# A New Family of Extended Lindley Models: Properties, Estimation and Applications

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Generalization Based on Mixing Weights

**Definition**

**1.**

**Proposition**

**1.**

^{th}moment is given by

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 2.1. Some Special Cases of $AK(\alpha ,\beta ,\delta ,\theta )$

- Case 1.
- For $\alpha =2$ and $\beta =0$, $AK(\alpha ,\beta ,\delta ,\theta )$ reduces to the well-known Lindley model.
- Case 2.
- When $\alpha =3$ and $\beta =0$, it reduces to the Akash distribution proposed and studied by Shanker [13].
- Case 3.
- For $\alpha =6$ and $\beta =0$, we have the Prakaamy distribution, which was introduced and discussed by Shukla [14].
- Case 4.
- For $\alpha =2$ and $\beta =1$, it reduces to the Shanker distribution, which was proposed and studied by Shanker [15].
- Case 5.
- For $\alpha =3$ and $\beta =1$, it is the Isheta distribution introduced and studied by Shanker and Shukla [16].
- Case 6.
- Case 7.
- For $\alpha =6$ and $\beta =1$, we have the Ram Awadh distribution, which was introduced and discussed by Shukla [18].

#### 2.2. Power Generalization

**Definition**

**2.**

**Proposition**

**3.**

^{th}moment is

**Proof.**

## 3. Estimation of Parameters for the Complete and Right Censored Data

## 4. Simulation Study

- Generate one sample of multinomial distribution with parameters v, $p=\frac{{\delta}^{\alpha +\beta -1}}{{\Gamma}\left(\alpha \right)+{\delta}^{\alpha +\beta -1}}$ and $1-p$. Suppose that the generated instance be denoted by ${k}_{1}$ and ${k}_{2}$, corresponding respectively to probabilities p and $1-p$. Note that ${k}_{1}+{k}_{2}=v$.
- Simulate one sample from gamma model $G(1,\theta )$ with size ${k}_{1}$ and another sample from $G(\alpha ,\theta )$ with size ${k}_{2}$. We merge these two samples to have one sample of $AK(\alpha ,\beta ,\delta ,\theta )$ with size v.

- Absolute means of biases ($A{B}_{\delta},A{B}_{\theta}$), have greater values for censored samples.
- ($A{B}_{\delta},A{B}_{\theta}$) and ($MS{E}_{\delta},MS{E}_{\theta}$) show smaller for larger sample size, v.
- ${B}_{\delta}$, $A{B}_{\delta}$ and $MS{E}_{\delta}$ increase with $\delta $ and a similar statement holds for $\theta $.

- $(A{B}_{\delta},A{B}_{\theta},A{B}_{\gamma})$ are greater for censored samples.
- $(A{B}_{\delta},A{B}_{\theta},A{B}_{\gamma})$ and $(MS{E}_{\delta},MS{E}_{\theta},MS{E}_{\gamma})$ are smaller for larger sample size, v.

## 5. Applications

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Density function for $AK(\alpha ,\beta ,\delta ,\theta )$ for some $\alpha $, $\beta $, $\delta $ and $\theta $ values.

**Figure 3.**(

**Left**): Empirical and fitted $AK(2,5,0.081045,0.202490)$ for data set of Example 1. (

**Right**): Empirical and fitted $AK(13,4,0.707794,0.421921)$ for data set of Example 2.

**Table 1.**Simulation results for $AK(\alpha ,\beta ,\delta ,\theta )$. Every cell consists of pairs $({B}_{\delta},{B}_{\theta})$, $(A{B}_{\delta},A{B}_{\theta})$ and $(MS{E}_{\delta},MS{E}_{\theta})$ from top to bottom.

$\mathit{p}=0$ | $\mathit{p}=0.20$ | ||||
---|---|---|---|---|---|

$\mathit{v}$ | $(\mathbf{\alpha},\mathbf{\beta})$ | $\mathbf{\delta}=\mathbf{0.2},\mathbf{\theta}=\mathbf{1.2}$ | $\mathbf{\delta}=\mathbf{0.4},\mathbf{\theta}=\mathbf{0.05}$ | $\mathbf{\delta}=\mathbf{0.2},\mathbf{\theta}=\mathbf{1.2}$ | $\mathbf{\delta}=\mathbf{0.4},\mathbf{\theta}=\mathbf{0.05}$ |

30 | (2,1) | (0.013000, −0.007520) | (0.013888, 0.001214) | (0.313307, −0.021339) | (0.458239, −0.000243) |

(0.237531, 0.145039) | (0.376828, 0.006787) | (0.541801, 0.164034) | (0.821119, 0.007860) | ||

(0.092244, 0.034001) | (1.440460, 0.000074) | (7.131440, 0.057312) | (7.109937, 0.000109) | ||

(3,2) | (−0.044634, -.004039) | (−0.190346, 0.000018) | (−0.045337, −0.007946) | (−0.163909, −0.000265) | |

(0.239059, 0.105796) | (0.326939, 0.004271) | (0.234000, 0.111629) | (0.355852, 0.005062) | ||

(0.089102, 0.018373) | (0.116500, 0.000029) | (0.066341, 0.019126) | (0.631660, 0.000043) | ||

50 | (2,1) | (−0.010284, 0.005392) | (−0.045135, 0.000987) | (0.030007, −0.004096) | (0.07146, −0.000103) |

(0.195604, 0.109608) | (0.244568, 0.005464) | (0.248331, 0.119299) | (0.363505, 0.005866) | ||

(0.055114, 0.019733) | (0.109090, 0.000048) | (0.977620, 0.023694) | (2.245982, 0.000057) | ||

(3,2) | (−0.049990, −0.002405) | (−0.187037, 0.000247) | (−0.054677, −0.006306) | (−0.176889, −0.000140) | |

(0.222270, 0.078619) | (0.301444, 0.003425) | (0.221826, 0.088594) | (0.301239, 0.003759) | ||

(0.056555, 0.009928) | (0.103816, 0.000018) | (0.058013, 0.012358) | (0.103287, 0.000022) |

**Table 2.**Simulation results for $PAK(\alpha ,\beta ,\delta ,\theta ,\gamma )$. Every cell consists of pairs $({B}_{\delta},{B}_{\theta},{B}_{\gamma})$, $(A{B}_{\delta},A{B}_{\theta},A{B}_{\gamma})$ and $(MS{E}_{\delta},MS{E}_{\theta},MS{E}_{\gamma})$ from top to bottom.

$\mathit{p}=0$ | $\mathit{p}=0.20$ | ||
---|---|---|---|

$\mathit{v}$ | $\mathbf{\gamma}$ | $\mathbf{\alpha}=\mathbf{3},\mathbf{\beta}=\mathbf{2},\mathbf{\delta}=\mathbf{0.2},\mathbf{\theta}=\mathbf{1.2}$ | $\mathbf{\alpha}=\mathbf{3},\mathbf{\beta}=\mathbf{2},\mathbf{\delta}=\mathbf{0.2},\mathbf{\theta}=\mathbf{1.2}$ |

50 | 1.2 | (0.042585, −0.140304, 0.144448) | (0.026979, −0.144273, 0.163527) |

(0.308414, 0.224418, 0.201279) | (0.318815, 0.227539, 0.229853) | ||

(0.182252, 0.111292, 0.116385) | (0.156089, 0.109475, 0.142901) | ||

1.5 | (−0.027510, −0.091166, 0.121848) | (−0.034820, −0.098860, 0.144941) | |

(0.258734, 0.175948, 0.191544) | (0.260512, 0.191275, 0.239627) | ||

(0.106731, 0.066488, 0.104755) | (0.117548, 0.077079, 0.143252) | ||

80 | 1.2 | (−0.003004, −0.086247, 0.093759) | (−0.010993, −0.101056, 0.106659) |

(0.262675, 0.159261, 0.142750) | (0.262216, 0.173366, 0.170353) | ||

(0.114639, 0.061799, 0.069339) | (0.112765, 0.071323, 0.090398) | ||

1.5 | (−0.044956, −0.047861, 0.069793) | (−0.049756, −0.076235, 0.109912) | |

(0.236456, 0.138523, 0.144182) | (0.240175, 0.156068, 0.191108) | ||

(0.081771, 0.043508, 0.067589) | (0.083860, 0.053685, 0.108493) |

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 |

**Table 4.**Strength of glass of aircraft window, reported by Fuller [22].

18.83 | 20.8 | 21.657 | 23.03 | 23.23 | 24.05 | 24.321 | 25.5 | 25.52 | 25.8 | 26.69 |

26.77 | 26.78 | 27.05 | 27.67 | 29.9 | 31.11 | 33.2 | 33.73 | 33.76 | 33.89 | 34.76 |

35.75 | 35.91 | 36.98 | 37.08 | 37.09 | 39.58 | 44.045 | 45.29 | 45.381 |

0.55 | 0.93 | 1.25 | 1.36 | 1.49 | 1.52 | 1.58 | 1.61 | 1.64 | 1.68 | 1.73 |

1.81 | 2.00 | 0.74 | 1.04 | 1.27 | 1.39 | 1.49 | 1.53 | 1.59 | 1.61 | 1.66 |

1.68 | 1.76 | 1.82 | 2.01 | 0.77 | 1.11 | 1.28 | 1.42 | 1.50 | 1.54 | 1.60 |

1.62 | 1.66 | 1.69 | 1.76 | 1.84 | 2.24 | 0.81 | 1.13 | 1.29 | 1.48 | 1.50 |

1.55 | 1.61 | 1.62 | 1.66 | 1.70 | 1.77 | 1.84 | 0.84 | 1.24 | 1.30 | 1.48 |

1.51 | 1.55 | 1.61 | 1.63 | 1.67 | 1.70 | 1.78 | 1.89 |

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Abouammoh, A.; Kayid, M.
A New Family of Extended Lindley Models: Properties, Estimation and Applications. *Mathematics* **2020**, *8*, 2146.
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**AMA Style**

Abouammoh A, Kayid M.
A New Family of Extended Lindley Models: Properties, Estimation and Applications. *Mathematics*. 2020; 8(12):2146.
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Abouammoh, Abdulrahman, and Mohamed Kayid.
2020. "A New Family of Extended Lindley Models: Properties, Estimation and Applications" *Mathematics* 8, no. 12: 2146.
https://doi.org/10.3390/math8122146