# A New More Flexible Class of Distributions on (0,1): Properties and Applications to Univariate Data and Quantile Regression

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Generalized Unitary Weibull Family of Distribution

#### 2.1. Density Function

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

- 1.
- If ${\theta}_{1}={\theta}_{2}=\alpha =\beta =1$ then $Y\sim U(0,1)$, where U denotes the uniform distribution in (0,1).
- 2.
- If ${\theta}_{1}={\theta}_{2}=\theta $, and $\alpha =\beta =1$ then ${f}_{Y}$ is symmetric.
- 3.
- If ${\theta}_{1}={\theta}_{2}=\alpha =1$ then ${f}_{Y}\left(y\right)=\frac{\beta}{{[1+(\beta -1)y]}^{2}}$.

**Proof.**

- 1.
- The result is obtained by replacing ${\theta}_{1}={\theta}_{2}=\alpha =\beta =1$ in the distribution of Y then $Y\sim U(0,1)$.
- 2.
- If ${\theta}_{1}={\theta}_{2}=\theta $ and $\alpha =\beta =1$ then:$$\begin{array}{ccc}\hfill {f}_{Y}\left(y\right)& =& \frac{\theta {y}^{\theta -1}{(1-y)}^{\theta -1}}{{[{y}^{\theta}+{(1-y)}^{\theta}]}^{2}},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}0<y<1.\hfill \end{array}$$Then ${f}_{Y}\left(y\right)={f}_{Y}(1-y)$.
- 3.
- The result follows from plugging ${\theta}_{1}={\theta}_{2}=\alpha =1$ into the distribution of Y.

#### 2.2. Density Function of the Unitary Weibull Distribution Type 2

**Definition**

**1.**

**Proposition**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 2.3. The Reliability, Hazard Rate Functions and Increasing Failure Rate

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

#### 2.4. Moments

**Definition**

**2.**

**Proposition**

**6.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Corollary**

**2.**

**Proof.**

#### 2.5. Some Statistical Properties

#### 2.5.1. Entropy of $UW2$

#### 2.5.2. Mean Residual Life

#### 2.5.3. Incomplete Moments

#### 2.5.4. Lorenz Curve and the Gini Index

**Proposition**

**7.**

#### 2.6. Canonical Type 2 Unitary Weibull Distribution

- 1.
- The $cdf$ of Y is provided by:$$\begin{array}{c}\hfill {F}_{Y}\left(t\right)=\frac{\beta t}{1+\left(\right)open="("\; close=")">\beta -1},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{4pt}{0ex}}0t1.\end{array}$$
- 2.
- Quantile function of Y is:$$t=\frac{p\left(\right)open="("\; close=")">1-\beta}{+}p$$
- 3.
- The r-th moment of Y has the following expression:$$\begin{array}{ccc}\hfill {\mu}_{r}& =& E\left[{Y}^{r}\right]=1-r\beta \phantom{\rule{0.166667em}{0ex}}{\phantom{\rule{0.166667em}{0ex}}}_{2}{F}_{1}\left(\right)open="("\; close=")">1,r+1;r+2;-(\beta -1)\Gamma \left(\right)open="("\; close=")">r+1\hfill & ,,\phantom{\rule{4pt}{0ex}}r=1,\phantom{\rule{0.166667em}{0ex}}2,\dots \end{array}$$In particular, for $r=1,2,3,4$ we have:$$\begin{array}{ccc}\hfill {\mu}_{1}& =& \frac{\beta ln\left(\beta \right)+1-\beta}{{\left(\right)}^{\beta}}\phantom{\rule{4pt}{0ex}};\beta \ne 1\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\mu}_{2}& =& \frac{{\beta}^{2}-2\beta ln\left(\beta \right)-1}{{\left(\right)}^{\beta}}\phantom{\rule{4pt}{0ex}};\beta \ne 1\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\mu}_{3}& =& \frac{{\beta}^{3}-6{\beta}^{2}+3\beta +6\beta ln\left(\beta \right)+2}{2{\left(\right)}^{\beta}4}\phantom{\rule{4pt}{0ex}};\beta \ne 1\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\mu}_{4}& =& \frac{{\beta}^{4}-6{\beta}^{3}+18{\beta}^{2}-10\beta -12\beta ln\left(\beta \right)-3}{3{\left(\right)}^{\beta}5}\phantom{\rule{4pt}{0ex}};\beta \ne 1,\hfill \end{array}$$
- 4.
- Kurtosis coefficient is provided by following expression.$$\begin{array}{ccc}\hfill {\beta}_{2}& =& \frac{6{\left(\right)}^{\beta}3}{\left(\right)}-4{\left(\right)}^{\beta}2\hfill & \left(\right)open="("\; close=")">\beta ln\left(\beta \right)+1-\beta \\ \left(\right)open="("\; close=")">{\beta}^{3}-6{\beta}^{2}+3\beta +6\beta ln\left(\beta \right)+2\end{array}{\left(\right)}^{{\left(\right)}^{\beta}}\left(\right)open="("\; close=")">{\beta}^{2}-2\beta ln\left(\beta \right)-1$$Figure 7 shows the graphic behavior of the kurtosis for the canonical distribution $UW2$ for different values of $\beta $.
- 5.
- The Lorenz curve of Y is:$$\begin{array}{c}\hfill L\left(\right)open="("\; close=")">p,1,\beta \\ =& \frac{p-p\beta -\beta (ln\left(\left(\right)open="|"\; close="|">\right(p-1)\beta -p)}{+}\beta ln\beta -\beta +1\hfill & .\end{array}$$
- 6.
- The expression for the Gini index of Y is provided by:$$\begin{array}{c}\hfill G\left(\right)open="("\; close=")">1,\beta \\ =& \frac{\beta \left(\right)open="["\; close="]">\left(\right)open="("\; close=")">1+\beta}{ln}\hfill & \left(\right)open="("\; close=")">\beta -1\left(\right)open="["\; close="]">1-\beta \left(\right)open="("\; close=")">1-ln\left(\beta \right)\end{array}$$Figure 8 shows the Lorenz curve and the Gini index of the canonical $UW2$ distribution for different values of $\beta $ in which the parameter $\beta $ is directly proportional to the Gini index.
- 7.
- Entropy of Y:$$\begin{array}{c}\hfill H(1,\beta )=2-\frac{\beta +1}{\beta -1}ln\left(\beta \right).\end{array}$$Figure 9 shows the graph of the entropy of the canonical $UW2$ distribution for different values of $\beta $.

## 3. Inference

#### 3.1. Maximum Likelihood Estimate

#### 3.2. Simulation Study

## 4. Analysis of Real Data

#### 4.1. Example 1: Application to Medical Data

#### 4.2. Example 2: An Application to Environment Data

#### 4.3. Example 3: An Application to Quantile Regression

#### 4.3.1. One-Dimensional Quantile Regression

#### 4.3.2. Quantile Regression Unitary Weibull Type 2

#### 4.3.3. An Application of Quantile Regression to Praters Gas Mileage Data

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**Lorenz curve and Gini index of the canonical $UW2$ distribution for different values of $\beta $.

**Figure 9.**Graph of the entropy of the canonical $UW2$ distribution for different values of $\beta $.

**Figure 10.**Histogram for LBMD data with Densities $UW2$ (solid line), $UW$ (dashed line), $Beta$ (dotted line), and $KW$ (dashed dotted line) (

**left**) and tails (

**right**).

**Figure 12.**Comparison of cumulative distributions for the LBMD data set for $UW2$ (blue line), $UW$ (red line), $Beta$ (green line), and $KW$ (orange line).

**Figure 13.**Histogram for percent dissolved oxygen data (

**left**) with densities of $UW2$ (solid line), $UW$ (dashed line), $Beta$ (dotted line), and $KW$ (dashed line) and tails (

**right**).

**Figure 15.**Quantile regression for Yield and Temperature data with $UW2$ density (

**left**) and $Beta$ density (

**right**).

**Figure 16.**Quantile regression for Yield and Temperature data with $UW2$ density (

**left**) and $UW$ density (

**right**).

${\mathit{S}}_{\mathit{Y}}\left(\mathit{t}\right)=\mathit{P}(\mathit{Y}>\mathit{t})$ | ||||
---|---|---|---|---|

$\mathit{t}$ | $\mathit{UW}\mathbf{2}(\mathbf{1},\mathbf{5})$ | $\mathit{UW}(\mathbf{1},\mathbf{5})$ | $\mathit{KW}(\mathbf{1},\mathbf{5})$ | $\mathit{Beta}(\mathbf{1},\mathbf{5})$ |

0.70 | 0.0789474 | 0.0057559 | 0.0024300 | 0.0024300 |

0.75 | 0.0625000 | 0.0019685 | 0.0009766 | 0.0009766 |

0.80 | 0.0476191 | 0.0005531 | 0.0003200 | 0.0003200 |

0.85 | 0.0340909 | 0.0001134 | 0.0000759 | 0.0000759 |

0.90 | 0.0217391 | 0.0000130 | 0.0000100 | 0.0000100 |

0.95 | 0.0104167 | 0.0000004 | 0.0000003 | 0.0000003 |

**Table 2.**Skewness and kurtosis values of the

**$UW2$**model with different values of $\theta $ and $\beta $.

$\mathit{Skewness}$ | $\mathit{Kurtosis}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{\theta}$ | $\mathbf{\beta}=\mathbf{1}/\mathbf{2}$ | $\mathbf{\beta}=\mathbf{1}/\mathbf{3}$ | $\mathbf{\beta}=\mathbf{1}$ | $\mathbf{\beta}=\mathbf{2}$ | $\mathbf{\beta}=\mathbf{3}$ | $\mathbf{\beta}=\mathbf{1}/\mathbf{2}$ | $\mathbf{\beta}=\mathbf{1}/\mathbf{3}$ | $\mathbf{\beta}=\mathbf{1}$ | $\mathbf{\beta}=\mathbf{2}$ | $\mathbf{\beta}=\mathbf{3}$ |

1 | −0.4861 | −0.7849 | 0 | 0.4861 | 0.7849 | 2.0928 | 2.5644 | 1.8000 | 2.0928 | 2.5644 |

2 | −0.5980 | −0.9739 | 0 | 0.5980 | 0.9739 | 3.0459 | 3.9578 | 2.5013 | 3.0459 | 3.9578 |

3 | −0.5744 | −0.9279 | 0 | 0.5744 | 0.9279 | 3.5809 | 4.5497 | 2.9939 | 3.5809 | 4.5497 |

4 | −0.5176 | −0.8255 | 0 | 0.5176 | 0.8255 | 3.8564 | 4.7050 | 3.3240 | 3.8564 | 4.7050 |

5 | −0.4592 | −0.7237 | 0 | 0.4592 | 0.7237 | 3.9995 | 4.6954 | 3.5460 | 3.9994 | 4.6954 |

6 | −0.4077 | −0.6363 | 0 | 0.4077 | 0.6363 | 4.0765 | 4.6377 | 3.6984 | 4.0765 | 4.6377 |

7 | −0.3642 | −0.5641 | 0 | 0.3642 | 0.5641 | 4.1201 | 4.5736 | 3.8058 | 4.1201 | 4.5736 |

8 | −0.3278 | −0.5048 | 0 | 0.3278 | 0.5048 | 4.1460 | 4.5161 | 3.8836 | 4.1460 | 4.5161 |

9 | −0.2972 | −0.4557 | 0 | 0.2972 | 0.4557 | 4.1620 | 4.4678 | 3.9412 | 4.1620 | 4.4678 |

10 | −0.2715 | −0.4148 | 0 | 0.2715 | 0.4148 | 4.1723 | 4.4281 | 3.9848 | 4.1723 | 4.4281 |

11 | −0.2496 | −0.3802 | 0 | 0.2496 | 0.3802 | 4.1793 | 4.3957 | 4.0186 | 4.1793 | 4.3957 |

12 | −0.2307 | −0.3508 | 0 | 0.2307 | 0.3508 | 4.1840 | 4.3692 | 4.0472 | 4.1840 | 4.3693 |

13 | −0.2144 | −0.3254 | 0 | 0.2144 | 0.3254 | 4.1874 | 4.3475 | 4.0665 | 4.1874 | 4.3475 |

14 | −0.2002 | −0.3033 | 0 | 0.2002 | 0.3033 | 4.1899 | 4.3295 | 4.0765 | 4.1899 | 4.3295 |

15 | −0.1877 | −0.2840 | 0 | 0.1877 | 0.2840 | 4.1917 | 4.3144 | 4.0979 | 4.1917 | 4.3145 |

16 | −0.1766 | −0.2670 | 0 | 0.1766 | 0.2670 | 4.1932 | 4.3018 | 4.1096 | 4.1931 | 4.3018 |

17 | −0.1667 | −0.2518 | 0 | 0.1667 | 0.2518 | 4.1942 | 4.2911 | 4.1194 | 4.1942 | 4.2908 |

18 | −0.1578 | −0.2393 | 0 | 0.1578 | 0.2382 | 4.1951 | 4.4034 | 4.1278 | 4.1951 | 4.2837 |

19 | −0.1498 | −0.2261 | 0 | 0.1498 | 0.2263 | 4.1958 | 4.2741 | 4.1350 | 4.1958 | 4.2692 |

20 | −0.1426 | −0.2405 | 0 | 0.1426 | 0.2149 | 4.1963 | 5.7680 | 4.1411 | 4.1963 | 4.2746 |

**Table 3.**Entropy values for the distribution $UW2$($\theta ,\beta $) for different values of $\theta $ and $\beta $.

$\mathit{\theta}$ | $\mathit{\beta}=1/3$ | $\mathit{\beta}=1/2$ | $\mathit{\beta}=1$ | $\mathit{\beta}=2$ | $\mathit{\beta}=3$ |
---|---|---|---|---|---|

1 | −0.1976 | −0.0798 | 0.0000 | −0.0781 | −0.1939 |

2 | −0.5145 | −0.3657 | −0.2640 | −0.3657 | −0.5146 |

3 | −0.8398 | −0.6808 | −0.5714 | −0.6808 | −0.8398 |

4 | −1.0984 | −0.9350 | −0.8223 | −0.9350 | −1.0984 |

5 | −1.3079 | −1.1423 | −1.0279 | −1.1423 | −1.3079 |

6 | −1.4828 | −1.3159 | −1.2006 | −1.3159 | −1.4828 |

7 | −1.6324 | −1.4648 | −1.3488 | −1.4648 | −1.6324 |

8 | −1.7630 | −1.5949 | −1.4785 | −1.5949 | −1.7630 |

9 | −1.8788 | −1.7103 | −1.5936 | −1.7103 | −1.8788 |

10 | −1.9827 | −1.8139 | −1.6971 | −1.8139 | −1.9827 |

11 | −2.0770 | −1.9080 | −1.7910 | −1.9080 | −2.0770 |

12 | −2.1632 | −1.9940 | −1.8769 | −1.9940 | −2.1632 |

13 | −2.2426 | −2.0733 | −1.9561 | −2.0733 | −2.2426 |

14 | −2.3162 | −2.1469 | −2.0295 | −2.1469 | −2.3162 |

15 | −2.3848 | −2.2154 | −2.0980 | −2.2154 | −2.3848 |

16 | −2.4490 | −2.2795 | −2.1621 | −2.2795 | −2.4490 |

17 | −2.5093 | −2.3398 | −2.2223 | −2.3398 | −2.5093 |

18 | −2.5663 | −2.3967 | −2.2792 | −2.3967 | −2.5663 |

19 | −2.6201 | −2.4505 | −2.3330 | −2.4505 | −2.6201 |

20 | −2.6713 | −2.5016 | −2.3841 | −2.5016 | −2.6713 |

n | $\mathit{\beta}$ | $\mathit{\theta}$ | $\widehat{\mathit{\beta}}$ | $\mathit{sd}(\widehat{\mathit{\beta}})$ | $\mathit{c}(\widehat{\mathit{\beta}})$ | $\widehat{\mathit{\theta}}$ | $\mathit{sd}(\widehat{\mathit{\theta}})$ | $\mathit{c}(\widehat{\mathit{\theta}})$ |
---|---|---|---|---|---|---|---|---|

50 | 2 | 0.5 | 2.2512 | 1.1014 | 90.6 | 0.5105 | 0.0603 | 94.6 |

100 | 2 | 0.5 | 2.1292 | 0.7380 | 94.3 | 0.5043 | 0.0422 | 94.0 |

200 | 2 | 0.5 | 2.0346 | 0.4975 | 94.2 | 0.5022 | 0.0297 | 94.5 |

50 | 2 | 1 | 2.0559 | 0.5021 | 92.8 | 1.0209 | 0.1207 | 94.6 |

100 | 2 | 1 | 2.0340 | 0.3523 | 95.1 | 1.0086 | 0.0844 | 94.0 |

200 | 2 | 1 | 2.0029 | 0.2450 | 95.0 | 1.0043 | 0.0594 | 94.5 |

50 | 2 | 2 | 2.0118 | 0.2455 | 94.3 | 2.0419 | 0.2413 | 94.6 |

100 | 2 | 2 | 2.0097 | 0.1740 | 95.4 | 2.0172 | 0.1687 | 94.0 |

200 | 2 | 2 | 1.9979 | 0.1222 | 95.2 | 2.0087 | 0.1188 | 94.4 |

50 | 2 | 4 | 2.0020 | 0.1221 | 94.1 | 4.0837 | 0.4826 | 94.6 |

100 | 2 | 4 | 2.0031 | 0.0867 | 95.8 | 4.0344 | 0.3374 | 94.0 |

200 | 2 | 4 | 1.9981 | 0.0611 | 95.3 | 4.0173 | 0.2376 | 94.5 |

50 | 0.5 | 4 | 0.5005 | 0.0305 | 94.2 | 4.0837 | 0.4826 | 94.6 |

100 | 0.5 | 4 | 0.5008 | 0.0217 | 95.8 | 4.0344 | 0.3374 | 94.0 |

200 | 0.5 | 4 | 0.4995 | 0.0153 | 95.3 | 4.0173 | 0.2376 | 94.5 |

50 | 0.5 | 2 | 0.5030 | 0.0614 | 94.2 | 2.0419 | 0.2413 | 94.6 |

100 | 0.5 | 2 | 0.5024 | 0.0435 | 95.4 | 2.0172 | 0.1687 | 94.0 |

200 | 0.5 | 2 | 0.4995 | 0.0306 | 95.2 | 2.0086 | 0.1188 | 94.5 |

50 | 1 | 2 | 1.0059 | 0.1228 | 94.3 | 2.0419 | 0.2413 | 94.6 |

100 | 1 | 2 | 1.0049 | 0.0870 | 95.4 | 2.0172 | 0.1687 | 94.0 |

200 | 1 | 2 | 0.9989 | 0.0611 | 95.2 | 2.0087 | 0.1188 | 94.5 |

n | $\overline{\mathit{w}}$ | $\mathit{sd}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

66 | 0.5864 | 0.1339 | 0.04085 | 3.5395 |

Parameter Estimates | $\mathit{KW}\left(\mathit{sd}\right)$ | $\mathit{Beta}\left(\mathit{sd}\right)$ | $\mathit{UW}\left(\mathit{sd}\right)$ | $\mathit{UW}2\left(\mathit{sd}\right)$ |
---|---|---|---|---|

$\widehat{\alpha}$ | 5.0241 (1.0507) | 4.4717 (0.7554) | 2.3068 (0.2055) | - |

$\widehat{\beta}$ | 3.8972 (0.4386) | 6.4115 (1.1016) | 2.8807 (0.3867) | 0.6992 (0.0513) |

$\widehat{\theta}$ | - | - | - | 2.9370 (0.3041) |

Log-likelihood | 33.9586 | 35.5639 | 36.333 | 38.0463 |

AIC | −61.712 | −67.1278 | −68.666 | −72.0926 |

BIC | −59.333 | −62.748 | −64.281 | −67.713 |

KS Statistic | 0.1212 | 0.1515 | 0.1212 | 0.0909 |

W* | 0.1380 | 0.1029 | 0.08197 | 0.06354 |

A* | 0.9276 | 0.7149 | 0.6091 | 0.4204 |

n | $\overline{\mathit{w}}$ | $\mathit{sd}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

210 | 0.8294 | 0.1283 | −2.3702 | 11.3423 |

Parameter Estimates | $\mathit{Beta}\left(\mathit{sd}\right)$ | $\mathit{KW}\left(\mathit{sd}\right)$ | $\mathit{UW}\left(\mathit{sd}\right)$ | $\mathit{UW}2\left(\mathit{sd}\right)$ |
---|---|---|---|---|

$\widehat{\alpha}$ | 7.3538 (0.7436) | 6.9905 (0.5687) | 5.4931 (0.4721) | - |

$\widehat{\beta}$ | 1.6043 (0.1435) | 1.8721 (0.0.2003) | 1.1210 (0.0504) | 0.1616 (0.0089) |

$\widehat{\theta}$ | - | - | - | 2.1259 (0.1740) |

AIC | −344.234 | −352.005 | −326.509 | −394.225 |

BIC | −337.540 | −345.311 | −319.814 | −387.530 |

$\mathit{Data}$ | n | $\overline{\mathit{w}}$ | $\mathit{sd}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|---|

Yield | 52 | 332.0938 | 69.7559 | −0.2657 | 1.3058 |

Temp | 52 | 0.1965 | 0.1070 | 0.3687 | 2.1997 |

**Table 10.**Parameters estimates and standard error for the quantile regression coefficients $UW2$, $UW$, and $Beta$ models for the dataset and the quantile of 0.5.

$\mathit{UW}2$ | $\mathit{UW}$ | $\mathit{Beta}$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{Coef}.$ | $\mathit{Est}.$ | $\mathit{sd}$ | t-Value | p-Value | $\mathit{Est}.$ | $\mathit{sd}$ | t-Value | p-Value | $\mathit{Est}.$ | $\mathit{sd}$ | t-Value | p-Value |

${\alpha}_{0}$ | −0.1702 | 0.0800 | −2.1256 | 0.0418 | −0.1733 | 0.0963 | −1.7992 | 0.0820 | −0.1339 | 0.0528 | −2.5333 | 0.0167 |

${\alpha}_{1}$ | 0.0011 | 0.0002 | 4.1976 | 0.0002 | 0.0012 | 0.0003 | 3.6745 | 0.0009 | 0.0009 | 0.0001 | 5.1823 | 0.0000 |

Model | AIC | BIC |
---|---|---|

$UW2$ | −58.07373 | −53.67652 |

$UW$ | −53.75805 | −49.36084 |

$Beta$ | −43.69310 | −39.29589 |

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## Share and Cite

**MDPI and ACS Style**

Reyes, J.; Rojas, M.A.; Cortés, P.L.; Arrué, J.
A New More Flexible Class of Distributions on (0,1): Properties and Applications to Univariate Data and Quantile Regression. *Symmetry* **2023**, *15*, 267.
https://doi.org/10.3390/sym15020267

**AMA Style**

Reyes J, Rojas MA, Cortés PL, Arrué J.
A New More Flexible Class of Distributions on (0,1): Properties and Applications to Univariate Data and Quantile Regression. *Symmetry*. 2023; 15(2):267.
https://doi.org/10.3390/sym15020267

**Chicago/Turabian Style**

Reyes, Jimmy, Mario A. Rojas, Pedro L. Cortés, and Jaime Arrué.
2023. "A New More Flexible Class of Distributions on (0,1): Properties and Applications to Univariate Data and Quantile Regression" *Symmetry* 15, no. 2: 267.
https://doi.org/10.3390/sym15020267