# Parametric Distributions for Survival and Reliability Analyses, a Review and Historical Sketch

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## Abstract

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## 1. Introduction

## 2. Fundamentals of Survival Analysis

#### 2.1. Definition of Survival Time

#### 2.2. Survival Function

#### 2.3. Hazard Function

#### 2.4. Shape of Hazard Function

## 3. Parametric Models

#### 3.1. Exponential Distribution and Its Variants

#### 3.2. Weibull Distribution

#### 3.3. Lognormal Distribution

#### 3.4. Log-Logistic Distribution

#### 3.5. Gamma Distribution

#### 3.6. Generalized Gamma Distribution

#### 3.7. Burr Distributions

#### 3.8. Pareto Distributions

#### 3.9. Spline Distributions

#### 3.10. Other Distributions

## 4. Data Analysis

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. How We Searched a List of Parametric Distributions?

## Appendix B. Five-Parameter Spline Basis Functions

## Appendix C. Maximum Likelihood Estimator (MLE)

- The exponential distribution:$$\ell \left(\lambda \right)=n\mathrm{ln}\lambda -\lambda {\displaystyle \sum}_{i=1}^{n}{x}_{i},\widehat{\lambda}=\frac{n}{{{\displaystyle \sum}}_{i=1}^{n}{x}_{i}}.$$

- The Weibull distribution:$$\ell \left(\lambda ,\alpha \right)=n\mathrm{ln}\left(\lambda \right)+n\mathrm{ln}\left(\alpha \right)+\left(\alpha -1\right){\displaystyle \sum}_{i=1}^{n}\mathrm{ln}\left({x}_{i}\right)-\lambda {\displaystyle \sum}_{i=1}^{n}{x}_{i}^{\alpha},$$$$\left(\widehat{\lambda},\widehat{\alpha}\right)={\mathrm{argmax}}_{\left(\lambda ,\alpha \right)}\ell \left(\lambda ,\alpha \right)\mathrm{via}\mathrm{a}\mathrm{numerical}\mathrm{optimization}\mathrm{method}.$$

- The lognormal distribution:$$\ell \left(\mu ,{\sigma}^{2}\right)=-\frac{n}{2}\mathrm{ln}\left(2\pi \right)-\frac{n}{2}\mathrm{ln}\left({\sigma}^{2}\right)-{\displaystyle \sum}_{i=1}^{n}\mathrm{ln}\left({x}_{i}\right)-\frac{1}{2{\sigma}^{2}}{\displaystyle \sum}_{i=1}^{n}{\left\{\mathrm{ln}\left({x}_{i}\right)-\mu \right\}}^{2},\phantom{\rule{0ex}{0ex}}\widehat{\mu}=\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}\mathrm{ln}\left({x}_{i}\right),\widehat{{\sigma}^{2}}=\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}{\left\{\mathrm{ln}\left({x}_{i}\right)-\widehat{\mu}\right\}}^{2}.$$

- The Pareto type I distribution:$$\ell \left(\lambda ,\alpha \right)=n\mathrm{ln}\left(\alpha \right)-n\alpha \mathrm{ln}\left(\lambda \right)-\left(\alpha +1\right){\displaystyle \sum}_{i=1}^{n}\mathrm{ln}\left({x}_{i}\right),\underset{i}{\mathrm{min}}\left({x}_{i}\right)\ge \frac{1}{\lambda},\phantom{\rule{0ex}{0ex}}\widehat{\alpha}=\frac{n}{{{\displaystyle \sum}}_{i=1}^{n}\mathrm{ln}\left({x}_{i}/{x}_{\left(1\right)}\right)},\widehat{\lambda}=\frac{1}{{x}_{\left(1\right)}},{x}_{\left(1\right)}=\underset{i}{\mathrm{min}}\left({x}_{i}\right).$$

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**Figure 1.**A survival function $S\left(x\right)=1/x$, $x\ge 1$. The area under $S\left(x\right)$ is shown in red color.

**Figure 2.**Hazard functions estimated by data from the residents of the United States. The left panel uses the whole range (age 1 to 85), while the right panel uses the restricted range (age 1 to 12).

**Figure 5.**Hazard functions of the log-logistic distribution with parameters $\alpha $ and $\lambda $. The mode ${\left\{\left(\alpha -1\right)/\lambda \right\}}^{1/\alpha}$ is denoted by circles.

**Figure 7.**Hazard functions defined by the spline functions $h\left(x\right)={\displaystyle \sum}_{\ell =1}^{5}{h}_{\ell}{M}_{\ell}\left(x\right)$, where the parameters were estimated by the breast cancer data analysis of Emura et al. [122].

**Figure 8.**Empirical survival function and fitted parametric survival functions based on a dataset consisting of failure times (in hours) of transmission on n = 15 caterpillar tractors.

Distribution | Parameter | Hazard Function | Survival Function | Expectation |
---|---|---|---|---|

Exponential | $\lambda >0$ | $\lambda $ | $\mathrm{exp}\left(-\lambda x\right)$ | $1/\lambda $ |

Piecewise Exponential | ${\lambda}_{i}>0,$ $j=1,\dots ,m$ | $\sum}_{j=1}^{m}{\lambda}_{j}{I}_{\left({a}_{j-1},{a}_{j}\right]}\left(x\right)$ | $\mathrm{exp}\left\{-{\lambda}_{j}\left(x-{a}_{j-1}\right)-{\displaystyle \sum}_{k=1}^{j-1}{\lambda}_{k}\Delta {a}_{k}\right\},x\in \left({a}_{j-1},{a}_{j}\right]$ | - |

Weibull | $\alpha ,\lambda >0$ | $\alpha \lambda {x}^{\alpha -1}$ | $\mathrm{exp}\left(-\lambda {x}^{\alpha}\right)$ | ${\lambda}^{-1/\alpha}\mathsf{\Gamma}\left(1+1/\alpha \right)$ |

Rayleigh | $\lambda >0$ | $2\lambda x$ | $\mathrm{exp}\left(-\lambda {x}^{2}\right)$ | ${\lambda}^{-1/2}\sqrt{\pi}/2$ |

Gamma | $\beta ,\lambda >0$ | $\frac{-dS\left(x\right)/dx}{S\left(x\right)}$ | $\underset{x}{\overset{\infty}{{\displaystyle \int}}}\frac{{\lambda}^{\beta}{t}^{\beta -1}\mathrm{exp}\left(-\lambda t\right)}{\mathsf{\Gamma}\left(\beta \right)}dt$ | $\frac{\beta}{\lambda}$ |

Lognormal | $\sigma >0$ | $\frac{1}{S\left(x\right)\sigma x}\varphi \left(\frac{\mathrm{ln}x-\mu}{\sigma}\right)$ | $1-\mathsf{\Phi}\left(\frac{\mathrm{ln}x-\mu}{\sigma}\right)$ | $\mathrm{exp}\left(\mu +\frac{{\sigma}^{2}}{2}\right)$ |

Log-logistic | $\alpha $,$\lambda >0$ | $\frac{\alpha \lambda {x}^{\alpha -1}}{1+\lambda {x}^{\alpha}}$ | $\frac{1}{1+\lambda {x}^{\alpha}}$ | ${\lambda}^{-\frac{1}{\alpha}}\mathsf{\Gamma}\left(1+\frac{1}{\alpha}\right)\mathsf{\Gamma}\left(1-\frac{1}{\alpha}\right)$ |

Pareto I | $\alpha ,\lambda >0$ | $\frac{\alpha}{x}I\left(x\ge \frac{1}{\lambda}\right)$ | $\frac{1}{{\left(\lambda x\right)}^{\alpha}}I\left(x\ge \frac{1}{\lambda}\right)$ | $\frac{\alpha}{\lambda \left(\alpha -1\right)},\alpha 1$ |

Pareto II | $\alpha ,\lambda >0$ | $\alpha \lambda /\left(1+\lambda x\right)$ | ${\left(1+\lambda x\right)}^{-\alpha}$ | $1/\left[\lambda \left(\alpha -1\right)\right],\alpha 1$ |

Pareto IV | $\alpha ,\lambda 0;$ $\delta \ge 0$ | $\frac{\alpha \lambda}{\delta}\frac{{\left(\lambda x\right)}^{1/\delta -1}}{1+{\left(\lambda x\right)}^{1/\delta}}$ | ${\left\{1+{\left(\lambda x\right)}^{\frac{1}{\delta}}\right\}}^{-\alpha}$ | $\alpha {\lambda}^{-1}\mathrm{B}\left(\alpha -\delta ,1+\delta \right),$ $\alpha >\delta >0$ |

Hjorth | $\alpha ,\delta \ge 0;$ $\lambda >0$ | $\delta x+\frac{\alpha}{1+\lambda x}$ | $\frac{\mathrm{exp}\left(-\frac{\delta {x}^{2}}{2}\right)}{{\left(1+\lambda x\right)}^{\frac{\alpha}{\lambda}}}$ | $\left(\frac{1}{\lambda}\right)\mathrm{H}\left(\frac{\delta}{{\lambda}^{2}},\frac{\delta}{\lambda}\right)$ |

Burr III | $\alpha ,\delta >0$ | $\frac{-dS\left(x\right)/dx}{S\left(x\right)}$ | $1-{\left(1+{x}^{-\delta}\right)}^{-\alpha}$ | ${\alpha}^{\frac{1}{\delta}}\mathrm{B}\left(1-\frac{1}{\delta},\alpha +\frac{1}{\delta}\right)$ |

Burr XII | $\alpha ,\delta >0$ | $\frac{\alpha \delta {x}^{\delta -1}}{\left(1+{x}^{\delta}\right)}$ | ${\left(1+{x}^{\delta}\right)}^{-\alpha}$ | $\alpha \mathrm{B}\left(\alpha -\frac{1}{\delta},1+\frac{1}{\delta}\right)$ |

Exponential power | $\alpha ,\lambda >0$ | $\alpha \lambda {\left(\lambda x\right)}^{\alpha -1}{e}^{{\left(\lambda x\right)}^{\alpha}}$ | $\mathrm{exp}\left\{1-{e}^{{\left(\lambda x\right)}^{\alpha}}\right\}$ | - |

Gompertz | $\alpha ,\lambda >0$ | $\alpha \lambda {e}^{\lambda x}$ | $\mathrm{exp}\left\{-\alpha \left({e}^{\lambda x}-1\right)\right\}$ | $\left(1/\lambda \right)\mathrm{exp}\left(\alpha \right)\mathrm{Ei}\left(\alpha \right)$ |

Generalized Gamma | $\alpha ,\beta ,\lambda >0$ | $\frac{-dS\left(x\right)/dx}{S\left(x\right)}$ | $\underset{x}{\overset{\infty}{{\displaystyle \int}}}\frac{\alpha {\lambda}^{\beta}{t}^{\alpha \beta -1}\mathrm{exp}\left(-\lambda {t}^{\alpha}\right)}{\mathsf{\Gamma}\left(\beta \right)}dt$ | $\underset{0}{\overset{\infty}{{\displaystyle \int}}}S\left(x\right)dx$ |

Birnbaum–Saunders | $\beta ,\sigma >0$ | $\frac{-dS\left(x\right)/dx}{S\left(x\right)}$ | $\mathsf{\Phi}\left[\frac{1}{\sigma}\left({\left(\frac{x}{\beta}\right)}^{-\frac{1}{2}}-{\left(\frac{x}{\beta}\right)}^{\frac{1}{2}}\right)\right]$ | $\beta \left(1+\frac{{\sigma}^{2}}{2}\right)$ |

Exponential-logarithmic | $\lambda >0;$ $0<p<1$ | $\frac{-dS\left(x\right)/dx}{S\left(x\right)}$ | $\frac{\mathrm{ln}\left[1-\left(1-p\right){\mathrm{e}}^{-\lambda x}\right]}{\mathrm{ln}\left(p\right)}$ | $-\frac{\mathrm{polylog}\left(2,1-p\right)}{\lambda \mathrm{ln}\left(p\right)}$ |

Generalized-Exponential | $\beta ,\lambda >0$ | $\left[-dS\left(x\right)/dx\right]/S\left(x\right)$ | $1-{\left[1-\mathrm{exp}\left(-\lambda x\right)\right]}^{\beta}$ | Gupta and Kundu [36] |

Exponentiated-Weibull | $\beta ,\lambda ,\gamma >0$ | $\left[-dS\left(x\right)/dx\right]/S\left(x\right)$ | $1-{\left[1-\mathrm{exp}\left(-\lambda {x}^{\gamma}\right)\right]}^{\beta}$ | Nadarajah and Gupta [37] |

G-modified Weibull | $\alpha ,\beta ,\gamma ,\lambda >0$ | $\left[-dS\left(x\right)/dx\right]/S\left(x\right)$ | $1-{\left[1-\mathrm{exp}\left(-\alpha {x}^{\gamma}{\mathrm{e}}^{\lambda x}\right)\right]}^{\beta}$ | Carrasco et al. [38] |

M-spline | ${h}_{\ell}\ge 0,$ $\ell =1,\dots ,L$ | $\sum}_{\ell =1}^{L}{h}_{\ell}{M}_{\ell}\left(x\right)$ | $\mathrm{exp}\left\{-{\displaystyle \sum}_{\ell =1}^{L}{h}_{\ell}{M}_{\ell}\left(x\right)\right\}$ | - |

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**MDPI and ACS Style**

Taketomi, N.; Yamamoto, K.; Chesneau, C.; Emura, T.
Parametric Distributions for Survival and Reliability Analyses, a Review and Historical Sketch. *Mathematics* **2022**, *10*, 3907.
https://doi.org/10.3390/math10203907

**AMA Style**

Taketomi N, Yamamoto K, Chesneau C, Emura T.
Parametric Distributions for Survival and Reliability Analyses, a Review and Historical Sketch. *Mathematics*. 2022; 10(20):3907.
https://doi.org/10.3390/math10203907

**Chicago/Turabian Style**

Taketomi, Nanami, Kazuki Yamamoto, Christophe Chesneau, and Takeshi Emura.
2022. "Parametric Distributions for Survival and Reliability Analyses, a Review and Historical Sketch" *Mathematics* 10, no. 20: 3907.
https://doi.org/10.3390/math10203907