# Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Epsilon–Positive Family

Algorithm 1 Algorithm to generate observations from an epsilon–positive distribution. |

Require: Initialize the algorithm fixing $\mathsf{\Psi}$ and $\epsilon $ |

1: Generate Y from ${g}_{Y}(\xb7)$ and U from $\mathcal{B}er\left(p=\frac{1+\epsilon}{2}\right)$ |

2: if $U=1$ then |

3: ${U}_{\epsilon}\leftarrow 1+\epsilon $ |

4: else |

5: ${U}_{\epsilon}\leftarrow 1-\epsilon $ |

6: end if |

7: return $X={U}_{\epsilon}Y.$ |

#### 2.1. Reliability Properties

#### 2.2. Mean Residual Life

#### 2.3. Stress-Strength Parameter

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

#### 2.4. Maximum Likelihood Estimation

#### 2.5. MLE via the EM Algorithm

**E–step:**For $i=1,\cdots ,n$, compute$${w}_{ij}^{\left(s\right)}=\frac{g\left(\frac{{x}_{i}}{1+{\epsilon}^{\left(s\right)}}\right)}{g\left(\frac{{x}_{i}}{1+{\epsilon}^{\left(s\right)}}\right)+g\left(\frac{{x}_{i}}{1-{\epsilon}^{\left(s\right)}}\right)},\phantom{\rule{1.em}{0ex}}{\xi}_{j}=1+\epsilon .$$**M–step:**Given ${\epsilon}^{\left(s\right)}$ and ${\mathsf{\Psi}}^{\left(s\right)}$, compute$$\begin{array}{cc}\hfill {\epsilon}^{(s+1)}& =\frac{2}{n}\sum _{i=1}^{n}{w}_{ij}^{\left(s\right)}-1,\phantom{\rule{1.em}{0ex}}j=1+\epsilon ,\hfill \\ \hfill {\mathsf{\Psi}}^{(s+1)}& =arg\underset{\mathsf{\Psi}}{max}\mathcal{Q}(\theta ,{\widehat{\theta}}^{\left(s\right)}).\hfill \end{array}$$

## 3. The Epsilon–Exponential Distribution

#### Numerical Experiments

## 4. Survival and Reliability Analysis

#### 4.1. Estimation Using the EM Algorithm

**E–step:**For $i=1,\cdots ,n$, compute$${w}_{ij}^{\left(s\right)}=\frac{{w}_{i,+}}{{w}_{i,+}+{w}_{i,-}},\phantom{\rule{1.em}{0ex}}for\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\xi}_{j}=1+\epsilon ,$$$${w}_{i,+}={\left[g\left(\frac{{t}_{i}}{1+{\epsilon}^{\left(s\right)}}\right)\right]}^{{d}_{i}}{\left[(1+{\epsilon}^{\left(s\right)})S\left(\frac{{t}_{i}}{1+{\epsilon}^{\left(s\right)}}\right)\right]}^{1-{d}_{i}}$$$${w}_{i,-}={\left[g\left(\frac{{t}_{i}}{1-{\epsilon}^{\left(s\right)}}\right)\right]}^{{d}_{i}}{\left[(1-{\epsilon}^{\left(s\right)})S\left(\frac{{t}_{i}}{1-{\epsilon}^{\left(s\right)}}\right)\right]}^{1-{d}_{i}}.$$**M–step:**Given ${\epsilon}^{\left(s\right)}$ and ${\mathsf{\Psi}}^{\left(s\right)}$, compute$$\begin{array}{cc}\hfill {\epsilon}^{(s+1)}& =\frac{2}{n}\sum _{i=1}^{n}{w}_{ij}^{\left(s\right)}-1,\phantom{\rule{1.em}{0ex}}{\xi}_{j}=1+\epsilon ,\hfill \\ \hfill {\mathsf{\Psi}}^{(s+1)}& =arg\underset{\mathsf{\Psi}}{max}\mathcal{Q}(\theta ,{\widehat{\theta}}^{\left(s\right)}).\hfill \end{array}$$

#### EM Algorithm for the Epsilon-Exponential Distribution

**E–step:**For $i=1,\cdots ,n$, compute$${w}_{ij}^{\left(s\right)}=\frac{{w}_{i,+}}{{w}_{i,+}+{w}_{i,-}},\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\xi}_{j}=1+\epsilon ,$$$${w}_{i,+}={\left[\frac{1}{\sigma}{e}^{-{t}_{i}/(1+{\epsilon}^{\left(s\right)}){\sigma}^{\left(s\right)}}\right]}^{{d}_{i}}{\left[(1+{\epsilon}^{\left(s\right)}){e}^{-{t}_{i}/(1+{\epsilon}^{\left(s\right)}){\sigma}^{\left(s\right)}}\right]}^{1-{d}_{i}}$$$${w}_{i,-}={\left[\frac{1}{\sigma}{e}^{-{t}_{i}/(1-{\epsilon}^{\left(s\right)}){\sigma}^{\left(s\right)}}\right]}^{{d}_{i}}{\left[(1-{\epsilon}^{\left(s\right)}){e}^{-{t}_{i}/(1-{\epsilon}^{\left(s\right)}){\sigma}^{\left(s\right)}}\right]}^{1-{d}_{i}}.$$**M–step:**Given ${\epsilon}^{\left(s\right)}$ and ${\sigma}^{\left(s\right)}$, compute$$\begin{array}{cc}\hfill {\epsilon}^{(s+1)}& =\frac{2}{n}\sum _{i=1}^{n}{w}_{ij}^{\left(s\right)}-1,\phantom{\rule{1.em}{0ex}}{\xi}_{j}=1+\epsilon ,\hfill \\ \hfill {\sigma}^{(s+1)}& =\sum _{i=1}^{n}\sum _{j\in \mathcal{J}\left({\epsilon}^{\left(s\right)}\right)}{w}_{ij}^{\left(s\right)}\left(\frac{{t}_{i}}{j}\right)/\sum _{i=1}^{n}\sum _{j\in \mathcal{J}\left({\epsilon}^{\left(s\right)}\right)}{w}_{ij}^{\left(s\right)}{d}_{i}.\hfill \end{array}$$

## 5. Real Data Examples

#### 5.1. Example 1: Maintenance Data

#### 5.2. Example 2: Recidivism Data

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. The EE–MLE

## Appendix B. An EM–Type Algorithm for the EE–MLE

**E–step:**For $i=1,\cdots ,n$, compute$${w}_{ij}^{\left(s\right)}=\frac{{e}^{-{x}_{i}/(1+{\epsilon}^{\left(s\right)}){\sigma}^{\left(s\right)}}}{{e}^{-{x}_{i}/(1+{\epsilon}^{\left(s\right)}){\sigma}^{\left(s\right)}}+{e}^{-{x}_{i}/(1-{\epsilon}^{\left(s\right)}){\sigma}^{\left(s\right)}}},\phantom{\rule{1.em}{0ex}}{\xi}_{j}=1+\epsilon .$$**M–step:**Given ${\epsilon}^{\left(s\right)}$ and ${\sigma}^{\left(s\right)}$, compute$$\begin{array}{cc}\hfill {\epsilon}^{(s+1)}& =\frac{2}{n}\sum _{i=1}^{n}{w}_{ij}^{\left(s\right)}-1,\phantom{\rule{1.em}{0ex}}{\xi}_{j}=1+\epsilon ,\hfill \\ \hfill {\sigma}^{(s+1)}& =\frac{1}{n}\sum _{i=1}^{n}\sum _{j=1}^{m\left(s\right)}{w}_{ij}^{\left(s\right)}\left(\frac{{x}_{i}}{{\xi}_{j}}\right).\hfill \end{array}$$

## References

- Mudholkar, G.S.; Srivastava, D.K.; Kollia, G.D. A generalization of the weibull distribution with application to the analysis of survival data. J. Am. Stat. Assoc.
**1996**, 91, 1575–1583. [Google Scholar] [CrossRef] - Gupta, R.D.; Kundu, D. Exponentiated exponential family: An alternative to gamma and weibull distributions. Biom. J.
**2001**, 43, 117–130. [Google Scholar] [CrossRef] - Cooray, K. Analyzing lifetime data with long-tailed skewed distribution: The logistic-sinh family. Stat. Model.
**2005**, 5, 343–358. [Google Scholar] [CrossRef] - Fernández, C.; Steel, M.F. On Bayesian modeling of fat tails and skewness. J. Am. Stat. Assoc.
**1998**, 93, 359–371. [Google Scholar] - Mudholkar, G.S.; Hutson, A.D. The epsilon skew normal distribution for analyzing near normal data. J. Statist. Plann. Inference
**2000**, 83, 291–309. [Google Scholar] [CrossRef] - Arellano-Valle, R.B.; Gómez, H.W.; Quintana, F.A. Statistical inference for a general class of asymmetric distributions. J. Statist. Plann. Inference
**2005**, 128, 427–443. [Google Scholar] [CrossRef] - Jones, M. A note on rescalings, reparametrizations and classes of distributions. J. Statist. Plann. Inference
**2006**, 136, 3730–3733. [Google Scholar] [CrossRef] - Dempster, A.P.; Laird, N.M.; Rubin, D.B. Maximum likelihood from incomplete data via the em algorithm. J. R. Stat. Soc. Ser. B Stat. Methodol.
**1977**, 39, 1–38. [Google Scholar] - Givens, G.H.; Hoeting, J.A. Computational Statistics, 2nd ed.; John Wiley & Sons: New York, NY, USA, 2012. [Google Scholar]
- Chhikara, R.S.; Folks, J.L. The inverse gaussian distribution as a lifetime model. Technometrics
**1977**, 19, 461–468. [Google Scholar] [CrossRef] - Akaike, H. A new look at the statistical model identification. IEEE Trans. Automat. Contr.
**1974**, 19, 716–723. [Google Scholar] [CrossRef]

**Figure 1.**Examples of the probability density $f\left(x\right)$, survival $S\left(x\right)$ and hazard $\lambda \left(x\right)$ functions of epsilon-exponential distribution, $EE(\sigma ,\epsilon )$, and epsilon-Weibull distribution, $EW(\alpha ,\sigma ,\epsilon )$. Please note that the exponential and Weibull distributions correspond to the case $\epsilon =0$.

**Figure 2.**Examples of the probability density $f\left(x\right)$, survival $S\left(x\right)$ and hazard $\lambda \left(x\right)$ functions of epsilon-log-logistic distribution, $ELL(\sigma ,\epsilon )$, and epsilon-gamma distribution, $EG(\alpha ,\sigma ,\epsilon )$. Please note that the log-logistic and gamma distributions correspond to the case $\epsilon =0$.

**Figure 4.**The density functions of the fitted epsilon exponential, exponential, Weibull and exponentiated exponential distributions.

**Figure 5.**Fit of the survival functions: Kaplan–Meier estimator (solid line), exponential (dashed line red) and epsilon–exponential (dashed line blue) distributions.

**Table 1.**Hazard rate, $\lambda (\xb7)$, Survival, $S(\xb7)$, and density, $f(\xb7)$, functions of some probability models that can be generalized using the definition in (2) In the table $I(a,\beta )={\int}_{0}^{a}\mathsf{\Gamma}{\left(\beta \right)}^{-1}{u}^{\beta -1}{e}^{-u}du$.

$\mathit{\lambda}\left(\mathit{y}\right)$ | $\mathit{S}\left(\mathit{y}\right)$ | $\mathit{f}\left(\mathit{y}\right)$ | |
---|---|---|---|

Exponential | $\frac{1}{\sigma}\phantom{\rule{0.166667em}{0ex}}(>0)$ | $exp(-\frac{y}{\sigma})$ | $\frac{1}{\sigma}exp(-\frac{y}{\sigma})$ |

Weibull | $\left(\frac{\beta}{{\sigma}^{\beta}}\right){y}^{(\beta -1)}\phantom{\rule{0.166667em}{0ex}}(\beta ,\sigma >0)$ | $exp(-{\left[\frac{y}{\sigma}\right]}^{\beta})$ | $\frac{\beta {y}^{(\beta -1)}}{{\sigma}^{\beta}}exp(-{\left[\frac{y}{\sigma}\right]}^{\beta})$ |

Log-logistic | $\frac{\left(\frac{\beta}{\sigma}\right){\left(\frac{y}{\sigma}\right)}^{\beta -1}}{1+{\left(\frac{y}{\sigma}\right)}^{\beta}}\phantom{\rule{0.166667em}{0ex}}(\beta ,\sigma >0)$ | ${(1+{\left[\frac{y}{\sigma}\right]}^{\beta})}^{-1}$ | $\frac{\left(\frac{\beta}{\sigma}\right){\left(\frac{y}{\sigma}\right)}^{\beta -1}}{{(1+{\left(\frac{y}{\sigma}\right)}^{\beta})}^{2}}$ |

Gamma | $\frac{f\left(y\right)}{S\left(y\right)}$ | $1-I(y/\sigma ,\beta )$ | $\frac{{y}^{(\beta -1)}exp(-\frac{y}{\sigma})}{{\sigma}^{\beta}\mathsf{\Gamma}\left(\beta \right)}$ |

True Value | n | $\mathit{\sigma}$ | $\mathit{\epsilon}$ | |||
---|---|---|---|---|---|---|

$\mathit{\sigma}$ | $\mathit{\epsilon}$ | Estimate | SD | Estimate | SD | |

n = 50 | 0.286 | 0.047 | 0.376 | 0.126 | ||

n = 100 | 0.292 | 0.037 | 0.350 | 0.127 | ||

0.3 | n = 200 | 0.292 | 0.030 | 0.335 | 0.118 | |

n = 500 | 0.295 | 0.022 | 0.317 | 0.106 | ||

n = 1000 | 0.299 | 0.017 | 0.302 | 0.089 | ||

n = 50 | 0.287 | 0.054 | 0.558 | 0.133 | ||

n = 100 | 0.290 | 0.043 | 0.542 | 0.129 | ||

0.3 | 0.5 | n = 200 | 0.294 | 0.037 | 0.527 | 0.123 |

n = 500 | 0.297 | 0.030 | 0.512 | 0.111 | ||

n = 1000 | 0.298 | 0.023 | 0.506 | 0.091 | ||

n = 50 | 0.299 | 0.059 | 0.818 | 0.126 | ||

n = 100 | 0.299 | 0.047 | 0.809 | 0.122 | ||

0.8 | n = 200 | 0.301 | 0.038 | 0.801 | 0.108 | |

n = 500 | 0.303 | 0.029 | 0.794 | 0.088 | ||

n = 1000 | 0.302 | 0.022 | 0.794 | 0.069 | ||

n = 50 | 0.476 | 0.079 | 0.379 | 0.125 | ||

n = 100 | 0.484 | 0.062 | 0.355 | 0.125 | ||

0.3 | n = 200 | 0.486 | 0.049 | 0.337 | 0.121 | |

n = 500 | 0.494 | 0.037 | 0.316 | 0.107 | ||

n = 1000 | 0.497 | 0.029 | 0.303 | 0.088 | ||

n = 50 | 0.477 | 0.088 | 0.558 | 0.133 | ||

n = 100 | 0.484 | 0.074 | 0.542 | 0.131 | ||

0.5 | 0.5 | n = 200 | 0.488 | 0.059 | 0.528 | 0.124 |

n = 500 | 0.494 | 0.048 | 0.512 | 0.109 | ||

n = 1000 | 0.498 | 0.039 | 0.502 | 0.091 | ||

n = 50 | 0.484 | 0.104 | 0.838 | 0.129 | ||

n = 100 | 0.499 | 0.076 | 0.809 | 0.119 | ||

0.8 | n = 200 | 0.503 | 0.064 | 0.799 | 0.110 | |

n = 500 | 0.505 | 0.049 | 0.793 | 0.089 | ||

n = 1000 | 0.504 | 0.038 | 0.793 | 0.070 | ||

n = 50 | 0.762 | 0.125 | 0.374 | 0.124 | ||

n = 100 | 0.771 | 0.099 | 0.356 | 0.125 | ||

0.3 | n = 200 | 0.779 | 0.080 | 0.336 | 0.120 | |

n = 500 | 0.789 | 0.058 | 0.314 | 0.105 | ||

n = 1000 | 0.796 | 0.046 | 0.301 | 0.089 | ||

n = 50 | 0.765 | 0.139 | 0.555 | 0.134 | ||

n = 100 | 0.771 | 0.116 | 0.543 | 0.134 | ||

0.8 | 0.5 | n = 200 | 0.782 | 0.096 | 0.528 | 0.123 |

n = 500 | 0.789 | 0.076 | 0.515 | 0.110 | ||

n = 1000 | 0.797 | 0.062 | 0.503 | 0.091 | ||

n = 50 | 0.791 | 0.153 | 0.815 | 0.130 | ||

n = 100 | 0.800 | 0.126 | 0.804 | 0.122 | ||

0.8 | n = 200 | 0.798 | 0.103 | 0.798 | 0.106 | |

n = 500 | 0.806 | 0.079 | 0.794 | 0.089 | ||

n = 1000 | 0.807 | 0.060 | 0.793 | 0.070 |

0.2 | 0.3 | 0.5 | 0.5 | 0.5 | 0.5 | 0.6 | 0.6 | 0.7 | 0.7 |

0.7 | 0.8 | 0.8 | 1.0 | 1.0 | 1.0 | 1.0 | 1.1 | 1.3 | 1.5 |

1.5 | 1.5 | 1.5 | 2.0 | 2.0 | 2.2 | 2.5 | 2.7 | 3.0 | 3.0 |

3.3 | 3.3 | 4.0 | 4.0 | 4.5 | 4.7 | 5.0 | 5.4 | 5.4 | 7.0 |

7.5 | 8.8 | 9.0 | 10.3 | 22.0 | 24.5 |

Time after Release | % Observed Recidivists |
---|---|

1 month | 4.7% |

3 month | 11.8% |

6 month | 19.9% |

12 month | 30.0% |

18 month | 35.9% |

24 month | 40.8% |

36 month | 46.6% |

48 month | 50.4% |

64 month | 52.2% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Celis, P.; de la Cruz, R.; Fuentes, C.; Gómez, H.W. Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications. *Symmetry* **2021**, *13*, 908.
https://doi.org/10.3390/sym13050908

**AMA Style**

Celis P, de la Cruz R, Fuentes C, Gómez HW. Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications. *Symmetry*. 2021; 13(5):908.
https://doi.org/10.3390/sym13050908

**Chicago/Turabian Style**

Celis, Perla, Rolando de la Cruz, Claudio Fuentes, and Héctor W. Gómez. 2021. "Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications" *Symmetry* 13, no. 5: 908.
https://doi.org/10.3390/sym13050908