# New Lifetime Distribution for Modeling Data on the Unit Interval: Properties, Applications and Quantile Regression

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bounded Truncated Cauchy Power Exponential Distribution

## 3. Some Important Properties

#### 3.1. Distribution Inequalities

**Proposition 1.**

**Proof.**

#### 3.2. Quantile Function

#### 3.3. Moments and Moments Generating Function

**Proposition 2.**

**Proof.**

**Proposition 3.**

**Proof.**

**Proposition 4.**

**Proof.**

#### 3.4. Order Statistics

## 4. Bivariate Extension

- (a)
- $\alpha =3.5,\lambda =8.2,{\delta}_{1}=0.3,{\delta}_{2}=0.1,{\delta}_{3}=0.3$;
- (b)
- $\alpha =2.5,\lambda =0.8,{\delta}_{1}=0.5,{\delta}_{2}=0.4,{\delta}_{3}=0.2$ and
- (c)
- $\alpha =0.5,\lambda =4.8,{\delta}_{1}=-0.3,{\delta}_{2}=-0.7,{\delta}_{3}=-0.1$.

- (a)
- $\alpha =3.5,\lambda =8.2,{\delta}_{1}=0.3,{\delta}_{2}=0.1,{\delta}_{3}=0.3$;
- (b)
- $\alpha =2.5,\lambda =0.8,{\delta}_{1}=0.5,{\delta}_{2}=0.4,{\delta}_{3}=0.2$ and
- (c)
- $\alpha =0.5,\lambda =4.8,{\delta}_{1}=-0.3,{\delta}_{2}=-0.7,{\delta}_{3}=-0.1$.

## 5. Parameter Estimation Methods

#### 5.1. Maximum Likelihood Estimation

#### 5.2. Ordinary and Weighted Least Squares Estimation

#### 5.3. Cramér–Von Mises Estimation

#### 5.4. Anderson–Darling Estimation

#### 5.5. Percentile Estimation

#### 5.6. Maximum and Minimum Product Spacing Estimation

## 6. Simulation

## 7. Applications

#### 7.1. UK COVID-19 Mortality

#### 7.2. Canada COVID-19 Mortality

#### 7.3. Spain COVID-19 Recovery Rate

## 8. Quantile Regression

#### 8.1. Residual Analysis

#### 8.2. Monte Carlo Simulation for Quantile Regression

#### 8.3. Application

^{2}), sex (female or male) and IPAQ (sedentary (S), insufficiently active (I), or active (A)). In this study, the response variable body fat percentage at arms is regressed on age (${z}_{i1}$), body mass index (${z}_{i2}$) and sex (${z}_{i3}$, 0 for female and 1 for male). The response variable is regressed on the covariates using the relationship $\mathrm{logit}({\rho}_{i})={\theta}_{0}+{\theta}_{1}{z}_{i1}+{\theta}_{2}{z}_{i2}+{\theta}_{3}{z}_{i3},i=1,2,\dots ,298$. Table 9 presents ML estimates, standard errors, and p-values for the parameters of the fitted models for the different quantiles. The estimates are all significant at the 5% level of significance.

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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${\mathit{\mu}}_{\mathit{r}}^{\mathbf{\prime}}$ | $\mathit{\alpha}\mathbf{=}\mathbf{0.4}\mathbf{,}\mathit{\lambda}\mathbf{=}\mathbf{2.5}$ | $\mathit{\alpha}\mathbf{=}\mathbf{4.5}\mathbf{,}\mathit{\lambda}\mathbf{=}\mathbf{3.1}$ | $\mathit{\alpha}\mathbf{=}\mathbf{20.0}\mathbf{,}\mathit{\lambda}\mathbf{=}\mathbf{1.5}$ |
---|---|---|---|

${\mu}_{1}^{\prime}$ | 0.8799 | 0.5602 | 0.1339 |

${\mu}_{2}^{\prime}$ | 0.8021 | 0.3401 | 0.0242 |

${\mu}_{3}^{\prime}$ | 0.7457 | 0.2185 | 0.0053 |

${\mu}_{4}^{\prime}$ | 0.7020 | 0.1465 | 0.0013 |

${\mu}_{5}^{\prime}$ | 0.6667 | 0.1017 | 0.0004 |

${\mu}_{6}^{\prime}$ | 0.6373 | 0.0726 | 0.0001 |

SD | 0.1668 | 0.1619 | 0.0794 |

CV | 0.1896 | 0.2890 | 0.5931 |

CS | −1.9527 | −0.3403 | 0.7713 |

CK | 6.6850 | 2.7084 | 3.5390 |

Parameter | $\mathit{n}$ | AB | RMSE | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MLE | MPS | MADS | MALDS | OLS | WLS | CVM | AD | PC | MLE | MPS | MADS | MALDS | OLS | WLS | CVM | AD | PC | ||

$\alpha $ | 25 | 0.7980 | 2.2327 | −2.3189 | 0.531 | 0.3530 | −2.6423 | 1.1477 | 0.4713 | −0.3155 | 2.2233 | 3.5728 | 2.9322 | 3.3391 | 2.4457 | 2.6729 | 3.5377 | 1.9964 | 1.4972 |

75 | 0.2140 | 0.7443 | −1.8442 | 0.0634 | 0.1365 | −3.1632 | 0.3415 | 0.1180 | −0.1694 | 0.9157 | 1.3139 | 2.4241 | 1.1330 | 1.1489 | 3.1657 | 1.2820 | 0.9535 | 0.8506 | |

125 | 0.1342 | 0.4372 | −1.3088 | −0.0031 | 0.0472 | −3.3268 | 0.1337 | 0.0713 | −0.0783 | 0.6843 | 0.8313 | 1.9860 | 0.7795 | 0.8149 | 3.3279 | 0.8159 | 0.7025 | 0.6641 | |

175 | 0.0914 | 0.2987 | −0.8738 | −0.0272 | 0.0460 | −2.1721 | 0.1164 | 0.0657 | −0.0484 | 0.5365 | 0.6791 | 1.5544 | 0.6323 | 0.6845 | 2.1990 | 0.6955 | 0.5941 | 0.5393 | |

225 | 0.0677 | 0.2509 | −0.6976 | 0.0062 | 0.0301 | 3.0791 | 0.1096 | 0.0365 | −0.0623 | 0.4841 | 0.5505 | 1.3266 | 0.5926 | 0.5906 | 3.3377 | 0.6147 | 0.5240 | 0.4860 | |

$\lambda $ | 25 | 0.1871 | 0.5436 | −1.1017 | 0.0300 | −0.0075 | −1.3344 | 0.2060 | 0.0670 | −0.1737 | 0.6038 | 0.8201 | 1.3862 | 0.7382 | 0.6538 | 1.3687 | 0.7340 | 0.5749 | 0.5401 |

75 | 0.0478 | 0.2197 | −0.8939 | −0.0079 | 0.0078 | −2.0003 | 0.0802 | 0.1185 | −0.0672 | 0.3089 | 0.4026 | 1.2090 | 0.3996 | 0.3692 | 2.0029 | 0.3886 | 0.3379 | 0.3175 | |

125 | 0.0378 | 0.1293 | −0.6271 | −0.0160 | −0.0055 | −2.1461 | 0.0305 | 0.0146 | −0.0448 | 0.2407 | 0.2740 | 0.9681 | 0.2987 | 0.2752 | 2.1472 | 0.2798 | 0.2560 | 0.2466 | |

175 | 0.0233 | 0.0866 | −0.3986 | −0.0139 | 0.0034 | −1.6394 | 0.0280 | 0.0114 | −0.0267 | 0.1959 | 0.2314 | 0.7264 | 0.2315 | 0.2372 | 1.6455 | 0.2383 | 0.2134 | 0.2008 | |

225 | 0.0208 | 0.0820 | −0.3101 | 0.0021 | 0.0003 | 0.1937 | 0.0230 | 0.0007 | −0.0225 | 0.1810 | 0.1939 | 0.6057 | 0.2196 | 0.2079 | 0.3066 | 0.2124 | 0.1874 | 0.1835 |

Parameter | $\mathit{n}$ | AB | RMSE | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MLE | MPS | MADS | MALDS | OLS | WLS | CVM | AD | PC | MLE | MPS | MADS | MALDS | OLS | WLS | CVM | AD | PC | ||

$\alpha $ | 25 | 0.5120 | 1.5281 | −1.3158 | 0.3712 | 0.2359 | −1.9121 | 0.7701 | 0.2725 | −0.6748 | 1.4793 | 2.5597 | 2.0817 | 1.9689 | 1.6086 | 1.9366 | 2.4325 | 1.3231 | 1.4204 |

75 | 0.2081 | 0.5190 | −0.9456 | 0.0477 | 0.0264 | −2.2924 | 0.1973 | 0.0989 | −0.3302 | 0.6848 | 0.8671 | 1.5913 | 0.7865 | 0.7088 | 2.2949 | 0.8122 | 0.6294 | 0.8145 | |

125 | 0.0994 | 0.3218 | −0.6895 | 0.0570 | 0.0153 | −2.4255 | 0.1020 | 0.0757 | −0.2704 | 0.4778 | 0.6228 | 1.2854 | 0.5657 | 0.5532 | 2.4266 | 0.5644 | 0.5109 | 0.6262 | |

175 | 0.0867 | 0.2259 | −0.5478 | 0.0107 | 0.0240 | −1.3857 | 0.0882 | 0.0554 | −0.2187 | 0.4077 | 0.4554 | 1.0538 | 0.4821 | 0.4813 | 1.4810 | 0.4940 | 0.4163 | 0.5242 | |

225 | 0.0461 | 0.1719 | −0.4192 | 0.0007 | 0.0195 | 2.2166 | 0.0555 | 0.0200 | −0.1735 | 0.3331 | 0.3880 | 0.8460 | 0.4021 | 0.4133 | 2.3785 | 0.4184 | 0.3477 | 0.4768 | |

$\lambda $ | 25 | 0.5725 | 1.9602 | −3.5282 | 0.1675 | 0.1107 | −4.6443 | 0.7293 | 0.2368 | −1.6432 | 2.0951 | 3.1555 | 4.8358 | 2.6753 | 2.3883 | 4.7703 | 2.6517 | 2.1203 | 2.6346 |

75 | 0.2957 | 0.8211 | −2.4563 | −0.0292 | −0.0416 | −6.9243 | 0.2449 | 0.0825 | −0.7220 | 1.1627 | 1.4130 | 3.9050 | 1.3676 | 1.2655 | 6.9330 | 1.3246 | 1.1047 | 1.4877 | |

125 | 0.1098 | 0.4964 | −1.6975 | 0.0128 | −0.0435 | −7.3960 | 0.1015 | 0.0784 | −0.5259 | 0.8837 | 1.0321 | 3.0631 | 1.0694 | 0.9899 | 7.3994 | 0.9670 | 0.9327 | 1.1482 | |

175 | 0.1182 | 0.3504 | −1.2348 | −0.0232 | 0.0022 | −5.5678 | 0.1061 | 0.0622 | −0.4011 | 0.7361 | 0.7786 | 2.4368 | 0.8864 | 0.8608 | 5.5937 | 0.8687 | 0.7734 | 0.9365 | |

225 | 0.0631 | 0.2843 | −0.9196 | −0.0034 | 0.0015 | 0.6715 | 0.043 | 0.0249 | −0.3171 | 0.6223 | 0.7091 | 1.9241 | 0.7604 | 0.7452 | 1.0898 | 0.7515 | 0.6590 | 0.8305 |

Country | Minimum | Maximum | Mean | Skewness | Kurtosis |
---|---|---|---|---|---|

UK | 0.0807 | 0.5331 | 0.2888 | 0.0476 | −1.1034 |

Canada | 0.1159 | 0.3347 | 0.2305 | −0.0850 | −0.4402 |

Spain | 0.4286 | 0.8628 | 0.7240 | −0.6890 | −0.4761 |

Model | Parameter | $\mathit{\ell}$ | AIC | BIC | AD | CVM |
---|---|---|---|---|---|---|

BTCPE | $\alpha =16.6904(5.2798)$ $\lambda =2.3884(0.2865)$ | 45.4400 | −86.8726 | −82.6840 | 0.6494 | 0.1049 |

Beta | $\alpha =4.0502(0.7128)$ $\beta =10.0132(1.8287)$ | 45.4000 | −86.7958 | −82.6071 | 0.7356 | 0.1280 |

UBIII | $\alpha =0.0757(0.0383)$ $\beta =13.3804(6.5631)$ | 38.9000 | −73.8075 | −69.6188 | 2.8948 | 0.5248 |

BMOEE | $\alpha =105.2655(59.9004)$ $\beta =3.5949(0.4092)$ | 40.7200 | −77.4396 | −73.2509 | 1.1465 | 0.1698 |

UW | $\alpha =0.2834(0.0602)$ $\beta =3.1228(0.3047)$ | 42.5600 | −81.1208 | −76.9322 | 1.0656 | 0.1820 |

UG | $\alpha =686.3600(2.2295\times {10}^{-10})$ $\beta =0.0011(1.4051\times {10}^{-4})$ | 2.8400 | −1.6760 | 2.5127 | 12.2290 | 2.4707 |

UL | $\alpha =2.8293(0.3029)$ | 32.3800 | −62.7533 | −60.6590 | 4.4878 | 0.7574 |

UISDL | $\alpha =3.4259(0.3151)$ | 33.6100 | −65.2142 | −63.1198 | 3.9972 | 0.6545 |

Model | Parameter | $\mathit{\ell}$ | AIC | BIC | AD | CVM |
---|---|---|---|---|---|---|

BTCPE | $\alpha =622.2064(399.8188)$ $\lambda =4.5085(0.4837)$ | 86.4400 | −168.8806 | −164.8299 | 0.3767 | 0.0689 |

Beta | $\alpha =14.5128(2.7128)$ $\beta =48.4900(9.1745)$ | 85.9400 | −167.8800 | −163.8293 | 0.4398 | 0.0692 |

UBIII | $\alpha =0.0080(0.0011)$ $\beta =101.7700(8.4127\times {10}^{-8})$ | 30.8900 | −57.7749 | −53.7242 | 14.8770 | 3.1113 |

BMOEE | $\alpha =2822.9776(3.3087\times {10}^{-5})$ $\beta =5.4444(0.1439)$ | 80.6700 | −157.3394 | −153.2887 | 1.5514 | 0.2327 |

UW | $\alpha =0.0552(0.0193)$ $\beta =6.1602(0.5868)$ | 79.9500 | −155.9080 | −151.8573 | 1.4890 | 0.2389 |

UG | $\alpha =628.3885(2.4072\times {10}^{-10})$ $\beta =0.0011(1.4212\times {10}^{-4})$ | 5.2500 | −6.4901 | −2.4393 | 18.5180 | 3.9712 |

UL | $\alpha =3.9381(0.4506)$ | 41.1400 | −80.2707 | −78.2453 | 12.7090 | 2.5936 |

UISDL | $\alpha =3.4259(0.3151)$ | 42.2000 | −82.3913 | −80.3660 | 12.3010 | 2.4925 |

Model | Parameter | $\mathit{\ell}$ | AIC | BIC | AD | CVM |
---|---|---|---|---|---|---|

BTCPE | $\alpha =7.1385(1.7764)$ $\lambda =7.1961(0.9033)$ | 58.7500 | −113.4953 | −109.1160 | 0.8770 | 0.1363 |

Beta | $\alpha =12.7943(2.2291)$ $\beta =4.8994(0.8270)$ | 57.5700 | −111.1489 | −106.7692 | 1.0520 | 0.1783 |

UBIII | $\alpha =5.4398(0.7948)$ $\beta =2.0613(0.1723)$ | 53.8000 | −103.5927 | −99.2134 | 1.3725 | 0.2209 |

BMOEE | $\alpha =22.1286(9.9041)$ $\beta =10.0043(1.2381)$ | 51.4600 | −98.9276 | −94.5483 | 1.4958 | 0.2100 |

UW | $\alpha =8.6445(1.6973)$ $\beta =2.2320(0.2036)$ | 53.9700 | −103.9316 | −99.5523 | 1.3830 | 0.2238 |

UG | $\alpha =0.2792(0.1059)$ $\beta =3.8482(0.6025)$ | 46.0300 | −88.0569 | −83.6776 | 2.4709 | 0.3691 |

UL | $\alpha =0.5200(0.0466)$ | 46.1100 | −90.2298 | −88.0402 | 4.2480 | 0.6736 |

UISDL | $\alpha =0.7403(0.0539)$ | 52.0400 | −102.0717 | −99.8820 | 2.3450 | 0.3194 |

I | II | III | |||||
---|---|---|---|---|---|---|---|

Parameter | n | AB | RMSE | AB | RMSE | AB | RMSE |

${\theta}_{0}$ | 50 | 0.1949 | 0.2235 | 0.3599 | 0.3753 | 0.2609 | 0.2969 |

100 | 0.1946 | 0.1961 | 0.3551 | 0.3726 | 0.2178 | 0.2579 | |

250 | 0.1919 | 0.1941 | 0.3465 | 0.3673 | 0.1525 | 0.1926 | |

350 | 0.1898 | 0.1928 | 0.3271 | 0.3544 | 0.1320 | 0.1700 | |

500 | 0.1838 | 0.1927 | 0.3109 | 0.3482 | 0.1101 | 0.1431 | |

600 | 0.1779 | 0.1886 | 0.3051 | 0.3434 | 0.0998 | 0.1318 | |

700 | 0.1761 | 0.1850 | 0.2908 | 0.3333 | 0.0908 | 0.1196 | |

${\theta}_{1}$ | 50 | 0.2826 | 0.3067 | 0.3485 | 0.3807 | 0.8194 | 0.8276 |

100 | 0.2605 | 0.2904 | 0.3181 | 0.3486 | 0.8142 | 0.8238 | |

250 | 0.2290 | 0.2651 | 0.3171 | 0.3363 | 0.8013 | 0.8134 | |

350 | 0.2176 | 0.2539 | 0.3138 | 0.3342 | 0.7872 | 0.8041 | |

500 | 0.2097 | 0.2454 | 0.3083 | 0.3305 | 0.7727 | 0.7945 | |

600 | 0.2079 | 0.2433 | 0.3020 | 0.3272 | 0.7188 | 0.7610 | |

700 | 0.2053 | 0.2389 | 0.2978 | 0.3253 | 0.6862 | 0.7447 | |

${\theta}_{2}$ | 50 | 1.5889 | 1.5959 | 1.7104 | 1.7153 | 0.5212 | 0.5338 |

100 | 1.5835 | 1.5913 | 1.7046 | 1.7102 | 0.5140 | 0.5291 | |

250 | 1.5818 | 1.5910 | 1.6938 | 1.7006 | 0.5073 | 0.5250 | |

350 | 1.5698 | 1.5815 | 1.6751 | 1.6845 | 0.4893 | 0.5130 | |

500 | 1.5566 | 1.5723 | 1.6432 | 1.6578 | 0.4753 | 0.5046 | |

600 | 1.4749 | 1.5132 | 1.5559 | 1.5917 | 0.4601 | 0.4999 | |

700 | 1.3803 | 1.4520 | 1.4593 | 1.5264 | 0.4535 | 0.4921 | |

$\alpha $ | 50 | 0.0792 | 0.0998 | 0.0842 | 0.1110 | 0.1091 | 0.1520 |

100 | 0.0577 | 0.0745 | 0.0570 | 0.0747 | 0.0872 | 0.1382 | |

250 | 0.0352 | 0.0463 | 0.0339 | 0.0437 | 0.0523 | 0.0859 | |

350 | 0.0295 | 0.0378 | 0.0287 | 0.0366 | 0.0427 | 0.0650 | |

500 | 0.0246 | 0.0316 | 0.0239 | 0.0317 | 0.0340 | 0.0467 | |

600 | 0.0227 | 0.0287 | 0.0217 | 0.0290 | 0.0317 | 0.0449 | |

700 | 0.0210 | 0.0267 | 0.0201 | 0.0259 | 0.0287 | 0.0375 |

$\mathit{u}$ | ${\widehat{\mathit{\theta}}}_{0}$ | ${\widehat{\mathit{\theta}}}_{1}$ | ${\widehat{\mathit{\theta}}}_{2}$ | ${\widehat{\mathit{\theta}}}_{3}$ | $\widehat{\mathit{\alpha}}$ | |
---|---|---|---|---|---|---|

0.10 | Estimates | −3.6699 | 0.0076 | 0.0905 | −1.004 | 308.7724 |

Standard error | 0.1681 | $1.1670\times {10}^{-3}$ | $7.5355\times {10}^{-3}$ | $4.3797\times {10}^{-2}$ | $9.3305\times {10}^{-5}$ | |

p-value | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | |

0.25 | Estimates | −3.2544 | 0.0071 | 0.0845 | −0.9326 | 325.4705 |

Standard error | 0.1545 | $1.0687\times {10}^{-3}$ | $6.9379\times {10}^{-3}$ | $4.0103\times {10}^{-2}$ | $4.6137\times {10}^{-5}$ | |

p-value | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | |

0.50 | Estimates | −2.8977 | 0.0067 | 0.0792 | −0.8732 | 340.4285 |

Standard error | 0.1436 | $9.9065\times {10}^{4}$ | $6.4570\times {10}^{-3}$ | $3.7166\times {10}^{-2}$ | $1.3990\times {10}^{-5}$ | |

p-value | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | |

0.75 | Estimates | −2.6424 | 0.0064 | 0.0766 | −0.8384 | 281.1611 |

Standard error | 0.1405 | $9.7128\times {10}^{-4}$ | $6.3363\times {10}^{-3}$ | $3.6303\times {10}^{-2}$ | $6.4012\times {10}^{-6}$ | |

p-value | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | |

0.90 | Estimates | −2.4030 | 0.0061 | 0.0731 | −0.7987 | 273.9968 |

Standard error | 0.1353 | $9.3470\times {10}^{-4}$ | $6.1047\times {10}^{-3}$ | $3.4900\times {10}^{-2}$ | $2.9792\times {10}^{-5}$ | |

p-value | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ | $<0.0001$ |

$\mathit{u}$ | $-2\mathit{\ell}$ | AIC | BIC |
---|---|---|---|

0.10 | −885.3517 | −875.3517 | −856.8663 |

0.25 | −887.4067 | −877.4067 | −858.9212 |

0.50 | −889.1990 | −879.1990 | −860.7136 |

0.75 | −889.8634 | −879.8634 | −861.3779 |

0.90 | −890.8307 | −880.8307 | −862.3453 |

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## Share and Cite

**MDPI and ACS Style**

Nasiru, S.; Abubakari, A.G.; Chesneau, C.
New Lifetime Distribution for Modeling Data on the Unit Interval: Properties, Applications and Quantile Regression. *Math. Comput. Appl.* **2022**, *27*, 105.
https://doi.org/10.3390/mca27060105

**AMA Style**

Nasiru S, Abubakari AG, Chesneau C.
New Lifetime Distribution for Modeling Data on the Unit Interval: Properties, Applications and Quantile Regression. *Mathematical and Computational Applications*. 2022; 27(6):105.
https://doi.org/10.3390/mca27060105

**Chicago/Turabian Style**

Nasiru, Suleman, Abdul Ghaniyyu Abubakari, and Christophe Chesneau.
2022. "New Lifetime Distribution for Modeling Data on the Unit Interval: Properties, Applications and Quantile Regression" *Mathematical and Computational Applications* 27, no. 6: 105.
https://doi.org/10.3390/mca27060105