# A New Parametric Life Distribution with Modified Bagdonavičius–Nikulin Goodness-of-Fit Test for Censored Validation, Properties, Applications, and Different Estimation Methods

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## Abstract

**:**

## 1. Introduction

## 2. The BXW Model

## 3. Properties

#### 3.1. Some Moments

#### 3.2. Generating Function

#### 3.3. Probability Weighted Moments (PWMs)

#### 3.4. Order Statistics

#### 3.5. Renyi and $\delta $ −Entropies

## 4. Classical Parameter Estimation

**I.**- The maximum likelihood method;
**II.**- Method of Cramer-Von-Mises estimation;
**III.**- Method of percentile estimation;
**IV.**- Method of L-moments.

#### 4.1. The Maximum Likelihood Method

#### 4.2. Cramer-Von-Mises Estimation Method

#### 4.3. Method of Percentile Estimation

#### 4.4. Method of L-Moments

## 5. Simulation Studies

#### 5.1. Simulation Study for Assessing the Maximum Likilihood Method

#### 5.1.1. Numerical Assessment

#### 5.1.2. Graphical Assessment

- Use Equation (13) to generate 1000 samples of size n from the BXW distribution;
- Compute the MLEs for the 1000 samples;
- Compute the standard errors (SEs) of the MLEs for the 1000 samples (the standard errors (SEs) were computed by inverting the observed information matrix).
- Compute the biases and mean square errors given for $\theta ,\beta $.

#### 5.2. Simulation Studies for Comparing Non-Bayesian Estimation Methods

Parameters | I | II | III |

θ | 2 | 0.6 | 6 |

β | 0.5 | 0.4 | 0.1 |

## 6. Non-Bayesian Uncensored Applications

#### 6.1. Non-Bayesian Uncensored Applications for Comparing Models

#### 6.2. Uncensored Applications for Comparing the Non-Bayesian Methods

## 7. Censored Maximum Likelihood Estimation

## 8. Modified Chi-Squared Type Test for Right Censored Data

#### 8.1. Choice of Random Grouping Intervals

#### 8.2. Quadratic Form $Q$

#### 8.3. Estimated Information Matrix $\hat{I}$

## 9. Simulations

#### 9.1. Censored Maximum Likelihood Estimation for BXW

#### 9.2. Test Statistic ${Y}^{2}$

## 10. Data Analysis

## 11. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Biases (

**left panels**) and MSEs (

**right panels**) for $\theta ,\beta ,$ and n = 50, 100, …, 1000 for the BXW model.

**Figure 3.**Total time on test (TTT) plot for the first dataset (

**left figure**) and for the second dataset (

**right figure**).

**Figure 5.**Fitted cumulative distribution functions (CDFs) on the empirical CDF of the first data set.

$\left(\mathit{\theta},\mathit{\beta}\right)$ | $\mathit{\theta}$ | $\mathit{\beta}$ |
---|---|---|

n = 100 | ||

(0.4,2.5) | 0.018 (0.018) | 0.180 (0.946) |

(3,0.2) | −0.114 (0.345) | 0.008 (0.001) |

(0.6,0.6) | 0.002 (0.033) | 0.053 (0.040) |

(0.19,2.5) | 0.038 (0.009) | −0.257 (0.449) |

n = 200 | ||

(0.4,2.5) | −0.004 (0.010) | 0.180 (0.424) |

(3,0.2) | −0.089 (0.172) | 0.002 (5e^{−4}) |

(0.6,0.6) | −0.001 (0.019) | 0.031 (0.018) |

(0.19,2.5) | 0.015 (0.004) | −0.206 (0.248) |

n = 500 | ||

(0.4,2.5) | 0.002 (0.003) | 0.036 (0.125) |

(3,0.2) | −0.026 (0.068) | −0.002 (3e^{−4}) |

(0.6,0.6) | 0.001 (0.007) | 0.008 (0.005) |

(0.19,2.5) | 0.006 (0.002) | −0.164 (0.141) |

Parameters | MLE | CVM | PerEs | L-Moment |
---|---|---|---|---|

Θ = 2 | 2.123510 | 2.088200 | 2.043230 | 2.09791 |

(0.08387) | (0.32986) | (0.27218) | (0.35417) | |

Β = 0.5 | 0.51657 | 0.51770 | 0.50701 | 0.517620 |

(0.00877) | (0.06045) | (0.01224) | (0.01580) | |

Θ = 0.6 | 0.64940 | 0.63391 | 0.66041 | 0.620200 |

(0.02499) | (0.03179) | (0.05275) | (0.06030) | |

Β = 0.4 | 0.413460 | 0.41341 | 0.432070 | 0.402780 |

(0.00592) | (0.00965) | (0.01652) | (0.01904) | |

Θ = 6 | 6.55660 | 6.251720 | 6.284390 | 6.695530 |

(5.97827) | (2.75560) | (7.71943) | (23.64994) | |

Β = 0.1 | 0.103750 | 0.108850 | 0.095230 | 0.122000 |

Parameters | MLE | CVM | PerEs | L-Moment |
---|---|---|---|---|

Θ = 2 | 2.04042 | 2.020530 | 1.9875600 | 2.028160 |

(0.11254) | (0.11274) | (0.09788) | (0.13328) | |

Β = 0.5 | 0.50549 | 0.50440 | 0.49716 | 0.50573 |

(0.00276) | (0.00471) | (0.00384) | (0.00685) | |

Θ = 0.6 | 0.61713 | 0.61232 | 0.62687 | 0.609540 |

(0.00771) | (0.01104) | (0.01714) | (0.02187) | |

Β = 0.4 | 0.40422 | 0.40531 | 0.41483 | 0.40260 |

(0.00216) | (0.00299) | (0.00533) | (0.00790) | |

Θ = 6 | 6.22870 | 6.14691 | 5.96776 | 6.212280 |

(1.82638) | (1.04639) | (4.01295) | (8.14331) | |

Β = 0.1 | 0.10169 | 0.10262 | 0.09246 | 0.11227 |

Parameters | MLE | CVM | PerEs | L-Moment |
---|---|---|---|---|

Θ = 2 | 2.00870 | 2.01118 | 1.99028 | 2.01254 |

(0.03709) | (0.03613) | (0.03488) | (0.03911) | |

Β = 0.5 | 0.50107 | 0.50228 | 0.49783 | 0.50274 |

(0.00095) | (0.00149) | (0.00124) | (0.00198) | |

Θ = 0.6 | 0.60668 | 0.60501 | 0.60609 | 0.60683 |

(0.00262) | (0.00312) | (0.00607) | (0.00641) | |

Β = 0.4 | 0.40272 | 0.40228 | 0.40332 | 0.40313 |

(0.00072) | (0.00086) | (0.00179) | (0.00236) | |

Θ = 6 | 6.08709 | 6.00740 | 5.75105 | 6.08802 |

(0.49247) | (0.28755) | (2.26499) | (2.31531) | |

Β = 0.1 | 0.10068 | 0.10020 | 0.09448 | 0.10431 |

Parameters | MLE | CVM | PerEs | L-Moment |
---|---|---|---|---|

Θ = 2 | 2.01328 | 2.01143 | 1.99373 | 2.00475 |

(0.01739) | (0.01740) | (0.01622) | (0.01930) | |

Β = 0.5 | 0.50197 | 0.50233 | 0.49867 | 0.50085 |

(0.00045) | (0.00071) | (0.00056) | (0.00101) | |

Θ = 0.6 | 0.60395 | 0.60365 | 0.60186 | 0.60209 |

(0.00134) | (0.00164) | (0.00271) | (0.00311) | |

Β = 0.4 | 0.40181 | 0.40173 | 0.40109 | 0.40074 |

(0.00036) | (0.00046) | (0.00077) | (0.00115) | |

θ = 6 | 6.02037 | 6.02330 | 5.56693 | 6.05655 |

(0.25345) | (0.15614) | (1.65906) | (0.94734) | |

β = 0.1 | 0.10015 | 0.10042 | 0.09423 | 0.101650 |

Model | Estimates | Log-Likelihood |
---|---|---|

BXW (θ, β) | 40.768 (7.32) 0.095 (10 × e^{−3}) | 73.565 |

Weibull (α, β) | 1.227 (0.160) 4.557 (0.666) | 74.788 |

Gamma (α, λ) | 1.487 (0.184) 0.350 (0.051) | 74.459 |

GE (α, λ) | 1.560 (0.280) 0.309 (0.045) | 74.396 |

EG (λ, p) | 0.234 (0.042) 0.010 (0.280) | 75.802 |

EP (λ, β) | 0.011 (0.622) 0.235 (0.042) | 75.795 |

CEG (λ, θ) | 0.297 (0.047) 0.618 (0.190) | 75.454 |

Model | Estimates | Log-Likelihood |
---|---|---|

BXW (θ, β) | 14.347 (2.46) 0.104 (90.01) | 55.049 |

Weibull (α, β) | 1.010 (0.125) 1.887 (0.320) | 55.449 |

Gamma (α, λ) | 1.062 (0.139) 0.565 (0.094) | 55.413 |

GE (α, λ) | 1.076 (0.184) 0.558 (0.092) | 55.401 |

EG (λ, p) | 0.481 (0.086) 0.177 (0.242) | 55.395 |

EP (λ, β) | 0.427 (0.596) 0.476 (0.085) | 55.392 |

CEG (λ, θ) | 0.532 (0.091) 0.999 (0.289) | 55.453 |

Model | Goodness of Fit Criteria | |||||
---|---|---|---|---|---|---|

AIC | BIC | HQIC | CAIC | ${\mathit{W}}^{*}$ | ${\mathit{A}}^{*}$ | |

BXW | 151.131 | 153.999 | 152.066 | 151.559 | 0.061 | 0.395 |

Weibull | 153.577 | 156.445 | 154.512 | 154.006 | 0.118 | 0.713 |

Gamma | 152.918 | 155.786 | 153.853 | 153.347 | 0.122 | 0.713 |

GE | 152.793 | 155.661 | 153.728 | 153.222 | 0.120 | 0.705 |

EG | 155.604 | 158.472 | 156.539 | 156.032 | 0.095 | 0.751 |

EP | 155.590 | 158.458 | 156.525 | 156.019 | 0.095 | 0.749 |

Model | Goodness of Fit Criteria | |||||
---|---|---|---|---|---|---|

AIC | BIC | HQIC | CAIC | ${\mathit{W}}^{*}$ | ${\mathit{A}}^{*}$ | |

BXW | 114.098 | 117.151 | 115.139 | 114.485 | 0.032 | 0.227 |

Weibull | 114.899 | 117.952 | 115.940 | 115.286 | 0.043 | 0.282 |

Gamma | 114.826 | 117.879 | 115.867 | 115.213 | 0.050 | 0.312 |

GE | 114.803 | 117.856 | 115.844 | 115.190 | 0.052 | 0.317 |

EG | 114.791 | 117.844 | 115.832 | 115.178 | 0.032 | 0.240 |

EP | 114.785 | 117.837 | 115.826 | 115.172 | 0.032 | 0.239 |

Method | θ | β | ${\mathit{W}}^{*}$ | ${\mathit{A}}^{*}$ |
---|---|---|---|---|

ML | 40.768 | 0.095 | 0.05782 | 0.37572 |

CVM | 35.997 | 0.087 | 0.05909 | 0.38163 |

PerEs | 55.730 | 0.101 | 0.05551 | 0.36612 |

L-moment | 42.097 | 0.098 | 0.05747 | 0.37407 |

Method | θ | β | ${\mathit{W}}^{*}$ | ${\mathit{A}}^{*}$ |
---|---|---|---|---|

ML | 14.347 | 0.105 | 0.02847 | 0.20972 |

CVM | 17.784 | 0.097 | 0.02876 | 0.21271 |

PerEs | 16.690 | 0.106 | 0.02910 | 0.21539 |

L-moment | 14.672 | 0.109 | 0.02846 | 0.20934 |

N = 10,000 | n₁ = 20 | n₂ = 50 | n₃ = 150 | n₄ = 300 |
---|---|---|---|---|

θ = 1.5 | 1.4838 (0.0076) | 1.4884 (0.0062) | 1.4922 (0.0045) | 1.4983 (0.0023) |

β = 0.7 | 0.7192 (0.0089) | 0.7137 (0.0077) | 0.7096 (0.0057) | 0.7043 (0.0034) |

θ = 0.8 | 0.8213 (0.0082) | 0.8126 (0.0058) | 0.8084 (0.0032) | 0.8012 (0.0016) |

β = 0.5 | 0.4828 (0.0076) | 0.4877 (0.0052) | 0.4912 (0.0037) | 0.4996 (0.0018) |

θ = 3 | 2.9696 (0.0094) | 2.9776 (0.0066) | 2.9894 (0.0042) | 2.9982 (0.0027) |

β = 0.4 | 0.4331 (0.0068) | 0.4284 (0.0044) | 0.4167 (0.0029) | 0.4024 (0.0013) |

**Table 13.**Simulated levels of significance for the ${Y}_{n}^{2}\left(\mathsf{\varphi}\right)$ test for the BXW model against their theoretical values (ε = 0.01, 0.05, 0.10).

N = 10,000 | n = 20 | n = 50 | n = 150 | n = 300 |
---|---|---|---|---|

ε = 1% | 0.0055 | 0.0064 | 0.0085 | 0.0094 |

ε = 5% | 0.0443 | 0.0452 | 0.0468 | 0.0486 |

ε = 10% | 0.0931 | 0.0943 | 0.0959 | 0.0974 |

**Table 14.**Values of ${\hat{a}}_{j},{e}_{j},{U}_{j},{\hat{C}}_{1j},\mathrm{and}\text{}{\hat{C}}_{2j}$.

${\hat{a}}_{j}$ | 189.6 | 214.9 | 237.7 | 304 |

${U}_{J}$ | 4 | 5 | 6 | 4 |

${\hat{C}}_{1j}$ | 0.9463 | 1.2416 | 0.8863 | 0.7648 |

${\hat{C}}_{2j}$ | 1.1346 | 0.9946 | 1.2476 | 0.9263 |

${e}_{j}$ | 0.4859 | 0.4859 | 0.4859 | 0.4859 |

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**MDPI and ACS Style**

Mansour, M.; Rasekhi, M.; Ibrahim, M.; Aidi, K.; Yousof, H.M.; Abd Elrazik, E.
A New Parametric Life Distribution with Modified Bagdonavičius–Nikulin Goodness-of-Fit Test for Censored Validation, Properties, Applications, and Different Estimation Methods. *Entropy* **2020**, *22*, 592.
https://doi.org/10.3390/e22050592

**AMA Style**

Mansour M, Rasekhi M, Ibrahim M, Aidi K, Yousof HM, Abd Elrazik E.
A New Parametric Life Distribution with Modified Bagdonavičius–Nikulin Goodness-of-Fit Test for Censored Validation, Properties, Applications, and Different Estimation Methods. *Entropy*. 2020; 22(5):592.
https://doi.org/10.3390/e22050592

**Chicago/Turabian Style**

Mansour, Mahmoud, Mahdi Rasekhi, Mohamed Ibrahim, Khaoula Aidi, Haitham M. Yousof, and Enayat Abd Elrazik.
2020. "A New Parametric Life Distribution with Modified Bagdonavičius–Nikulin Goodness-of-Fit Test for Censored Validation, Properties, Applications, and Different Estimation Methods" *Entropy* 22, no. 5: 592.
https://doi.org/10.3390/e22050592