# A New Flexible Generalized Lindley Model: Properties, Estimation and Applications

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Definition and Basic Properties

**Definition**

**1.**

#### Comparison of GLOm for Small Ms

## 3. Estimation of the Parameters

#### Right Censored Data

## 4. Simulation

- Simulate one random variable of multinomial distribution with parameters n and ${w}_{i}$, $i=1,2,\dots ,m$, ${\sum}_{i=1}^{m}{w}_{i}=1$. Assume the generated instance be denoted by ${k}_{1},{k}_{2},\dots ,{k}_{m}$ corresponding respectively to probabilities ${w}_{1},{w}_{2},\dots ,{w}_{m}$. Please note that ${\sum}_{i=1}^{m}{k}_{i}=n$.
- Simulate samples of sizes ${k}_{i}$, $i=1,2,\dots ,m$ from gamma distribution with parameters $(m-i+1,\theta )$. Then, we can merge these samples to provide one sample of size n from GLOm model.

- As m increases, B, ${B}^{*}$ and $MSE$ of $\widehat{\theta}$ decrease.
- Also, B, ${B}^{*}$ and $MSE$ increase with $\theta $.

#### Censored Data

- The mean of the absolute bias (${B}^{*}$) is slightly greater than that for the complete (non-censored) data and increases with rising the portion of the censored part of the sample.
- For small $\theta $s, in the presence of censorship, the bias exhibits more fluctuation around zero and is more skewed to left. It may, therefore, cause smaller values for B and larger values for ${B}^{*}$. For large $\theta $s, $\widehat{\theta}$ is less than the actual value of the parameter and it causes B to be negative and ${B}^{*}$ being its absolute value.
- Analogously as dealing with non-censored data, when m increases ($\theta $ decreases), B, ${B}^{*}$ and $MSE$ of $\widehat{\theta}$ decrease.

## 5. Real Data Examples

#### 5.1. Failure of Yarn

#### 5.2. Ovarian Cancer

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Lindley, D.V. Fiducial distributions and Bayes’ theorem. J. R. Stat. Soc. Ser. (Methodol.)
**1958**, 20, 102–107. [Google Scholar] [CrossRef] - Cakmakyapan, S.; Ozel, G. The Lindley family of distributions: Properties and applications. Hacet. J. Math. Stat.
**2017**, 46, 1113–1137. [Google Scholar] [CrossRef] - Ghitany, M.E.; Atieh, B.; Nadarajah, S. Lindley distribution and its application. Math. Comput. Simul.
**2008**, 78, 493–506. [Google Scholar] [CrossRef] - Sankaran, M. The discrete poisson-Lindley distribution. Biometrics
**1970**, 26, 145–149. [Google Scholar] [CrossRef] - Ghitany, M.E.; Al-Mutairi, D.K.; Nadarajah, S. Zero-truncated Poisson-Lindley distribution and its application. Math. Comput. Simul.
**2008**, 79, 279–287. [Google Scholar] [CrossRef] - Zamani, H.; Ismail, N. Negative Binomial-Lindley Distribution and Its Application. J. Math. Stat.
**2010**, 6, 4–9. [Google Scholar] [CrossRef] [Green Version] - Ghitany, M.E.; Al-Mutairi, D.K.; Awadhi, F.A.; Alburais, M. Marshall-Olkin extended Lindley distribution and its application. Int. J. Appl. Math.
**2012**, 25, 709–721. [Google Scholar] - Ghitany, M.E.; Al-Mutairi, D.K.; Balakrishnan, N.; Al-Enezi, L.J. Power Lindley distribution and associated inference. Comput. Stat. Data Anal.
**2013**, 64, 20–33. [Google Scholar] [CrossRef] - Al-Mutairi, D.K.; Ghitany, M.E.; Kundu, D. Inferences on stress-strength reliability from Lindley distribution. Commun. Stat.-Theory Methods
**2013**, 42, 1443–1463. [Google Scholar] [CrossRef] - Al-babtain, A.A.; Eid, H.A.; A-Hadi, N.A.; Merovci, F. The five parameter Lindley distribution. Pak. J. Stat.
**2014**, 31, 363–384. [Google Scholar] - Ghitany, M.E.; Al-Mutairi, D.K.; Aboukhamseen, S.M. Estimation of the reliability of a stress-strength system from power Lindley distributions. Commun. Stat.-Simul. Comput.
**2015**, 44, 118–136. [Google Scholar] [CrossRef] - Al-Mutairi, D.K.; Ghitany, M.E.; Kundu, D. Inferences on stress-strength reliability from weighted lindley distributions. Commun. Stat.-Theory Methods
**2015**, 44, 4096–4113. [Google Scholar] [CrossRef] - Abouammoh, A.M.; Alshangiti Arwa, M.; Ragab, I.E. A new generalized Lindley distribution. J. Stat. Comput. Simul.
**2015**, 85, 3662–3678. [Google Scholar] [CrossRef] - Shanker, R.; Fesshaye, H.; Sharma, S. On Two-Parameter Lindley Distribution and its Applications to Model Lifetime Data. Biom. Biostat. Int. J.
**2016**, 1, 9–15. [Google Scholar] [CrossRef] - Shanker, R.; Mishra, A. A quasi Lindley distribution. Afr. J. Math. Comput. Sci. Res.
**2013**, 6, 64–71. [Google Scholar] - Shanker, R.; Ghebretsadik, A.H. A New Quasi Lindley Distribution. Int. J. Stat. Syst.
**2013**, 8, 143–156. [Google Scholar] - Merovci, F.; Sharma, V.K. The Beta-Lindley Distribution: Properties and Applications. J. Appl. Math.
**2014**, 2014, 198951. [Google Scholar] [CrossRef] - Zakerzadeh, H.; Dolati, A. Generalized Lindley distribution. J. Math. Ext.
**2009**, 3, 13–25. [Google Scholar] - Ibrahim, E.; Merovci, F.; Elgarhy, M. A new generalized Lindley distribution. Math. Theory Model.
**2013**, 3, 30–47. [Google Scholar] - Shanker, R.; Shukla, K.K.; Shanker, R.; Leonida, T.A. A Three-Parameter Lindley Distribution. Am. J. Math. Stat.
**2017**, 7, 15–26. [Google Scholar] - Broderick, O.O.; Tiantian, Y. A new class of generalized Lindley distributions with applications. J. Stat. Comput. Simul.
**2015**, 85, 2072–2100. [Google Scholar] [CrossRef] - Wolstenholme, L.C. Reliability Modelling: A Statistical Approach; Chapman and Hall/CRC: London, UK, 1999; ISBN 9781584880141. [Google Scholar]
- McPherson, J.W. Reliability Physics and Engineering: Time-To-Failure Modeling; Springer: Berlin/Heidelberg, Germany, 2010; ISBN 978-1-4419-6348-2. [Google Scholar]
- Cheng, R.; Feast, G. Some Simple Gamma Variate Generators. J. R. Stat. Soc. Ser. C (Appl. Stat.)
**1979**, 28, 290–295. [Google Scholar] [CrossRef] - Bulmer, M.G. Principles of Statistics; Dover: New York, NY, USA, 1979; ISBN 0-486-63760-3. [Google Scholar]
- Lai, C.D.; Xie, M. Stochastic Aging and Dependence for Reliability; Springer: New York, NY, USA, 2006; ISBN 978-0-387-29742-2. [Google Scholar]
- Shao, J. Mathematical Statistics; Springer: New York, NY, USA, 2003. [Google Scholar] [CrossRef]
- Fleming, T.R.; Harrington, D.P. Counting Processes and Survival Analysis; Wiley: Hoboken, NJ, USA, 2011; ISBN 978-1-118-15066-5. [Google Scholar]
- Lawless, J.F. Statistical Models and Methods for Lifetime Data, 2nd ed.; Wiley: Hoboken, NJ, USA, 2003. [Google Scholar]
- Edmunson, J.H.; Fleming, T.R.; Decker, D.G.; Malkasian, G.D.; Jefferies, J.A.; Webb, M.J.; Kvols, L.K. Different Chemotherapeutic Sensitivities and Host Factors Affecting Prognosis in Advanced Ovarian Carcinoma vs. Minimal Residual Disease. Cancer Treat. Rep.
**1979**, 63, 241–247. [Google Scholar]

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

**Figure 1.**Density function for GLO2, GLO3, GLO4 and GLO5 and $\theta =2,1$ and $0.5$ from left to right respectively.

**Figure 2.**Survival function for GLO2, GLO3, GLO4 and GLO5 and $\theta =2,1$ and $0.5$ from left to right respectively.

**Figure 3.**Failure rate function for GLO2, GLO3, GLO4 and GLO5 and $\theta =2,1$ and $0.5$ from left to right respectively.

**Table 1.**PDF, CDF, mean and failure rate for generalized Lindley random variables of orders 2, 3 and 4.

GLO2 | GLO3 | GLO4 | |
---|---|---|---|

$\frac{{\theta}^{2}}{1+\theta}(1+x){e}^{-\theta x}$ | $\frac{{\theta}^{3}}{1+\theta +{\theta}^{2}}(1+x+\frac{1}{2}{x}^{2}){e}^{-\theta x}$ | $\frac{{\theta}^{4}}{1+\theta +{\theta}^{2}+{\theta}^{3}}(1+x+\frac{1}{2}{x}^{2}+\frac{1}{6}{x}^{3}){e}^{-\theta x}$ | |

CDF | $1-\left(\frac{1+\theta +\theta x}{1+\theta}\right){e}^{-\theta x}$ | $1-\left(\frac{(1+\theta +{\theta}^{2})+(\theta +{\theta}^{2})x+\left({\theta}^{2}\right)\frac{{x}^{2}}{2}}{1+\theta +{\theta}^{2}}\right){e}^{-\theta x}$ | $1-\left(\frac{(1+\theta +{\theta}^{2}+{\theta}^{3})+(\theta +{\theta}^{2}+{\theta}^{3})x+({\theta}^{2}+{\theta}^{3})\frac{{x}^{2}}{2}+\left({\theta}^{3}\right)\frac{{x}^{3}}{6}}{1+\theta +{\theta}^{2}+{\theta}^{3}}\right){e}^{-\theta x}$ |

Mean | $\frac{2+\theta}{\theta +{\theta}^{2}}$ | $\frac{3+2\theta +{\theta}^{2}}{\theta +{\theta}^{2}+{\theta}^{3}}$ | $\frac{4+3\theta +2{\theta}^{2}+{\theta}^{3}}{\theta +{\theta}^{2}+{\theta}^{3}+{\theta}^{4}}$ |

Failure rate | $\frac{{\theta}^{2}(1+x)}{1+\theta +\theta x}$ | $\frac{{\theta}^{3}(1+x+\frac{{x}^{2}}{2})}{(1+\theta +{\theta}^{2})+(\theta +{\theta}^{2})x+\left({\theta}^{2}\right)\frac{{x}^{2}}{2}}$ | $\frac{{\theta}^{4}(1+x+\frac{{x}^{2}}{2}+\frac{{x}^{3}}{6})}{(1+\theta +{\theta}^{2}+{\theta}^{3})+(\theta +{\theta}^{2}+{\theta}^{3})x+({\theta}^{2}+{\theta}^{3})\frac{{x}^{2}}{2}+\left({\theta}^{3}\right)\frac{{x}^{3}}{6}}$ |

m | $\mathit{\theta}$ | B | ${\mathit{B}}^{*}$ | $\mathit{MSE}$ |
---|---|---|---|---|

2 | 0.01 | 0.000392 | 0.001285 | 0.000002 |

0.2 | 0.003687 | 0.027083 | 0.001223 | |

2 | 0.089519 | 0.303592 | 0.156634 | |

3 | 0.01 | 0.000212 | 0.001064 | 0.000001 |

0.2 | 0.003756 | 0.021395 | 0.000725 | |

2 | 0.065359 | 0.260616 | 0.113436 | |

4 | 0.01 | 0.000223 | 0.000956 | 0.000001 |

0.2 | 0.002531 | 0.018226 | 0.000538 | |

2 | 0.067082 | 0.234948 | 0.090523 | |

5 | 0.01 | 0.000036 | 0.000827 | 0.000001 |

0.2 | 0.001806 | 0.016431 | 0.000420 | |

2 | 0.048145 | 0.205068 | 0.072493 | |

6 | 0.01 | 0.000089 | 0.000762 | 0.000000 |

0.2 | 0.000261 | 0.014639 | 0.000346 | |

2 | 0.048055 | 0.203754 | 0.068692 |

**Table 3.**Simulation results for estimating $\widehat{\theta}$ when 25% of the sample has been censored from right.

n | m | $\mathit{\theta}$ | B | ${\mathit{B}}^{*}$ | $\mathit{MSE}$ |
---|---|---|---|---|---|

20 | 2 | 0.01 | 0.000228 | 0.001437 | 0.000003 |

0.2 | 0.005887 | 0.029532 | 0.001384 | ||

1.2 | −0.218137 | 0.218137 | 0.055005 | ||

3 | 0.01 | 0.000192 | 0.001251 | 0.000002 | |

0.2 | 0.001745 | 0.022546 | 0.000809 | ||

1.2 | −0.209398 | 0.209398 | 0.044895 | ||

4 | 0.01 | 0.000066 | 0.000936 | 0.000001 | |

0.2 | 0.002496 | 0.019698 | 0.000614 | ||

1.2 | −0.204693 | 0.204693 | 0.042309 | ||

5 | 0.01 | 0.000041 | 0.000888 | 0.000001 | |

0.2 | 0.001541 | 0.018230 | 0.000541 | ||

1.2 | −0.202124 | 0.202124 | 0.040991 | ||

40 | 2 | 0.01 | 0.000041 | 0.001018 | 0.000001 |

0.2 | 0.002479 | 0.020986 | 0.000699 | ||

1.2 | −0.206282 | 0.206282 | 0.043137 | ||

3 | 0.01 | 0.000012 | 0.000769 | 0.000001 | |

0.2 | 0.001470 | 0.015627 | 0.000389 | ||

1.2 | −0.202902 | 0.202902 | 0.041378 | ||

4 | 0.01 | 0.000029 | 0.000694 | 0.000000 | |

0.2 | 0.001004 | 0.014321 | 0.000313 | ||

1.2 | −0.201060 | 0.201060 | 0.040472 | ||

5 | 0.01 | 0.000002 | 0.000636 | 0.000000 | |

0.2 | −0.000065 | 0.012032 | 0.000226 | ||

1.2 | −0.200265 | 0.200265 | 0.040113 |

**Table 4.**Simulation results for estimating $\widehat{\theta}$ when 40% of the sample has been censored from right.

n | m | $\mathit{\theta}$ | B | ${\mathit{B}}^{*}$ | $\mathit{MSE}$ |
---|---|---|---|---|---|

20 | 2 | 0.01 | 0.000307 | 0.001546 | 0.000003 |

0.2 | 0.003416 | 0.032657 | 0.001677 | ||

1.2 | −0.221936 | 0.221936 | 0.053017 | ||

3 | 0.01 | 0.000057 | 0.001298 | 0.000002 | |

0.2 | 0.000150 | 0.024646 | 0.000981 | ||

1.2 | −0.216420 | 0.216420 | 0.049195 | ||

4 | 0.01 | −0.000017 | 0.001050 | 0.000002 | |

0.2 | 0.001599 | 0.021568 | 0.000731 | ||

1.2 | −0.206033 | 0.206033 | 0.043004 | ||

5 | 0.01 | −0.000027 | 0.000901 | 0.000001 | |

0.2 | 0.001679 | 0.018990 | 0.000585 | ||

1.2 | −0.203726 | 0.203726 | 0.041973 | ||

40 | 2 | 0.01 | 0.000093 | 0.001088 | 0.000002 |

0.2 | 0.000000 | 0.021989 | 0.000781 | ||

1.2 | −0.211623 | 0.211623 | 0.046143 | ||

3 | 0.01 | 0.000072 | 0.000867 | 0.000001 | |

0.2 | −0.000216 | 0.016417 | 0.000432 | ||

1.2 | −0.205086 | 0.205086 | 0.042572 | ||

4 | 0.01 | −0.000024 | 0.000739 | 0.000000 | |

0.2 | 0.000730 | 0.014284 | 0.000318 | ||

1.2 | −0.202220 | 0.202220 | 0.041028 | ||

5 | 0.01 | 0.000014 | 0.000652 | 0.000000 | |

0.2 | −0.000495 | 0.012862 | 0.000261 | ||

1.2 | −0.201258 | 0.201258 | 0.040596 |

15 | 20 | 38 | 42 | 61 | 76 | 86 | 98 | 121 | 146 | 149 | 157 | 175 |

176 | 180 | 180 | 198 | 220 | 224 | 251 | 264 | 282 | 321 | 325 | 653 |

m | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{var}}\left(\widehat{\mathit{\theta}}\right)$ | -log-likelihood |
---|---|---|---|

1 | 0.00560 | 0.00000125 | 154.5895 |

2 | 0.01116 | 0.00000249 | 152.5070 |

3 | 0.01674 | 0.00000373 | 154.5319 |

4 | 0.02230 | 0.00000497 | 158.1863 |

m | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{var}}\left(\widehat{\mathit{\theta}}\right)$ | -log-likelihood |
---|---|---|---|

1 | 0.00118 | 0.00000019 | 92.8365 |

2 | 0.00316 | 0.00000028 | 90.0980 |

3 | 0.00529 | 0.00000057 | 89.7396 |

4 | 0.00752 | 0.00000085 | 90.2579 |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abouammoh, A.; Kayid, M.
A New Flexible Generalized Lindley Model: Properties, Estimation and Applications. *Symmetry* **2020**, *12*, 1678.
https://doi.org/10.3390/sym12101678

**AMA Style**

Abouammoh A, Kayid M.
A New Flexible Generalized Lindley Model: Properties, Estimation and Applications. *Symmetry*. 2020; 12(10):1678.
https://doi.org/10.3390/sym12101678

**Chicago/Turabian Style**

Abouammoh, Abdulrahman, and Mohamed Kayid.
2020. "A New Flexible Generalized Lindley Model: Properties, Estimation and Applications" *Symmetry* 12, no. 10: 1678.
https://doi.org/10.3390/sym12101678