# New Method for Generating New Families of Distributions

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## Abstract

**:**

## 1. Introduction

## 2. The Alpha Power Weibull—Exponential Distribution

#### 2.1. Special Cases of the APWED

- The APWED reduces to the WED at $\alpha =1$.
- The APWED reduces to the Weibull distribution at $\alpha =\lambda =1$.
- The APWED reduces to the exponential distribution when $\alpha =\gamma =a=1$.

#### 2.2. Expansion for the PDF

## 3. Properties of the APWED

#### 3.1. Quantile Function

#### 3.2. Moments

#### 3.3. Moment Generating and Characteristic Functions

#### 3.4. Mean Residual Life and Mean Waiting Time

#### 3.5. Rényi and Shannon Entropies

#### 3.6. Order Statistics

## 4. Maximum Likelihood Estimates

## 5. Simulation Study

- Different sample sizes (30, 50, 100, 150, 200, and 500) were drawn from the APWED under 1000 replicates;
- Two different sets of parameters values were assignedSet 1 $(\alpha =3,\lambda =0.1,\gamma =5,a=9)$ andSet 2 $(\alpha =0.5,\lambda =2,\gamma =1.5,a=0.3)$.
- For each sample size, the average estimates of the parameters and mean square error (MSE) were calculated using the “optim” function in R.

## 6. Applications

#### 6.1. Survival Time Data

#### 6.2. Failure Time Data

#### 6.3. Strength Data

- Kumaraswamy–Weibull (Ku–W) by [22]$$f{(x)}_{Ku-W}=\alpha \beta \frac{c}{\gamma}{\left(\frac{x}{\gamma}\right)}^{c-1}{e}^{-{\left(\frac{x}{\gamma}\right)}^{c}}{\left[1-{e}^{-{\left(\frac{x}{\beta}\right)}^{c}}\right]}^{\alpha -1}{\left[1-{\left(1-{e}^{-{\left(\frac{x}{\beta}\right)}^{c}}\right)}^{\alpha}\right]}^{\beta -1},$$
- Exponentiated truncated inverse Weibull–inverse Weibull (ETIWIW) by [23]$$\begin{array}{cc}\hfill f{(x)}_{ETIWIW}& =ab\theta {\mu}^{\theta}{x}^{-\theta -1}{e}^{-{\left(\frac{\mu}{x}\right)}^{\theta}}{\left[1-{e}^{-{\left(\frac{\mu}{x}\right)}^{\theta}}\right]}^{-b-1}{e}^{1-{\left[1-{e}^{-{\left(\frac{\mu}{x}\right)}^{\theta}}\right]}^{-b}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{{\int}_{t}}{\left[1-{e}^{1-{\left[1-{e}^{-{\left(\frac{\mu}{x}\right)}^{\theta}}\right]}^{-b}}\right]}^{a-1},\hfill \end{array}$$
- Alpha power inverse Weibull (APIW) distribution by [24].$${f}_{APIW}(x)=\left\{\begin{array}{cc}\frac{log\alpha}{\alpha -1}\lambda \beta {x}^{-(\beta +1)}{e}^{-\lambda {x}^{\beta}}{\alpha}^{{e}^{-\lambda {x}^{\beta}}}\hfill & \mathrm{if}\alpha 0,\alpha \ne 1\hfill \\ \lambda \beta {x}^{-(\beta +1)}{e}^{-\lambda {x}^{\beta}}\hfill & \mathrm{if}\alpha =1,\hfill \end{array}\right.$$
- Alpha power exponential (APE) distribution by [11].$${f}_{APE}(x)=\left\{\begin{array}{cc}\frac{log\alpha}{\alpha -1}\lambda {e}^{-\lambda x}{\alpha}^{(1-{e}^{-\lambda x})}\hfill & \mathrm{if}\alpha 0,\alpha \ne 1\hfill \\ \lambda {e}^{-\lambda x}\hfill & \mathrm{if}\alpha =1,\hfill \end{array}\right.$$
- Weibull–Lomax (WL) distribution by [9].$$\begin{array}{cc}\hfill {f}_{WL}(x)& =\frac{a}{\gamma}\frac{\alpha}{\beta}\frac{{\left[1+(\frac{x}{\beta})\right]}^{-(\alpha +1)}}{{\left[1+(\frac{x}{\beta})\right]}^{-\alpha}}{\left(\frac{-log{\left[1+(\frac{x}{\beta})\right]}^{-\alpha}}{\gamma}\right)}^{a-1}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{{\int}_{t}}exp\left[-{\left(\frac{-log{\left[1+(\frac{x}{\beta})\right]}^{-\alpha}}{\gamma}\right)}^{a}\right],\hfill \end{array}$$
- Exponential (E) distribution.$$f{(x)}_{E}=\lambda {e}^{-\lambda x},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}x\ge 0,\phantom{\rule{4pt}{0ex}}\lambda >0.$$

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Partial Derivatives of (33), with Respect to Each Parameter

## References

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**Table 1.**The mean, variance, skewness, and kurtosis of the APWED for certain selected values of the parameters.

$\mathit{\alpha}$ | $\mathit{\lambda}$ | $\mathit{\gamma}$ | a | Mean | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

0.5 | 0.5 | 2 | 1 | 3.3378 | 13.1798 | 5.3935 | 7.3632 |

1 | 2 | 1.5962 | 0.7901 | 1.8985 | 2.1830 | ||

2 | 3 | 0.8298 | 0.1018 | 1.4105 | 1.5306 | ||

3 | 5 | 0.5847 | 0.0199 | 1.1602 | 1.2061 | ||

5.5 | 6 | 0.3246 | 0.0044 | 1.1148 | 1.1478 | ||

0.9 | 0.5 | 2 | 1 | 3.8953 | 15.5766 | 4.6210 | 6.1819 |

1 | 2 | 1.7452 | 0.8497 | 1.7866 | 2.0266 | ||

2 | 3 | 0.8833 | 0.1050 | 1.3672 | 1.4716 | ||

3 | 5 | 0.6079 | 0.0197 | 1.1457 | 1.1867 | ||

5.5 | 6 | 0.3354 | 0.0043 | 1.1049 | 1.1346 | ||

1.5 | 0.5 | 2 | 1 | 4.4136 | 17.5837 | 4.0788 | 5.3732 |

1 | 2 | 1.8783 | 0.8858 | 1.6952 | 1.9026 | ||

2 | 3 | 0.9303 | 0.1057 | 1.3291 | 1.4210 | ||

3 | 5 | 0.6281 | 0.0192 | 1.1322 | 1.1689 | ||

5.5 | 6 | 0.3448 | 0.0042 | 1.0954 | 1.1222 | ||

2 | 0.5 | 2 | 1 | 4.7150 | 18.6460 | 3.8192 | 4.9917 |

1 | 2 | 1.9534 | 0.8990 | 1.6473 | 1.8387 | ||

2 | 3 | 0.9566 | 0.1052 | 1.3082 | 1.3938 | ||

3 | 5 | 0.6392 | 0.0188 | 1.1244 | 1.1588 | ||

5.5 | 6 | 0.3499 | 0.0040 | 1.0899 | 1.1150 | ||

2.5 | 0.5 | 2 | 1 | 4.9516 | 19.4253 | 3.6379 | 4.7276 |

1 | 2 | 2.0114 | 0.9057 | 1.6121 | 1.7923 | ||

2 | 3 | 0.9767 | 0.1045 | 1.2925 | 1.3734 | ||

3 | 5 | 0.6477 | 0.0184 | 1.1184 | 1.1511 | ||

5.5 | 6 | 0.3538 | 0.0039 | 1.0856 | 1.1095 | ||

6 | 0.5 | 2 | 1 | 5.8843 | 22.0241 | 3.0690 | 3.9099 |

1 | 2 | 2.2316 | 0.9040 | 1.4921 | 1.6361 | ||

2 | 3 | 1.0519 | 0.0985 | 1.2367 | 1.3019 | ||

3 | 5 | 0.6790 | 0.0165 | 1.0962 | 1.1227 | ||

5.5 | 6 | 0.3683 | 0.0035 | 1.0696 | 1.0890 |

Set 1 | Set 2 | ||||
---|---|---|---|---|---|

Sample Size | Par. | MLE | MSE | MLE | MSE |

30 | $\alpha $ | 2.9422 | 2.1201 | 0.3806 | 4.1005 |

$\lambda $ | 0.0755 | 0.0010 | 4.9901 | 23.3080 | |

$\gamma $ | 3.7871 | 2.4981 | 2.7951 | 11.2943 | |

a | 9.5300 | 1.9036 | 0.3053 | 0.0034 | |

50 | $\alpha $ | 2.9373 | 1.5402 | 1.2925 | 3.5021 |

$\lambda $ | 0.0771 | 0.0010 | 4.4244 | 17.3286 | |

$\gamma $ | 3.8701 | 2.4263 | 2.6141 | 8.2774 | |

a | 9.4225 | 1.1767 | 0.2989 | 0.0021 | |

100 | $\alpha $ | 2.9961 | 0.8505 | 0.4001 | 0.0795 |

$\lambda $ | 0.1036 | 0.0004 | 3.9395 | 12.6731 | |

$\gamma $ | 5.1892 | 0.9504 | 2.4014 | 5.3823 | |

a | 9.0946 | 0.4844 | 0.2943 | 0.0014 | |

150 | $\alpha $ | 3.0051 | 0.7086 | 0.3960 | 0.0687 |

$\lambda $ | 0.1045 | 0.0004 | 3.6920 | 9.2768 | |

$\gamma $ | 5.2315 | 0.8841 | 2.3709 | 5.0723 | |

a | 9.0983 | 0.3791 | 0.2943 | 0.0011 | |

200 | $\alpha $ | 2.9866 | 0.5953 | 0.4138 | 0.0609 |

$\lambda $ | 0.1037 | 0.0003 | 3.4956 | 8.3503 | |

$\gamma $ | 5.1882 | 0.7747 | 2.3642 | 4.2086 | |

a | 9.0770 | 0.2935 | 0.2955 | 0.0009 | |

500 | $\alpha $ | 2.9896 | 0.3320 | 0.4547 | 0.0524 |

$\lambda $ | 0.1018 | 0.0002 | 2.9487 | 4.2672 | |

$\gamma $ | 5.0923 | 0.4310 | 2.1032 | 2.4497 | |

a | 9.0334 | 0.1393 | 0.2969 | 0.0005 |

Distribution | Estimated Parameters | |||
---|---|---|---|---|

APWED | 0.0097 | 0.0021 | 2.2258 | 1.0213 |

$(\widehat{\alpha}$,$\widehat{\lambda}$,$\widehat{\gamma}$,$\widehat{a}$) | (0.0134) | (0.0002) | (0.7427) | (0.0976) |

Ku-W | 0.9783 | 0.0848 | 0.3994 | 0.5239 |

$(\widehat{\alpha}$,$\widehat{\beta}$,$\widehat{c}$,$\widehat{\gamma}$) | (0.4010) | (0.0119) | (0.0019) | (0.0109) |

ETIWIW | 28.0765 | 0.0612 | 3.5071 | 0.1089 |

$(\widehat{a}$,$\widehat{b}$,$\widehat{\theta}$,$\widehat{\mu})$ | (7.3394) | (0.0025) | (0.0033) | (0.0033) |

APIW | 27.1117 | 19.1687 | 0.8835 | - |

$(\widehat{\alpha}$,$\widehat{\lambda}$, $\widehat{\beta})$ | (20.9187) | (6.9268) | (0.0772) | |

APE | 0.2756 | 0.0023 | - | - |

$(\widehat{\alpha}$,$\widehat{\lambda}$) | (0.2311) | (0.0004) | ||

WL | 5.1200 | 5.4303 | 0.7655 | 3.0234 |

$(\widehat{\gamma}$,$\widehat{a}$,$\widehat{\alpha}$,$\widehat{\beta}$) | (6.5560) | (0.5031) | (0.8938) | (0.5809) |

E | 0.0027 | - | - | - |

$(\widehat{\lambda})$ | (0.0003) |

Distribution | Estimated Parameters | |||
---|---|---|---|---|

APWED | 25.6685 | 2.6851 | 4.9713 | 1.5886 |

$(\widehat{\alpha}$,$\widehat{\lambda}$,$\widehat{\gamma}$,$\widehat{a}$) | (40.1965) | (199.6717) | (369.7458) | (0.3246) |

Ku-W | 1.3649 | 0.0812 | 1.9011 | 0.7197 |

$(\widehat{\alpha}$,$\widehat{\beta}$,$\widehat{c}$,$\widehat{\gamma}$) | (0.0095) | (0.0089) | (0.0040) | (0.0045) |

ETIWIW | 2.3249 | 3.5489 | 0.4390 | 5.7867 |

$(\widehat{a}$,$\widehat{b}$,$\widehat{\theta}$,$\widehat{\mu})$ | (0.8317) | (0.9676) | (0.0619) | (2.7310) |

APIW | 3.9152 | 0.1948 | 1.2339 | - |

$(\widehat{\alpha}$,$\widehat{\lambda}$, $\widehat{\beta})$ | (1.2192) | (0.0235) | (0.0695) | |

APE | 34.1428 | 0.7591 | - | - |

$(\widehat{\alpha}$,$\widehat{\lambda}$) | (13.3994) | (0.0592) | ||

WL | 9.8170 | 4.5320 | 5.6555 | 1.7000 |

$(\widehat{\gamma}$,$\widehat{a}$,$\widehat{\alpha}$,$\widehat{\beta}$) | (10.0656) | (0.3975) | (4.8035) | (0.0898) |

E | 0.3902 | - | - | - |

$(\widehat{\lambda}$) | (0.0423) |

Distribution | Estimated Parameters | |||
---|---|---|---|---|

APWED | 10.8494 | 10.4039 | 14.9854 | 4.4840 |

$(\widehat{\alpha}$,$\widehat{\lambda}$,$\widehat{\gamma}$,$\widehat{a}$) | (12.8014) | (1625.4758) | (2341.4107) | (0.7681) |

Ku-W | 8.3440 | 0.0923 | 2.8288 | 0.6285 |

$(\widehat{\alpha}$,$\widehat{\beta}$,$\widehat{c}$,$\widehat{\gamma}$) | (0.6533) | (0.0117) | (0.0024) | (0.0024) |

ETIWIW | 1.0768 | 23.9679 | 0.9791 | 5.8196 |

$(\widehat{a}$,$\widehat{b}$,$\widehat{\theta}$,$\widehat{\mu})$ | (0.4988) | (18.3488) | (0.2967) | (3.0173) |

APIW | 193.0604 | 0.6365 | 3.8769 | - |

$(\widehat{\alpha}$,$\widehat{\lambda}$, $\widehat{\beta})$ | (267.2453) | (0.1822) | (0.3096) | |

APE | 33.0483 | 1.2993 | - | - |

$(\widehat{\alpha}$,$\widehat{\lambda}$) | (11.9269) | (0.1080) | ||

WL | 2.7539 | 8.7108 | 2.8070 | 1.0327 |

$(\widehat{\gamma}$,$\widehat{a}$,$\widehat{\alpha}$,$\widehat{\beta}$) | (1.7144) | (0.8782) | (1.0768) | (0.0211) |

E | 0.6636 | - | - | - |

$(\widehat{\lambda}$) | (0.0836) |

Distribution | AIC | CAIC | HQIC | K-S | W | A | $-\mathit{\ell}\left(\widehat{\mathit{\theta}}\right)$ |
---|---|---|---|---|---|---|---|

APWED | 751.5332 | 752.3332 | 754.6382 | 0.1596 | 0.3468 | 1.8001 | 371.7666 |

Ku-W | 795.5555 | 796.3555 | 798.6605 | 0.3165 | 0.3808 | 1.9230 | 393.7778 |

ETIWIW | 753.0122 | 753.8122 | 756.1172 | 0.1940 | 0.3507 | 1.9772 | 372.5061 |

APIW | 752.8410 | 753.3116 | 755.1698 | 0.1927 | 0.4975 | 2.5184 | 373.4205 |

APE | 757.1062 | 757.3369 | 758.6587 | 0.1730 | 0.3904 | 2.1997 | 376.5531 |

WL | 752.0366 | 752.8366 | 755.1416 | 0.1620 | 0.3766 | 1.8987 | 372.0183 |

E | 764.0156 | 764.0911 | 764.7919 | 0.2763 | 0.9638 | 4.5921 | 381.0078 |

Distribution | AIC | CAIC | HQIC | K-S | W | A | $-\mathit{\ell}\left(\widehat{\mathit{\theta}}\right)$ |
---|---|---|---|---|---|---|---|

APWED | 267.4350 | 267.9350 | 271.3650 | 0.0652 | 0.0478 | 0.4565 | 129.7175 |

Ku-W | 278.3159 | 278.8159 | 282.2459 | 0.1288 | 0.0987 | 0.9209 | 135.158 |

ETIWIW | 303.1870 | 303.6870 | 307.1170 | 0.1626 | 0.5833 | 3.4447 | 147.5935 |

APIW | 354.8962 | 355.1925 | 357.8437 | 0.2619 | 1.2682 | 7.6828 | 174.4481 |

APE | 281.9912 | 282.1375 | 283.9562 | 0.1603 | 0.6024 | 3.4923 | 138.9956 |

WL | 285.1424 | 285.6424 | 289.0724 | 0.0916 | 0.1492 | 1.2825 | 138.5712 |

E | 331.9754 | 332.0236 | 332.9579 | 0.3032 | 2.3607 | 11.81 | 164.9877 |

Distribution | AIC | CAIC | HQIC | K-S | W | A | $-\mathit{\ell}\left(\widehat{\mathit{\theta}}\right)$ |
---|---|---|---|---|---|---|---|

APWE | 34.9483 | 35.6379 | 38.3199 | 0.1225 | 0.1314 | 0.8554 | 13.4741 |

Ku-W | 66.1026 | 66.7922 | 69.4742 | 0.2845 | 0.6943 | 3.7613 | 29.0513 |

ETIWIW | 42.8534 | 43.5431 | 46.2251 | 0.1920 | 0.3909 | 1.9878 | 17.4267 |

APIW | 81.7724 | 82.1791 | 84.3011 | 0.2163 | 0.8679 | 4.9894 | 37.8862 |

APE | 124.7411 | 124.9411 | 126.4269 | 0.3388 | 2.4692 | 12.379 | 60.3706 |

WL | 40.5321 | 41.2217 | 43.9037 | 0.1728 | 0.2859 | 1.5654 | 16.2660 |

E | 179.6606 | 179.7262 | 180.5035 | 0.4180 | 3.8618 | 18.424 | 88.8303 |

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**MDPI and ACS Style**

Baharith, L.A.; Aljuhani, W.H.
New Method for Generating New Families of Distributions. *Symmetry* **2021**, *13*, 726.
https://doi.org/10.3390/sym13040726

**AMA Style**

Baharith LA, Aljuhani WH.
New Method for Generating New Families of Distributions. *Symmetry*. 2021; 13(4):726.
https://doi.org/10.3390/sym13040726

**Chicago/Turabian Style**

Baharith, Lamya A., and Wedad H. Aljuhani.
2021. "New Method for Generating New Families of Distributions" *Symmetry* 13, no. 4: 726.
https://doi.org/10.3390/sym13040726