# The Exponentiated Lindley Geometric Distribution with Applications

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## Abstract

**:**

## 1. Introduction

## 2. Properties of the ELG distribution

#### 2.1. Probability Density function

#### 2.2. Hazard Rate Function

#### 2.3. Quantile Function

#### 2.4. Order Statistics

#### 2.5. Moment Properties

#### 2.6. Residual Life Function

#### 2.7. Mean Deviations

#### 2.8. Bonferroni and Lorenz Curves

#### 2.9. Entropies

## 3. Estimation of Parameters

#### 3.1. Maximum Likelihood Estimation

#### 3.2. Expectation-Maximization Algorithm

#### 3.3. Censored Maximum Likelihood Estimation

- ${n}_{0}$ is known to have failed at the times ${t}_{1},\dots ,{t}_{{n}_{0}}$,
- ${n}_{1}$ is known to have failed into the interval $[{s}_{i-1},{s}_{i}]$ for $i=1,\dots ,{n}_{1}$,
- ${n}_{2}$ is known to have survived at a time ${r}_{i}$ for $i=1,\dots {n}_{2}$ but not observed any longer.

## 4. Two Real-Data Applications

- (i)
- Gamma$(\beta ,\alpha )$$$\begin{array}{cc}\hfill {f}_{1}\left(x\right)& =\frac{1}{\mathsf{\Gamma}\left(\beta \right)}{\alpha}^{\beta}{x}^{\beta -1}{e}^{-\alpha x},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\beta >0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\alpha >0;\hfill \end{array}$$
- (ii)
- Weibull$(\beta ,\lambda )$$$\begin{array}{cc}\hfill {f}_{2}\left(x\right)& =\frac{\alpha}{\beta}{\left(\frac{x}{\beta}\right)}^{\alpha -1}{e}^{-{(x/\beta )}^{\alpha}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\beta >0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\alpha >0;\hfill \end{array}$$
- (iii)
- LG$(\theta ,p)$$$\begin{array}{cc}\hfill {f}_{3}\left(x\right)& =\frac{{\theta}^{2}}{\theta +1}(1-p)(1+x){e}^{-\theta x}{\left[1-\frac{p(\theta +1+\theta x)}{\theta +1}{e}^{-\theta x}\right]}^{-2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\theta >0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}p<1,\hfill \end{array}$$
- (iv)
- WG$(\alpha ,\beta ,p)$$$\begin{array}{cc}\hfill {f}_{4}\left(x\right)& =\alpha {\beta}^{\alpha}(1-p){x}^{\alpha -1}{e}^{-{\left(\beta x\right)}^{\alpha}}{\left[1-p{e}^{-{\left(\beta x\right)}^{\alpha}}\right]}^{-2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\alpha >0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\beta >0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}p<1,\hfill \end{array}$$

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plots of the pdf of the ELG distribution for different values of $\alpha ,\phantom{\rule{3.33333pt}{0ex}}\theta $, and p.

**Figure 2.**Plots of the hf of the ELG distribution for different values of $\alpha ,\phantom{\rule{3.33333pt}{0ex}}\theta $, and p.

**Figure 3.**Plots of the Shannon entropy of the ELG distribution for different values of $\alpha ,\phantom{\rule{3.33333pt}{0ex}}\theta $, and p.

**Figure 4.**Plots of the estimated density and survival function of the fitted models for the first data set.

**Figure 5.**Plots of the estimated density and survival function of the fitted models for the second data set.

**Table 1.**The first data set: the remission time (in months) of a random sample of 128 bladder cancer patients.

0.08 | 2.09 | 3.48 | 4.87 | 6.94 | 8.66 | 13.11 | 23.63 | 0.20 | 2.23 |

3.52 | 4.98 | 6.97 | 9.02 | 13.29 | 0.40 | 2.26 | 3.57 | 5.06 | 7.09 |

9.22 | 13.80 | 25.74 | 0.50 | 2.46 | 3.64 | 5.09 | 7.26 | 9.47 | 14.24 |

25.82 | 0.51 | 2.54 | 3.70 | 5.17 | 7.28 | 9.74 | 14.76 | 26.31 | 0.81 |

2.62 | 3.82 | 5.32 | 7.32 | 10.06 | 14.77 | 32.15 | 2.64 | 3.88 | 5.32 |

7.39 | 10.34 | 14.83 | 34.26 | 0.90 | 2.69 | 4.18 | 5.34 | 7.59 | 10.66 |

15.96 | 36.66 | 1.05 | 2.69 | 4.23 | 5.41 | 7.62 | 10.75 | 16.62 | 43.01 |

1.19 | 2.75 | 4.26 | 5.41 | 7.63 | 17.12 | 46.12 | 1.26 | 2.83 | 4.33 |

5.49 | 7.66 | 11.25 | 17.14 | 79.05 | 1.35 | 2.87 | 5.62 | 7.87 | 11.64 |

17.36 | 1.40 | 3.02 | 4.34 | 5.71 | 7.93 | 11.79 | 18.10 | 1.46 | 4.40 |

5.85 | 8.26 | 11.98 | 19.13 | 1.76 | 3.25 | 4.50 | 6.25 | 8.37 | 12.02 |

2.02 | 3.31 | 4.51 | 6.54 | 8.53 | 12.03 | 20.28 | 2.02 | 3.36 | 6.76 |

12.07 | 21.73 | 2.07 | 3.36 | 6.93 | 8.65 | 12.63 | 22.69 |

Model | Parameters | AIC | BIC | AICc | ||
---|---|---|---|---|---|---|

Gamma | $\widehat{\alpha}$ = 0.1252 | $\widehat{\beta}=1.1726$ | 830.7356 | 836.4396 | 830.8316 | |

Weibull | $\widehat{\alpha}$ = 1.0478 | $\widehat{\beta}$ = 9.5607 | 832.1738 | 837.8778 | 832.2698 | |

LG | $\widehat{\theta}$ = 0.0742 | $\widehat{p}=0.8898$ | 823.1859 | 833.742 | 823.2819 | |

WG | $\widehat{\alpha}$ = 1.6042 | $\widehat{\beta}$ = 0.0286 | $\widehat{p}$ = 0.9362 | 826.1842 | 834.7403 | 826.3777 |

ELG | $\widehat{\alpha}$ = 1.0792 | $\widehat{\theta}$ = 0.0699 | $\widehat{p}$ = 0.9204 | 824.6214 | 833.1775 | 824.8149 |

Statistic | ||
---|---|---|

Model | ${\mathit{W}}^{*}$ | ${\mathit{A}}^{*}$ |

Gamma | 0.11988 | 0.71928 |

Weibull | 0.13136 | 0.78643 |

LG | 0.05374 | 0.33827 |

WG | 0.01493 | 0.09939 |

ELG | 0.01389 | 0.09498 |

0.8 | 0.8 | 1.3 | 1.5 | 1.8 | 1.9 | 1.9 | 2.1 | 2.6 | 2.7 |

2.9 | 3.1 | 3.2 | 3.3 | 3.5 | 3.6 | 4.0 | 4.1 | 4.2 | 4.2 |

4.3 | 4.3 | 4.4 | 4.4 | 4.6 | 4.7 | 4.7 | 4.8 | 4.9 | 4.9 |

5.0 | 5.3 | 5.5 | 5.7 | 5.7 | 6.1 | 6.2 | 6.2 | 6.2 | 6.3 |

6.7 | 6.9 | 7.1 | 7.1 | 7.1 | 7.1 | 7.4 | 7.6 | 7.7 | 8.0 |

8.2 | 8.6 | 8.6 | 8.6 | 8.8 | 8.8 | 8.9 | 8.9 | 9.5 | 9.6 |

9.7 | 9.8 | 10.7 | 10.9 | 11.0 | 11.0 | 11.1 | 11.2 | 11.2 | 11.5 |

11.9 | 12.4 | 12.5 | 12.9 | 13.0 | 13.1 | 13.3 | 13.6 | 13.7 | 13.9 |

14.1 | 15.4 | 15.4 | 17.3 | 17.3 | 18.1 | 18.2 | 18.4 | 18.9 | 19.0 |

19.9 | 20.6 | 21.3 | 21.4 | 21.9 | 23.0 | 27.0 | 31.6 | 33.1 | 38.5 |

Model | Parameters | AIC | BIC | AICc | ||
---|---|---|---|---|---|---|

Gamma | $\widehat{\alpha}$ = 0.2033 | $\widehat{\beta}=2.0089$ | 638.6002 | 643.8106 | 638.724 | |

Weibull | $\widehat{\alpha}$ = 1.4585 | $\widehat{\beta}$ = 10.9553 | 641.4614 | 646.6717 | 641.5851 | |

LG | $\widehat{\theta}$ = 0.2027 | $\widehat{p}=-0.2427$ | 641.8269 | 647.0372 | 641.9506 | |

WG | $\widehat{\alpha}$ = 1.9789 | $\widehat{\beta}$ = 0.0501 | $\widehat{p}$ = 0.82132 | 639.9084 | 647.7239 | 640.1584 |

ELG | $\widehat{\alpha}$ = 1.4602 | $\widehat{\theta}$ = 0.1725 | $\widehat{p}$ = 0.5385 | 640.3108 | 648.1263 | 640.5608 |

Statistic | ||
---|---|---|

Model | ${\mathit{W}}^{*}$ | ${\mathit{A}}^{*}$ |

Gamma | 0.02761 | 0.18225 |

Weibull | 0.06294 | 0.39624 |

LG | 0.05374 | 0.33827 |

WG | 0.01706 | 0.12365 |

ELG | 0.01801 | 0.12665 |

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**MDPI and ACS Style**

Peng, B.; Xu, Z.; Wang, M.
The Exponentiated Lindley Geometric Distribution with Applications. *Entropy* **2019**, *21*, 510.
https://doi.org/10.3390/e21050510

**AMA Style**

Peng B, Xu Z, Wang M.
The Exponentiated Lindley Geometric Distribution with Applications. *Entropy*. 2019; 21(5):510.
https://doi.org/10.3390/e21050510

**Chicago/Turabian Style**

Peng, Bo, Zhengqiu Xu, and Min Wang.
2019. "The Exponentiated Lindley Geometric Distribution with Applications" *Entropy* 21, no. 5: 510.
https://doi.org/10.3390/e21050510