0.1216 | 95.2 | 4.4275 | 1.5731 | 95.3 |

150 | 1 | 1 | 4 | 1.0115 | 0.0699 | 94.7 | 1.0015 | 0.0960 | 95.5 | 4.3927 | 1.2308 | 94.9 |

200 | 1 | 1 | 4 | 1.0109 | 0.0597 | 95.0 | 0.9999 | 0.0823 | 93.5 | 4.3101 | 0.9668 | 95.1 |

50 | 3 | 1 | 2 | 2.9590 | 0.4196 | 92.0 | 1.0179 | 0.2117 | 94.0 | 2.3601 | 0.7448 | 98.0 |

100 | 3 | 1 | 2 | 3.1021 | 0.3145 | 98.0 | 1.0034 | 0.1495 | 96.0 | 2.4675 | 0.5667 | 100.0 |

150 | 3 | 1 | 2 | 3.1885 | 0.2512 | 92.0 | 1.0075 | 0.1156 | 98.0 | 2.6076 | 0.4875 | 100.0 |

200 | 3 | 1 | 2 | 3.1785 | 0.2199 | 94.0 | 0.9806 | 0.0982 | 94.0 | 2.6012 | 0.4217 | 90.0 |

50 | 2 | 1 | 2 | 2.0166 | 0.2702 | 94.0 | 1.0699 | 0.2161 | 86.0 | 2.3286 | 0.7356 | 94.0 |

100 | 2 | 1 | 2 | 2.0362 | 0.1823 | 92.0 | 1.0260 | 0.1374 | 92.0 | 2.2804 | 0.4400 | 98.0 |

150 | 2 | 1 | 2 | 2.0972 | 0.1616 | 92.0 | 0.9978 | 0.1112 | 90.0 | 2.4041 | 0.4160 | 98.0 |

200 | 2 | 1 | 2 | 2.1220 | 0.1372 | 86.0 | 0.9856 | 0.0903 | 88.0 | 2.4925 | 0.3741 | 96.0 |

n | $\overline{\mathit{z}}$ | ${\mathit{s}}^{2}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

86 | 88.5698 | 56.4683 | 2.1948 | 13.4317 |

**Table 4.**Parameter estimates, standard error (SE), and values for PM and SPM models for Rubidium concentration.

Parameter Estimates | PM (SE) | SPM (SE) |
---|---|---|

$\beta $ | 81.6487 (6.6748) | 67.6313 (6.0901) |

$\gamma $ | 0.6311 (0.0468) | 0.6398 (0.0890) |

q | 4.7829 (1.9498) | |

AIC | 922.1212 | 910.826 |

BIC | 927.030 | 918.188 |

n | $\overline{\mathit{z}}$ | ${\mathit{s}}^{2}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

315 | 832.7143 | 347261.600 | 4.453 | 40.447 |

**Table 6.**Parameter estimates, (SE) indicates standard error estimates, and log-likelihood values for PM and SPM models for the dietary retinol concentration.

Parameter Estimates | PM (SE) | SPM (SE) |
---|---|---|

$\beta $ | 755.999 (35.916) | 566.988 (29.268) |

$\gamma $ | 0.566 (0.019) | 1.024 (0.078) |

q | − | 3.110 (0.437) |

AIC | 4810.924 | 4699.180 |

BIC | 4818.429 | 4710.438 |

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**MDPI and ACS Style**

Astorga, J.M.; Reyes, J.; Santoro, K.I.; Venegas, O.; Gómez, H.W.
A Reliability Model Based on the Incomplete Generalized Integro-Exponential Function. *Mathematics* **2020**, *8*, 1537.
https://doi.org/10.3390/math8091537

**AMA Style**

Astorga JM, Reyes J, Santoro KI, Venegas O, Gómez HW.
A Reliability Model Based on the Incomplete Generalized Integro-Exponential Function. *Mathematics*. 2020; 8(9):1537.
https://doi.org/10.3390/math8091537

**Chicago/Turabian Style**

Astorga, Juan M., Jimmy Reyes, Karol I. Santoro, Osvaldo Venegas, and Héctor W. Gómez.
2020. "A Reliability Model Based on the Incomplete Generalized Integro-Exponential Function" *Mathematics* 8, no. 9: 1537.
https://doi.org/10.3390/math8091537