# On a Modified Weighted Exponential Distribution with Applications

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## Abstract

**:**

## 1. Introduction

#### 1.1. State of Art

#### 1.2. Contributions

- (i)
- The cdf can be written as ${F}_{o}(x;\alpha ,\lambda )=aF(x;\lambda )+bF(x;\lambda (\alpha +1)),$ where $a=(\alpha +2)/(\alpha +1)>0$ and $b=-1/(\alpha +1)<0$, meaning that the MWE distribution also belongs to the family of generalized mixture of two exponential distributions, following the spirit of the distribution proposed by [16],
- (ii)
- The cdf is quite simple to manage and consequently, the MWE distribution can be studied in an-depth manner on all the theoretical and practical aspects,
- (iii)
- Thanks to the parameter $\alpha $, the related pdf can be decreasing or unimodal, and the related hrf can be constant or increasing as proven later,
- (iv)
- In some concrete scenarios, the MWE model can be more efficient in data fitting than the exponential or WE models, among other lifetime models.

#### 1.3. Paper Organization

## 2. Statistical Properties

#### 2.1. Quantile and Survival Functions

#### 2.2. Shapes of the Probability Density and Hazard Rate Functions

**Proposition**

**1.**

- if $\alpha \in [0,(\sqrt{5}-1)/2)$, then ${f}_{o}(x;\alpha ,\lambda )$ is decreasing.
- if $\alpha \ge (\sqrt{5}-1)/2$, ${f}_{o}(x;\alpha ,\lambda )$ is unimodal, with the mode:$${x}_{\ast}=\frac{1}{\alpha \lambda}log\left[\frac{{(\alpha +1)}^{2}}{\alpha +2}\right].$$

**Proof.**

- if $\alpha \in [0,(\sqrt{5}-1)/2)$, then$$\frac{d}{dx}{f}_{o}(x;\alpha ,\lambda )\le \frac{{\lambda}^{2}}{\alpha +1}{e}^{-\lambda (\alpha +1)x}[{(\alpha +1)}^{2}-(\alpha +2)]<0,$$
- In the case, $\alpha \ge (\sqrt{5}-1)/2$, we have ${(\alpha +1)}^{2}\ge (\alpha +2)$, and one value of x vanished $d{f}_{o}(x;\alpha ,\lambda )/dx$; it is given by $x={x}_{\ast}$. For $x<{x}_{\ast}$, we have $d{f}_{o}(x;\alpha ,\lambda )/dx>0$ and for $x>{x}_{\ast}$, $d{f}_{o}(x;\alpha ,\lambda )/dx<0$, implying that $x={x}_{\ast}$ is a maximal point; it corresponds to the mode of the MWE distribution.

**Proposition**

**2.**

**Proof.**

#### 2.3. Moments and Moment Generating Function

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

#### 2.4. Bonferroni and Lorenz Curves

#### 2.5. Rényi Entropy

**Proposition**

**6.**

**Proof.**

#### 2.6. Reliability Characteristics of the MWE Distribution

**Proposition**

**7.**

**Proof.**

#### 2.7. Mean Residual Life Function

## 3. Parameters Estimation

#### 3.1. Maximum Likelihood Estimates

#### 3.2. Method of Moments Estimates

#### 3.3. Least Squares and Weighted Least Squares Estimates

**Least Squares Estimates:**The least square function is defined by

**Weighted Least Squares Estimates:**The weighted least square function is defined by

#### 3.4. Cramér-von Mises Estimates

## 4. Simulation

## 5. Real Data Analysis

**Data set 1:**The failure times data can be found in [27]. The values are: 0.12, 0.43, 0.92, 1.14, 1.24, 1.61, 1.93, 2.38, 4.51, 5.09, 6.79, 7.64, 8.45, 11.9, 11.94, 13.01, 13.25, 14.32, 17.47, 18.1, 18.66, 19.23, 24.39, 25.01, 26.41, 26.8, 27.75, 29.69, 29.84, 31.65, 32.64, 35, 40.7, 42.34, 43.05, 43.4, 44.36, 45.4, 48.14, 49.1, 49.44, 51.17, 58.62, 60.29, 72.13, 72.22, 72.25, 72.29, 85.2, 89.52.

**Data set 2:**We also consider the bladder cancer data set by Aldeni et al. [28]. The values are: 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, 1.46, 18.10, 11.79, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 13.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 12.07, 6.76, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.

## 6. Concluding Remarks

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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$\mathit{\lambda}$ | $\mathit{\alpha}$ | ${\mathit{m}}_{1}$ | V | ${\mathit{\gamma}}_{1}$ | ${\mathit{\beta}}_{2}$ |
---|---|---|---|---|---|

0.5000 | 0.2000 | 2.2778 | 4.8488 | 1.8639 | 8.1028 |

0.5000 | 0.4000 | 2.4082 | 4.9996 | 1.7628 | 7.5801 |

0.5000 | 0.6000 | 2.4688 | 4.9521 | 1.7198 | 7.4207 |

0.5000 | 0.8000 | 2.4938 | 4.8535 | 1.7089 | 7.4265 |

0.5000 | 1.0000 | 2.5000 | 4.7500 | 1.7146 | 7.5042 |

0.5000 | 1.2000 | 2.4959 | 4.6557 | 1.7283 | 7.6100 |

0.5000 | 1.4000 | 2.4861 | 4.5739 | 1.7454 | 7.7230 |

$\mathbf{\alpha}$ | $\mathbf{\lambda}$ | ${\mathbf{m}}_{\mathbf{1}}$ | $\mathbf{V}$ | ${\mathbf{\gamma}}_{\mathbf{1}}$ | ${\mathbf{\beta}}_{\mathbf{2}}$ |

1.2000 | 0.4000 | 3.1198 | 7.2745 | 1.7283 | 7.6100 |

1.2000 | 0.6000 | 2.0799 | 3.2331 | 1.7283 | 7.6100 |

1.2000 | 0.8000 | 1.5599 | 1.8186 | 1.7283 | 7.6100 |

1.2000 | 1.0000 | 1.2479 | 1.1639 | 1.7283 | 7.6100 |

1.2000 | 1.2000 | 1.0399 | 0.8083 | 1.7283 | 7.6100 |

1.2000 | 1.4000 | 0.8914 | 0.5938 | 1.7283 | 7.6100 |

1.2000 | 1.6000 | 0.7800 | 0.4547 | 1.7283 | 7.6100 |

n | Estimate | Bias | MSE | ||
---|---|---|---|---|---|

$\widehat{\mathbf{\alpha}}$ | $\widehat{\mathbf{\lambda}}$ | $\widehat{\mathbf{\alpha}}$ | $\widehat{\mathbf{\lambda}}$ | ||

50 | MLE | 0.6411 | −0.0459 | 6.4825 | 0.0195 |

MOME | 0.7038 | −0.0356 | 0.9878 | 0.0217 | |

OLSE | 0.3002 | −0.0956 | 1.0082 | 0.0272 | |

WLSE | 0.3090 | −0.0840 | 1.2376 | 0.0246 | |

CME | 0.4610 | −0.0800 | 1.3098 | 0.0244 | |

100 | MLE | 0.2742 | −0.0446 | 0.7519 | 0.0109 |

MOME | 0.6530 | −0.0367 | 0.8751 | 0.0124 | |

OLSE | 0.1904 | −0.0828 | 0.4913 | 0.0172 | |

WLSE | 0.1782 | −0.0712 | 0.5446 | 0.0147 | |

CME | 0.2772 | −0.0727 | 0.6083 | 0.0152 | |

200 | MLE | 0.1165 | −0.0409 | 0.1834 | 0.0059 |

MOME | 0.5855 | −0.0330 | 0.7346 | 0.0058 | |

OLSE | 0.1391 | −0.0674 | 0.2210 | 0.0099 | |

WLSE | 0.1177 | −0.0573 | 0.1701 | 0.0080 | |

CME | 0.1794 | −0.0619 | 0.2406 | 0.0090 | |

500 | MLE | 0.0620 | −0.0365 | 0.0317 | 0.0028 |

MOME | 0.5134 | −0.0274 | 0.5407 | 0.0023 | |

OLSE | 0.0981 | −0.0527 | 0.0693 | 0.0047 | |

WLSE | 0.0822 | −0.0457 | 0.0532 | 0.0037 | |

CME | 0.1136 | −0.0506 | 0.0744 | 0.0044 |

n | Estimate | Bias | MSE | ||
---|---|---|---|---|---|

$\widehat{\mathbf{\alpha}}$ | $\widehat{\mathbf{\lambda}}$ | $\widehat{\mathbf{\alpha}}$ | $\widehat{\mathbf{\lambda}}$ | ||

50 | MLE | 0.7146 | −0.0072 | 4.7462 | 0.0201 |

MOME | 0.1304 | −0.0103 | 0.5513 | 0.0270 | |

OLSE | 0.1174 | −0.0410 | 1.9271 | 0.0243 | |

WLSE | 0.2331 | −0.0352 | 2.9491 | 0.0223 | |

CME | 0.3445 | −0.0250 | 2.2686 | 0.0227 | |

100 | MLE | 0.4174 | −0.0078 | 3.5071 | 0.0116 |

MOME | −0.0152 | −0.0182 | 0.5008 | 0.0158 | |

OLSE | 0.0858 | −0.0207 | 1.5188 | 0.0137 | |

WLSE | 0.1388 | −0.0166 | 2.0104 | 0.0126 | |

CME | 0.2204 | −0.0114 | 1.6787 | 0.0130 | |

200 | MLE | 0.0649 | −0.0017 | 1.5542 | 0.0062 |

MOME | −0.1548 | −0.0145 | 0.4094 | 0.0086 | |

OLSE | −0.0483 | −0.0052 | 0.9840 | 0.0072 | |

WLSE | −0.0602 | −0.0028 | 1.0602 | 0.0064 | |

CME | 0.0158 | −0.0002 | 0.9988 | 0.0069 | |

500 | MLE | −0.2141 | 0.0089 | 0.3936 | 0.0022 |

MOME | −0.3341 | −0.0051 | 0.3066 | 0.0030 | |

OLSE | −0.2070 | 0.0160 | 0.2761 | 0.0026 | |

WLSE | −0.2370 | 0.0141 | 0.2548 | 0.0023 | |

CME | −0.1802 | 0.0179 | 0.2866 | 0.0026 |

n | Estimate | Bias | MSE | ||
---|---|---|---|---|---|

$\widehat{\mathbf{\alpha}}$ | $\widehat{\mathbf{\lambda}}$ | $\widehat{\mathbf{\alpha}}$ | $\widehat{\mathbf{\lambda}}$ | ||

50 | MLE | 0.2723 | 0.0012 | 4.7454 | 0.0162 |

MOME | −0.0325 | 0.0342 | 0.3603 | 0.0200 | |

OLSE | 0.0888 | −0.0405 | 2.2201 | 0.0202 | |

WLSE | 0.2376 | −0.0312 | 2.8245 | 0.0182 | |

CME | 0.2884 | −0.0303 | 2.2121 | 0.0192 | |

100 | MLE | −0.0163 | 0.0078 | 2.0841 | 0.0083 |

MOME | −0.0247 | 0.0291 | 0.3181 | 0.0096 | |

OLSE | 0.1303 | −0.0244 | 1.7747 | 0.0103 | |

WLSE | 0.2232 | −0.0166 | 2.0862 | 0.0091 | |

CME | 0.2683 | −0.0205 | 1.8484 | 0.0099 | |

200 | MLE | −0.2227 | 0.0148 | 0.6364 | 0.0042 |

MOME | 0.0318 | 0.0267 | 0.2444 | 0.0046 | |

OLSE | 0.1333 | −0.0099 | 1.3872 | 0.0050 | |

WLSE | 0.1396 | −0.0039 | 1.3626 | 0.0044 | |

CME | 0.2233 | −0.0092 | 1.4512 | 0.0049 | |

500 | MLE | −0.3446 | 0.0196 | 0.2372 | 0.0017 |

MOME | 0.1420 | 0.0222 | 0.1582 | 0.0018 | |

OLSE | 0.0786 | 0.0008 | 0.8774 | 0.0021 | |

WLSE | −0.0197 | 0.0063 | 0.4987 | 0.0018 | |

CME | 0.1225 | 0.0006 | 0.9003 | 0.0021 |

Model | MLEs (Standard Errors) |
---|---|

MWE | $\widehat{\alpha}=0.4069\phantom{\rule{3.33333pt}{0ex}}\left(0.3763\right)$, $\widehat{\lambda}=0.0399\phantom{\rule{3.33333pt}{0ex}}\left(0.0056\right)$ |

W | $\widehat{\mu}=1.0149\phantom{\rule{3.33333pt}{0ex}}\left(0.1210\right)$, $\widehat{\sigma}=30.3358\phantom{\rule{3.33333pt}{0ex}}\left(4.4144\right)$ |

G | $\widehat{\mu}=0.9267\phantom{\rule{3.33333pt}{0ex}}(0.1621$, $\widehat{\sigma}=32.5640\phantom{\rule{3.33333pt}{0ex}}\left(7.4381\right)$ |

GE | $\widehat{\alpha}=0.9086\phantom{\rule{3.33333pt}{0ex}}\left(0.1622\right)$, $\widehat{\lambda}=0.0312\phantom{\rule{3.33333pt}{0ex}}\left(0.0058\right)$ |

WE | $\widehat{\alpha}=0.0097\phantom{\rule{3.33333pt}{0ex}}\left(0.6436\right)$, $\widehat{\lambda}=0.0660\phantom{\rule{3.33333pt}{0ex}}\left(0.0198\right)$ |

MOME | OLSE | WLSE | CME | |
---|---|---|---|---|

$\widehat{\alpha}$ | 1.8324 | 0.2399 | 0.3517 | 0.3289 |

$\widehat{\lambda}$ | 0.0417 | 0.0344 | 0.0373 | 0.0358 |

Model | AIC | KS | p(KS) | CVM | p(CVM) | AD | p(AD) |
---|---|---|---|---|---|---|---|

MWE | 443.8088 | 0.0955 | 0.7161 | 0.1015 | 0.5796 | 0.8336 | 0.4568 |

W | 444.6980 | 0.1113 | 0.5290 | 0.1269 | 0.4698 | 0.8907 | 0.4195 |

G | 444.5201 | 0.1226 | 0.4074 | 0.1477 | 0.3976 | 0.8702 | 0.4325 |

GE | 444.4178 | 0.1243 | 0.3903 | 0.1520 | 0.3844 | 0.8719 | 0.4314 |

WE | 468.9382 | 0.1544 | 0.1658 | 0.2851 | 0.1489 | 4.6374 | 0.0043 |

Distribution | MLEs (Standard Errors) |
---|---|

MWE | $\widehat{\alpha}=6.8800\phantom{\rule{3.33333pt}{0ex}}\left(5.6074\right)$, $\widehat{\lambda}=0.1180\phantom{\rule{3.33333pt}{0ex}}\left(0.0126\right)$ |

W | $\widehat{\mu}=1.0528\phantom{\rule{3.33333pt}{0ex}}\left(0.0680\right)$, $\widehat{\sigma}=9.6581\phantom{\rule{3.33333pt}{0ex}}\left(0.8574\right)$ |

G | $\widehat{\mu}=1.1782\phantom{\rule{3.33333pt}{0ex}}\left(0.1315\right)$, $\widehat{\sigma}=8.0157\phantom{\rule{3.33333pt}{0ex}}\left(1.1076\right)$ |

GE | $\widehat{\alpha}=1.2227\phantom{\rule{3.33333pt}{0ex}}\left(0.1493\right)$, $\widehat{\lambda}=0.1204\phantom{\rule{3.33333pt}{0ex}}\left(0.0135\right)$ |

WE | $\widehat{\alpha}=13.1550\phantom{\rule{3.33333pt}{0ex}}\left(10.6474\right)$, $\widehat{\lambda}=0.1134\phantom{\rule{3.33333pt}{0ex}}\left(0.0115\right)$ |

MOME | OLSE | WLSE | CME | |
---|---|---|---|---|

$\widehat{\alpha}$ | 0.0003 | 5.5633 | 6.4080 | 5.0124 |

$\widehat{\lambda}$ | 0.1003 | 0.1294 | 0.1263 | 0.1312 |

Model | $-2log\mathit{L}$ | AIC | KS | p(KS) | CVM | p(CVM) | AD | p(AD) |
---|---|---|---|---|---|---|---|---|

MWE | 827.2080 | 831.2080 | 0.0619 | 0.7100 | 0.0842 | 0.6689 | 0.5012 | 0.7453 |

W | 830.1968 | 834.1968 | 0.0663 | 0.6272 | 0.1380 | 0.4286 | 0.8743 | 0.4302 |

G | 828.7471 | 832.7471 | 0.0692 | 0.5722 | 0.1178 | 0.5049 | 0.6849 | 0.5713 |

GE | 828.1806 | 832.1806 | 0.0684 | 0.5877 | 0.1100 | 0.5389 | 0.6275 | 0.6221 |

WE | 828.3536 | 832.3536 | 0.0594 | 0.7579 | 0.0757 | 0.7182 | 0.4816 | 0.7653 |

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**MDPI and ACS Style**

Chesneau, C.; Kumar, V.; Khetan, M.; Arshad, M.
On a Modified Weighted Exponential Distribution with Applications. *Math. Comput. Appl.* **2022**, *27*, 17.
https://doi.org/10.3390/mca27010017

**AMA Style**

Chesneau C, Kumar V, Khetan M, Arshad M.
On a Modified Weighted Exponential Distribution with Applications. *Mathematical and Computational Applications*. 2022; 27(1):17.
https://doi.org/10.3390/mca27010017

**Chicago/Turabian Style**

Chesneau, Christophe, Vijay Kumar, Mukti Khetan, and Mohd Arshad.
2022. "On a Modified Weighted Exponential Distribution with Applications" *Mathematical and Computational Applications* 27, no. 1: 17.
https://doi.org/10.3390/mca27010017