# On Unit Exponential Pareto Distribution for Modeling the Recovery Rate of COVID-19

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## Abstract

**:**

## 1. Introduction

## 2. Unit Exponential Pareto Distribution

## 3. Statistical Properties

#### 3.1. Quantile Function

#### 3.2. Moments

#### 3.3. The Moment-Generating Function

#### 3.4. Different Types of Entropy

#### 3.4.1. Rényi Entropy

#### 3.4.2. Havrda and Charvat Entropy

#### 3.4.3. Tsallis Entropy

#### 3.4.4. Arimoto Entropy

#### 3.4.5. Mathai–Haubold Entropy

#### 3.5. The Order Statistics

#### 3.6. Stress–Strength

#### 3.7. Stochastic Ordering

- The ${S}_{O}$, ($X{\le}_{\left(sto\right)}Y$) when ${F}_{X}\left(x\right)\ge {F}_{Y}\left(x\right)$, for all $x$.
- The HF order, ($X{\le}_{\left(hro\right)}Y$) when ${h}_{X}\left(x\right)\ge {h}_{Y}\left(x\right)$, for all $x$.
- The MRL order, ($X{\le}_{\left(mrlo\right)}Y$) when ${m}_{X}\left(x\right)\le {m}_{Y}\left(x\right)$, for all $x$.
- The likelihood ratio order, ($X{\le}_{\left(lro\right)}Y$) when $\frac{{f}_{X}\left(x\right)}{{f}_{Y}\left(x\right)}\phantom{\rule{3.33333pt}{0ex}}$ is decreasing in x.

## 4. Maximum Likelihood Estimators

## 5. Maximum Product Spacings

## 6. Bayesian Estimation

- Start with initial values $\left({\alpha}^{\left(0\right)},{\beta}^{\left(0\right)},{\lambda}^{\left(0\right)}\right).$
- Let $j=1.$
- Use the M-H algorithm to generate ${\alpha}^{\left(j\right)},{\beta}^{\left(j\right)}$ and ${\lambda}^{\left(j\right)}\phantom{\rule{3.33333pt}{0ex}}$ from ${\pi}_{1}^{*}\left({\alpha}^{(j-1)}\mid {\beta}^{(j-1)},{\lambda}^{(j-1)},\underset{\xaf}{\mathrm{x}}\right)$$,\phantom{\rule{3.33333pt}{0ex}}{\pi}_{2}^{*}\left({\beta}^{(j-1)}\mid {\alpha}^{\left(j\right)},{\lambda}^{(j-1)},\underset{\xaf}{\mathrm{x}}\right)$ and ${\pi}_{3}^{*}\left({\lambda}^{(j-1)}\mid {\alpha}^{\left(j\right)},{\beta}^{\left(j\right)},\underset{\xaf}{\mathrm{x}}\right)$ with the normal distributions$$N\left({\alpha}^{(j-1)},var\left(\alpha \right)\right),\phantom{\rule{3.33333pt}{0ex}}N\left({\beta}^{(j-1)},var\left(\beta \right)\right)\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}N\left({\lambda}^{(j-1)},var\left(\lambda \right)\right),$$
- Generate a required ${\alpha}^{*}$ from $N\left({\alpha}^{(j-1)},var\left(\alpha \right)\right),\phantom{\rule{3.33333pt}{0ex}}{\beta}^{*}$ from $N\left({\beta}^{(j-1)},var\left(\alpha \right)\right)$ and ${\lambda}^{*}$ from $N\left({\lambda}^{(j-1)},var\left(\lambda \right)\right).$
- (i)
- Find the acceptance probabilities$$\left.\begin{array}{c}{\eta}_{\alpha}=min\left[1,\frac{{\pi}_{1}^{*}\left({\alpha}^{*}\mid {\beta}^{(j-1)},{\lambda}^{(j-1)},\underset{\xaf}{\mathrm{x}}\right)}{{\pi}_{1}^{*}\left({\alpha}^{(j-1)}\mid {\beta}^{(j-1)},{\lambda}^{(j-1)},\underset{\xaf}{\mathrm{x}}\right)}\right],\hfill \\ \\ {\eta}_{\beta}=min\left[1,\frac{{\pi}_{2}^{*}\left({\beta}^{*}\mid {\alpha}^{\left(j\right)},{\lambda}^{(j-1)},\underset{\xaf}{\mathrm{x}}\right)}{{\pi}_{2}^{*}\left({\beta}^{(j-1)}\mid {\alpha}^{\left(j\right)},{\lambda}^{(j-1)},\underset{\xaf}{\mathrm{x}}\right)}\right],\hfill \\ \\ {\eta}_{\lambda}=min\left[1,\frac{{\pi}_{3}^{*}\left({\lambda}^{*}\mid {\alpha}^{\left(j\right)},{\beta}^{\left(j\right)},\underset{\xaf}{\mathrm{x}}\right)}{{\pi}_{3}^{*}\left({\lambda}^{(j-1)}\mid {\alpha}^{\left(j\right)},{\beta}^{\left(j\right)},\underset{\xaf}{\mathrm{x}}\right)}\right].\hfill \end{array}\right\}$$
- (ii)
- From the uniform $(0,1)$ distribution, generate ${u}_{1}$, ${u}_{2}$ and ${u}_{3}$
- (iii)
- If ${u}_{1}<{\eta}_{\alpha}$, accept the proposal and set ${\alpha}^{\left(j\right)}={\alpha}^{*}$; else, set ${\alpha}^{\left(j\right)}={\alpha}^{(j-1)}$.
- (iv)
- If ${u}_{2}$$<{\eta}_{\beta}$, accept the proposal and set ${\beta}^{\left(j\right)}={\beta}^{*}$; else, set ${\beta}^{\left(j\right)}={\beta}^{(j-1)}$.
- (v)
- If ${u}_{3}<{\eta}_{\lambda}$, accept the proposal and set ${\lambda}^{\left(j\right)}={\lambda}^{*}$; else, set ${\lambda}^{\left(j\right)}={\lambda}^{(j-1)}.$

- Set $j=j+1.$
- Repeat steps $\left(3\right)\u2013\left(6\right)$ N times and obtain ${\alpha}^{\left(i\right)},{\beta}^{\left(i\right)}$ and ${\lambda}^{\left(i\right)},i=1,2,\dots N.$
- To compute the CRs of ${\theta}_{k}^{\left(i\right)}=\left({\theta}_{1},{\theta}_{2},{\theta}_{3}\right)=\left(\alpha ,\beta ,\lambda \right)$ as ${\theta}_{k}^{\left(1\right)}<{\theta}_{k}^{\left(2\right)}\dots <{\theta}_{k}^{\left(N\right)},k=1,2,3,$ then the $100(1-\gamma )\%$ CRIs of $\phantom{\rule{3.33333pt}{0ex}}{\theta}_{k}$ is

## 7. Applications

#### 7.1. Recovery Rate of COVID-19 in Turkey

#### 7.2. Milk Production Data

#### 7.3. The Failure Components Data

#### 7.4. Recovery Rate of COVID-19 in France

## 8. Results of Simulation

- In some instances, we notice that as the sample size grows larger, the AVSEs for all estimations decrease.
- This shows that various estimation techniques have good results for big sample sizes in terms of bias and AVSEs.
- $MP{S}_{p}$ Estimation methods has better measures than the ${M}_{L}E$ method.
- The Bayesian estimation method is the best estimation method to estimate the parameter of UEPD.
- The results of the simulation revealed empirical proof of the stability of our estimates.

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Estimated pdf, CDF, and P-P plots of UEP for the recovery rate of COVID-19 data from Turkey.

**Figure 6.**Trace plot of MCMC results for parameters of UEP for the recovery rate of COVID-19 data from Turkey.

**Figure 7.**Density plot of posterior MCMC results for parameters of UEP for the recovery rate of COVID-19 data from Turkey.

**Figure 12.**Density plot of posterior MCMC results for parameters of UEPD for the milk production data.

**Figure 16.**Trace plot of MCMC results for parameters of UEPD for the failures of 20 components data.

**Figure 17.**Density plot of posterior MCMC results for parameters of UEPD for the failures of 20 components data.

**Figure 22.**Density plot of posterior MCMC results for parameters of UEPD for the failure of 20 components data.

$\mathit{\beta}$ | $\mathit{\alpha}$ | $\mathit{E}\left(\mathit{X}\right)$ | $\mathit{E}\left({\mathit{X}}^{2}\right)$ | $\mathit{E}\left({\mathit{X}}^{3}\right)$ | $\mathit{E}\left({\mathit{X}}^{4}\right)$ | var | SK | KU | CV | ID |
---|---|---|---|---|---|---|---|---|---|---|

0.8 | 1.5 | 0.111 | 0.05 | 0.025 | 0.013 | 0.037 | 1.542 | 4.034 | 1.744 | 0.337 |

2.0 | 0.12 | 0.054 | 0.026 | 0.013 | 0.039 | 1.31 | 3.167 | 1.657 | 0.329 | |

2.5 | 0.1250 | 0.0560 | 0.0270 | 0.0130 | 0.0400 | 1.1750 | 2.7090 | 1.6100 | 0.3230 | |

3.0 | 0.1280 | 0.0570 | 0.0270 | 0.0130 | 0.0410 | 1.0890 | 2.4370 | 1.5810 | 0.3200 | |

4.0 | 0.1310 | 0.0590 | 0.0270 | 0.0130 | 0.0420 | 0.9920 | 2.1440 | 1.5510 | 0.3160 | |

1.2 | 1.5 | 0.0930 | 0.0470 | 0.0260 | 0.0160 | 0.0380 | 1.9760 | 5.5650 | 2.1080 | 0.4130 |

2.0 | 0.1000 | 0.0520 | 0.0290 | 0.0170 | 0.0420 | 1.7940 | 4.6600 | 2.0540 | 0.4200 | |

2.5 | 0.1030 | 0.0550 | 0.0300 | 0.0170 | 0.0440 | 1.6920 | 4.1790 | 2.0270 | 0.4250 | |

3.0 | 0.1060 | 0.0560 | 0.0310 | 0.0180 | 0.0450 | 1.6300 | 3.8930 | 2.0110 | 0.4270 | |

4.0 | 0.1080 | 0.0580 | 0.0320 | 0.0180 | 0.0470 | 1.5630 | 3.5860 | 1.9950 | 0.4310 | |

1.5 | 1.5 | 0.0820 | 0.0440 | 0.0260 | 0.0160 | 0.0370 | 2.2640 | 6.8140 | 2.3550 | 0.4540 |

2.0 | 0.0870 | 0.0480 | 0.0290 | 0.0180 | 0.0410 | 2.1160 | 5.9330 | 2.3280 | 0.4710 | |

2.5 | 0.0890 | 0.0510 | 0.0300 | 0.0190 | 0.0430 | 2.0370 | 5.4690 | 2.3180 | 0.4810 | |

3.0 | 0.0910 | 0.0530 | 0.0310 | 0.0190 | 0.0440 | 1.9900 | 5.1960 | 2.3130 | 0.4870 | |

4.0 | 0.1250 | 0.0450 | 0.0180 | 0.0008 | 0.0290 | 1.0900 | 2.9300 | 1.3630 | 0.2320 |

$\mathit{\beta}$ | $\mathit{\alpha}$ | $\mathit{\nu}=0.5$ | $\mathit{\nu}=0.8$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

RE | HCE | TE | AE | MHE | RE | HCE | TE | AE | MHE | ||

0.8 | 1.5 | −0.6470 | −1.8700 | −1.0500 | −0.7750 | 1.5870 | −2.2090 | −4.8400 | −3.1920 | −2.8790 | 3.6990 |

2.0 | −0.7050 | −1.9380 | −1.1120 | −0.8030 | 1.5510 | −2.2810 | −4.9160 | −3.2510 | −2.9240 | 3.6600 | |

2.5 | −0.7660 | −2.0010 | −1.1720 | −0.8290 | 1.5070 | −2.3520 | −4.9890 | −3.3080 | −2.9670 | 3.6130 | |

3.0 | −0.8240 | −2.0520 | −1.2260 | −0.8500 | 1.4630 | −2.4190 | −5.0540 | −3.3580 | −3.0060 | 3.5660 | |

4.0 | −0.9280 | −2.1290 | −1.3130 | −0.8820 | 1.3780 | −2.5320 | −5.1600 | −3.4420 | −3.0690 | 3.4810 | |

1.2 | 1.5 | −0.7500 | −1.9850 | −1.1570 | −0.8220 | 1.7540 | −2.7220 | −5.3210 | −3.5720 | −3.1650 | 4.1250 |

2.0 | −0.8200 | −2.0480 | −1.2220 | −0.8480 | 1.7430 | −2.8480 | −5.4200 | −3.6530 | −3.2240 | 4.1310 | |

2.5 | −0.8890 | −2.1020 | −1.2810 | −0.8710 | 1.7240 | −2.9530 | −5.4970 | −3.7170 | −3.2690 | 4.1170 | |

3.0 | −0.9540 | −2.1460 | −1.3330 | −0.8890 | 1.7030 | −3.0420 | −5.5570 | −3.7680 | −3.3060 | 4.0970 | |

4.0 | −1.0670 | −2.2070 | −1.4140 | −0.9140 | 1.6590 | −3.1820 | −5.6480 | −3.8450 | −3.3590 | 4.0540 | |

1.5 | 1.5 | −0.8200 | −2.0480 | −1.2220 | −0.8480 | 1.8190 | −3.0410 | −5.5570 | −3.7680 | −3.3050 | 4.3110 |

2.0 | −0.8990 | −2.1100 | −1.2900 | −0.8740 | 1.8160 | −3.2080 | −5.6640 | −3.8590 | −3.3690 | 4.3320 | |

2.5 | −0.9760 | −2.1590 | −1.3500 | −0.8940 | 1.8050 | −3.3400 | −5.7420 | −3.9260 | −3.4150 | 4.3310 | |

3.0 | −1.0460 | −2.1970 | −1.4000 | −0.9100 | 1.7920 | −3.4450 | −5.8000 | −3.9770 | −3.4500 | 4.3210 | |

4.0 | −1.1670 | −2.2500 | −1.4780 | −0.9320 | 1.7630 | −3.6060 | −5.8810 | −4.0500 | −3.4980 | 4.2950 |

$\mathit{\beta}$ | $\mathit{\alpha}$ | $\mathit{\nu}=1.2$ | $\mathit{\nu}=1.5$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

RE | HCE | TE | AE | MHE | RE | HCE | TE | AE | MHE | ||

0.8 | 1.5 | 2.9230 | 5.2090 | 3.6990 | 4.0460 | −3.1920 | 1.3700 | 2.2210 | 1.5870 | 1.9520 | −1.0500 |

2.0 | 2.8590 | 5.1470 | 3.6600 | 3.9970 | −3.2510 | 1.2970 | 2.1520 | 1.5510 | 1.8910 | −1.1120 | |

2.5 | 2.7850 | 5.0720 | 3.6130 | 3.9390 | −3.3080 | 1.2170 | 2.0720 | 1.5070 | 1.8210 | −1.1720 | |

3.0 | 2.7130 | 4.9970 | 3.5660 | 3.8810 | −3.3580 | 1.1420 | 1.9930 | 1.4630 | 1.7520 | −1.2260 | |

4.0 | 2.5870 | 4.8630 | 3.4810 | 3.7770 | −3.4420 | 1.0150 | 1.8480 | 1.3780 | 1.6240 | −1.3130 | |

1.2 | 1.5 | 3.7860 | 5.9180 | 4.1250 | 4.5960 | −3.5720 | 1.8210 | 2.5700 | 1.7540 | 2.2590 | −1.1570 |

2.0 | 3.7990 | 5.9270 | 4.1310 | 4.6040 | −3.6530 | 1.7830 | 2.5450 | 1.7430 | 2.2370 | −1.2220 | |

2.5 | 3.7650 | 5.9040 | 4.1170 | 4.5860 | −3.7170 | 1.7210 | 2.5030 | 1.7240 | 2.1990 | −1.2810 | |

3.0 | 3.7170 | 5.8700 | 4.0970 | 4.5590 | −3.7680 | 1.6560 | 2.4560 | 1.7030 | 2.1580 | −1.3330 | |

4.0 | 3.6160 | 5.7970 | 4.0540 | 4.5020 | −3.8450 | 1.5370 | 2.3650 | 1.6590 | 2.0780 | −1.4140 | |

1.5 | 1.5 | 4.3030 | 6.2440 | 4.3110 | 4.8490 | −3.7680 | 2.0880 | 2.7270 | 1.8190 | 2.3960 | −1.2220 |

2.0 | 4.3700 | 6.2810 | 4.3320 | 4.8790 | −3.8590 | 2.0740 | 2.7190 | 1.8160 | 2.3890 | −1.2900 | |

2.5 | 4.3670 | 6.2790 | 4.3310 | 4.8770 | −3.9260 | 2.0240 | 2.6920 | 1.8050 | 2.3650 | −1.3500 | |

3.0 | 4.3360 | 6.2620 | 4.3210 | 4.8640 | −3.9770 | 1.9650 | 2.6590 | 1.7920 | 2.3360 | −1.4000 | |

4.0 | 4.2520 | 6.2140 | 4.2950 | 4.8270 | −4.0500 | 1.8530 | 2.5910 | 1.7630 | 2.2760 | −1.4780 |

0.0074 | 0.0095 | 0.0113 | 0.015 | 0.018 | 0.0212 | 0.0229 |

0.0231 | 0.0328 | 0.0385 | 0.0439 | 0.0464 | 0.0483 | 0.0507 |

0.0515 | 0.0568 | 0.0605 | 0.0648 | 0.0737 | 0.0818 | 0.0955 |

0.1099 | 0.127 | 0.1388 | 0.1476 |

**Table 5.**${M}_{L}E$ and $SE$ with measures of goodness of fit for the recovery rate of COVID-19 data from Turkey.

Models | Estimates and SE | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\lambda}$ | KS | PVKS | CVM | AD |
---|---|---|---|---|---|---|---|---|

UEPD | estimates | 1.3419 | 0.1147 | 2.0628 | 0.1005 | 0.9407 | 0.0298 | 0.2290 |

SE | 0.2088 | 0.0540 | 1.0694 | |||||

UW | estimates | 0.0054 | 4.1597 | 0.1362 | 0.6923 | 0.0652 | 0.3860 | |

SE | 0.0031 | 0.4182 | ||||||

Kumaraswamy | estimates | 1.4164 | 50.9406 | 0.1022 | 0.9329 | 0.0310 | 0.2313 | |

SE | 0.2303 | 31.3225 | ||||||

MOK | estimates | 0.1377 | 1.8755 | 47.5473 | 0.1129 | 0.8722 | 0.0359 | 0.2371 |

SE | 0.1588 | 0.3459 | 52.9895 | |||||

UG | estimates | 0.0166 | 1.1455 | 0.1602 | 0.4929 | 0.1242 | 0.7260 | |

SE | 0.0125 | 0.1801 | ||||||

MOETL | estimates | 0.0062 | 2.0660 | 0.1058 | 0.9147 | 0.0559 | 0.3477 | |

SE | 0.0046 | 0.2976 | ||||||

UGLBXII | estimates | 1.8385 | 2.7239 | 3.6048 | 0.0989 | 0.9471 | 0.0375 | 0.2505 |

SE | 3.0788 | 1.1344 | 1.6640 | |||||

UGG | estimates | 1.2880 | 29.9588 | 0.8056 | 0.2189 | 0.1563 | 0.0393 | 0.2545 |

SE | 0.2831 | 14.0703 | 0.3997 | |||||

EPD | estimates | 1.4316 | 0.1171 | 2.5012 | 0.1027 | 0.9304 | 0.0303 | 0.2313 |

SE | 0.2244 | 0.9087 | 27.8069 |

**Table 6.**${M}_{L}E$, $MP{S}_{p}$, Bayesian estimation methods for the recovery rate of COVID-19 data from Turkey.

Methods | Estimates | SE | Lower | Upper | |
---|---|---|---|---|---|

${M}_{L}E$ | $\alpha $ | 1.2082 | 0.1996 | 0.8171 | 1.5994 |

$\beta $ | 0.1278 | 0.0545 | 0.0210 | 0.2345 | |

$\lambda $ | 2.1415 | 1.0056 | 0.1704 | 4.1126 | |

$MP{S}_{p}$ | $\alpha $ | 1.3419 | 0.2088 | 0.9327 | 1.7512 |

$\beta $ | 0.1147 | 0.0540 | 0.0088 | 0.2205 | |

$\lambda $ | 2.0628 | 1.0694 | 0.0108 | 4.1148 | |

Bayesian | $\alpha $ | 1.3257 | 0.2001 | 0.9363 | 1.7126 |

$\beta $ | 0.1307 | 0.0424 | 0.0638 | 0.2272 | |

$\lambda $ | 2.1702 | 0.6614 | 0.9412 | 3.5015 |

0.0168 | 0.1546 | 0.3188 | 0.3751 | 0.4332 | 0.4612 | 0.515 | 0.5553 | 0.6012 | 0.6768 | 0.7471 |

0.0609 | 0.216 | 0.3259 | 0.3821 | 0.4365 | 0.4675 | 0.5232 | 0.5627 | 0.6058 | 0.6789 | 0.7629 |

0.065 | 0.2303 | 0.3323 | 0.3891 | 0.4371 | 0.4694 | 0.5285 | 0.5629 | 0.6114 | 0.6844 | 0.7687 |

0.0671 | 0.2356 | 0.3383 | 0.3906 | 0.4438 | 0.4741 | 0.5349 | 0.5707 | 0.6174 | 0.686 | 0.7804 |

0.0776 | 0.2361 | 0.3406 | 0.3945 | 0.447 | 0.4752 | 0.535 | 0.5744 | 0.6196 | 0.6891 | 0.8147 |

0.0854 | 0.2605 | 0.3413 | 0.4049 | 0.4517 | 0.48 | 0.5394 | 0.577 | 0.622 | 0.6907 | 0.8492 |

0.1131 | 0.2681 | 0.348 | 0.4111 | 0.453 | 0.4823 | 0.5447 | 0.5853 | 0.6465 | 0.6927 | 0.8781 |

0.1167 | 0.2747 | 0.3598 | 0.4143 | 0.4553 | 0.499 | 0.5481 | 0.5878 | 0.6488 | 0.7131 | |

0.1479 | 0.3134 | 0.3627 | 0.4151 | 0.4564 | 0.5113 | 0.5483 | 0.5912 | 0.6707 | 0.7261 | |

0.1525 | 0.3175 | 0.3635 | 0.426 | 0.4576 | 0.514 | 0.5529 | 0.5941 | 0.675 | 0.729 |

Models | Estimates and SE | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\lambda}$ | $\mathit{\theta}$ | KS | PVKS | CVM | AD |
---|---|---|---|---|---|---|---|---|---|

UEPD | estimates | 1.2087 | 1.1238 | 0.8427 | 0.0787 | 0.5213 | 0.1088 | 0.6520 | |

SE | 0.0867 | 1.0180 | 0.4151 | ||||||

UW | estimates | 0.9846 | 1.5619 | 0.1206 | 0.0890 | 0.3963 | 2.4244 | ||

SE | 0.1015 | 0.1064 | |||||||

Kumaraswamy | estimates | 2.1949 | 3.4366 | 0.0796 | 0.5162 | 0.1561 | 1.0090 | ||

SE | 0.2224 | 0.5821 | |||||||

Beta | estimates | 2.4125 | 2.8297 | 0.0910 | 0.3384 | 0.2083 | 1.3263 | ||

SE | 0.3145 | 0.3744 | |||||||

KK | estimates | 0.3781 | 5.1808 | 3.8954 | 1.4834 | 0.0790 | 0.5165 | 0.1120 | 0.7195 |

SE | 0.2337 | 3.9516 | 9.2215 | 2.8696 | |||||

UG | estimates | 2.1191 | 0.3878 | 0.1835 | 0.0015 | 0.5206 | 3.0947 | ||

SE | 0.8684 | 0.1145 | |||||||

MOETL | estimates | 1.0535 | 2.0230 | 0.0968 | 0.2682 | 0.2246 | 1.4262 | ||

SE | 0.3475 | 0.4118 | |||||||

UGG | estimates | 4.1576 | 5.2380 | 0.4268 | 0.1067 | 0.1747 | 0.2195 | 1.4377 | |

SE | 1.0592 | 1.6032 | 0.1393 | ||||||

EPD | estimates | 2.6012 | 0.6366 | 1.6623 | 0.0832 | 0.4487 | 0.2318 | 1.5235 | |

SE | 0.2098 | 6.2038 | 42.1400 |

Methods | Estimates | SE | Lower | Upper | |
---|---|---|---|---|---|

MLE | $\alpha $ | 1.2087 | 0.0867 | 1.0362 | 1.3812 |

$\beta $ | 1.1238 | 1.0180 | 1.0180 | 1.2296 | |

$\lambda $ | 0.8427 | 0.4151 | 0.0290 | 1.6564 | |

MPS | $\alpha $ | 1.1531 | 0.5736 | 0.0289 | 2.2772 |

$\beta $ | 1.1827 | 0.7048 | 0.0269 | 2.5642 | |

$\lambda $ | 0.8923 | 0.3892 | 0.1294 | 1.6552 | |

Bayesian | $\alpha $ | 1.2113 | 0.0791 | 0.9851 | 1.4410 |

$\beta $ | 1.2587 | 0.3279 | 0.6771 | 1.8810 | |

$\lambda $ | 0.9900 | 0.3545 | 0.3666 | 1.6193 |

0.0009 | 0.004 | 0.0142 | 0.0221 | 0.0261 | 0.0418 | 0.0473 |

0.0834 | 0.1091 | 0.1252 | 0.1404 | 0.1498 | 0.175 | 0.2031 |

0.2099 | 0.2168 | 0.2918 | 0.3465 | 0.4035 | 0.6143 |

**Table 11.**${M}_{L}E$ and SE with different measures of goodness of fit for the failure of 20 components data.

Models | Estimates and SE | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\lambda}$ | CVMS | ADS | KSD | PVKS |
---|---|---|---|---|---|---|---|---|

UEPD | estimates | 0.7289 | 0.3755 | 1.5363 | 0.0265 | 0.1649 | 0.0928 | 0.9887 |

SE | 0.1272 | 0.4975 | 1.7629 | |||||

UW | estimates | 0.1598 | 1.7271 | 0.0567 | 0.3302 | 0.1319 | 0.8334 | |

SE | 0.0710 | 0.2875 | ||||||

Kumaraswamy | estimates | 0.7641 | 3.4347 | 0.0290 | 0.1692 | 0.1026 | 0.9700 | |

SE | 0.1752 | 1.3117 | ||||||

UG | estimates | 0.7718 | 0.5964 | 0.0912 | 0.5520 | 0.1491 | 0.7110 | |

SE | 0.2786 | 0.1202 | ||||||

TL | estimates | 0.5112 | 0.0301 | 0.1754 | 0.1848 | 0.4482 | ||

SE | 0.1143 | |||||||

MOTL | estimates | 0.3520 | 0.8346 | 0.0492 | 0.2785 | 0.1096 | 0.9485 | |

SE | 0.2539 | 0.2739 | ||||||

UGG | estimates | 1.3430 | 3.8518 | 0.4346 | 0.0401 | 0.2268 | 0.1571 | 0.6507 |

SE | 1.0710 | 2.3086 | 0.4485 | |||||

EPD | estimates | 0.8999 | 0.2647 | 1.6266 | 0.0428 | 0.2423 | 0.1193 | 0.9067 |

SE | 0.1654 | 0.7591 | 0.4333 |

Methods | Estimates | SE | Lower | Upper | |
---|---|---|---|---|---|

${M}_{L}E$ | $\alpha $ | 0.7289 | 0.1272 | 0.4796 | 0.9782 |

$\beta $ | 0.3755 | 0.4975 | 0.2696 | 0.4813 | |

$\lambda $ | 1.5363 | 1.7629 | 0.3198 | 3.3923 | |

$MP{S}_{p}$ | $\alpha $ | 0.6238 | 0.1530 | 0.3239 | 0.9237 |

$\beta $ | 0.4484 | 0.2066 | 0.0435 | 0.8532 | |

$\lambda $ | 1.5716 | 0.7804 | 0.0420 | 3.1011 | |

Bayesian | $\alpha $ | 0.7473 | 0.1613 | 0.4432 | 1.0613 |

$\beta $ | 0.4126 | 0.1197 | 0.1977 | 0.6520 | |

$\lambda $ | 1.6112 | 0.4591 | 0.8273 | 2.5929 |

0.195 | 0.2338 | 0.2368 | 0.1073 | 0.1592 | 0.2784 | 0.0689 | 0.1791 |

0.1121 | 0.1865 | 0.2631 | 0.0716 | 0.1411 | 0.1477 | 0.1874 | 0.0853 |

0.0922 | 0.1711 | 0.1962 | 0.2146 | 0.1041 | 0.1524 | 0.1811 | 0.0643 |

0.2698 | 0.1245 | 0.176 | 0.2363 | 0.0712 | 0.1361 | 0.1386 | |

0.3316 | 0.077 | 0.1367 | 0.1549 | 0.2178 | 0.0951 | 0.1346 |

**Table 14.**${M}_{L}E$ and SE with different measures of goodness of fit for the recovery rate of COVID-19 data in France.

Models | Estimates and SE | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\lambda}$ | CVMS | ADS | KSD | PVKS |
---|---|---|---|---|---|---|---|---|

UEPD | estimates | 2.1715 | 0.3544 | 2.6397 | 0.0365 | 0.2965 | 0.0722 | 0.9807 |

SE | 0.2651 | 0.1115 | 0.5027 | |||||

UW | estimates | 0.0280 | 4.8690 | 0.0844 | 0.5464 | 0.1203 | 0.5994 | |

SE | 0.0145 | 0.5965 | ||||||

MOK | estimates | 0.1490 | 3.8135 | 191.1284 | 0.1700 | 1.0407 | 0.1542 | 0.2952 |

SE | 0.1142 | 0.9003 | 88.8899 | |||||

UG | estimates | 0.0073 | 2.3191 | 0.1700 | 1.0407 | 0.1542 | 0.2952 | |

SE | 0.0039 | 0.2286 | ||||||

MOETL | estimates | 0.0042 | 4.2695 | 0.0639 | 0.4531 | 0.0769 | 0.9653 | |

SE | 0.0021 | 0.4101 | ||||||

UGLBXII | estimates | 2.0928 | 3.0810 | 2.2037 | 0.0429 | 0.3210 | 0.0788 | 0.9573 |

SE | 1.2820 | 1.0797 | 0.8284 | |||||

UGG | estimates | 2.5233 | 157.2439 | 1.3847 | 0.0631 | 0.4369 | 0.1534 | 0.3010 |

SE | 0.4258 | 77.1515 | 0.7092 |

**Table 15.**${M}_{L}E$, $MP{S}_{p}$, Bayesian estimation methods for pandemic coronavirus in France data.

Method | Estimates | SE | Lower | Upper | |
---|---|---|---|---|---|

MLE | $\alpha $ | 2.1715 | 0.2651 | 1.9222 | 2.4209 |

$\beta $ | 0.3544 | 0.1115 | 0.2486 | 0.4603 | |

$\lambda $ | 2.6397 | 0.5027 | 0.7837 | 4.4957 | |

MPS | $\alpha $ | 2.0015 | 0.9577 | 0.1244 | 3.8786 |

$\beta $ | 0.3849 | 0.1267 | 0.1366 | 0.6331 | |

$\lambda $ | 2.8257 | 0.5472 | 1.7531 | 3.8982 | |

Bayesian | $\alpha $ | 2.1585 | 0.2483 | 1.6877 | 2.6627 |

$\beta $ | 0.2981 | 0.1172 | 0.0673 | 0.5083 | |

$\lambda $ | 1.9924 | 1.3850 | 0.0208 | 4.7526 |

**Table 16.**${M}_{L}E$, $MP{S}_{p}$ and Bayesian estimation methods for parameters of UEPD for $\beta =0.5$.

$\mathit{\beta}=0.5$ | ${\mathit{M}}_{\mathit{L}}\mathit{E}$ | ${\mathbf{MPS}}_{\mathit{p}}$ | Bayesian | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\beta}$ | $\mathit{\lambda}$ | n | Bias | AVSE | L.CI | Bias | AVSE | L.CI | Bias | AVSE | L.CI | |

0.5 | 0.6 | 30 | $\alpha $ | 0.0776 | 0.0133 | 0.3347 | 0.0245 | 0.0064 | 0.2993 | 0.0285 | 0.0059 | 0.2787 |

$\beta $ | 0.0338 | 0.0053 | 0.2518 | 0.0243 | 0.0031 | 0.1958 | 0.0096 | 0.0049 | 0.2354 | |||

$\lambda $ | 0.0595 | 0.0128 | 0.3768 | −0.0429 | 0.0094 | 0.3399 | 0.0170 | 0.0099 | 0.3707 | |||

80 | $\alpha $ | 0.0583 | 0.0054 | 0.1738 | 0.0337 | 0.0029 | 0.1651 | 0.0235 | 0.0023 | 0.1611 | ||

$\beta $ | 0.0425 | 0.0036 | 0.1640 | 0.0297 | 0.0018 | 0.1185 | 0.0132 | 0.0035 | 0.1428 | |||

$\lambda $ | 0.0542 | 0.0061 | 0.2191 | −0.0510 | 0.0054 | 0.2056 | 0.0153 | 0.0049 | 0.2672 | |||

150 | $\alpha $ | 0.0559 | 0.0042 | 0.1284 | 0.0409 | 0.0027 | 0.1245 | 0.0187 | 0.0013 | 0.1189 | ||

$\beta $ | 0.0427 | 0.0031 | 0.1376 | 0.0315 | 0.0016 | 0.0940 | 0.0041 | 0.0023 | 0.1858 | |||

$\lambda $ | 0.0529 | 0.0046 | 0.1672 | −0.0528 | 0.0044 | 0.1596 | 0.0135 | 0.0021 | 0.1634 | |||

2.4 | 30 | $\alpha $ | 0.0763 | 0.0122 | 0.3123 | 0.0363 | 0.0057 | 0.2799 | 0.0392 | 0.0063 | 0.2646 | |

$\beta $ | 0.0222 | 0.0235 | 0.5955 | 0.0833 | 0.0392 | 0.7041 | 0.0049 | 0.0077 | 0.3312 | |||

$\lambda $ | 0.0676 | 0.0260 | 0.5744 | −0.0449 | 0.0099 | 0.3483 | 0.0058 | 0.0159 | 0.4865 | |||

80 | $\alpha $ | 0.0599 | 0.0057 | 0.1799 | 0.0354 | 0.0031 | 0.1707 | 0.0369 | 0.0033 | 0.1749 | ||

$\beta $ | 0.0257 | 0.0101 | 0.3805 | 0.0352 | 0.0135 | 0.4350 | 0.0019 | 0.0048 | 0.2683 | |||

$\lambda $ | 0.0863 | 0.0181 | 0.4044 | −0.0176 | 0.0026 | 0.1866 | 0.0063 | 0.0069 | 0.3350 | |||

150 | $\alpha $ | 0.0550 | 0.0041 | 0.1293 | 0.0340 | 0.0026 | 0.1254 | 0.0293 | 0.0018 | 0.1179 | ||

$\beta $ | 0.0298 | 0.0057 | 0.2722 | 0.0209 | 0.0064 | 0.3033 | 0.0035 | 0.0022 | 0.1819 | |||

$\lambda $ | 0.0897 | 0.0143 | 0.3112 | −0.0103 | 0.0011 | 0.1256 | 0.0018 | 0.0027 | 0.2055 | |||

2 | 0.6 | 30 | $\alpha $ | 0.3055 | 0.1881 | 1.2074 | 0.1922 | 0.0832 | 1.0718 | 0.0110 | 0.0154 | 0.4680 |

$\beta $ | 0.0247 | 0.0033 | 0.2047 | 0.0924 | 0.0025 | 0.1704 | 0.0129 | 0.0032 | 0.2004 | |||

$\lambda $ | 0.0216 | 0.0141 | 0.4587 | −0.0391 | 0.0004 | 0.0724 | 0.0044 | 0.0121 | 0.4076 | |||

80 | $\alpha $ | 0.2373 | 0.0863 | 0.6787 | 0.1848 | 0.0459 | 0.6412 | 0.0135 | 0.0061 | 0.2985 | ||

$\beta $ | 0.0269 | 0.0017 | 0.1223 | 0.0271 | 0.0014 | 0.1011 | 0.0092 | 0.0016 | 0.1169 | |||

$\lambda $ | 0.0181 | 0.0051 | 0.2716 | −0.0108 | 0.0002 | 0.0428 | 0.0082 | 0.0046 | 0.2531 | |||

150 | $\alpha $ | 0.2220 | 0.0667 | 0.5178 | 0.1613 | 0.0422 | 0.4987 | 0.0107 | 0.0028 | 0.2038 | ||

$\beta $ | 0.0263 | 0.0011 | 0.0780 | 0.0272 | 0.0011 | 0.0754 | 0.0077 | 0.0008 | 0.1099 | |||

$\lambda $ | 0.0174 | 0.0029 | 0.1982 | −0.0108 | 0.0002 | 0.0317 | 0.0062 | 0.0025 | 0.1904 | |||

2.4 | 30 | $\alpha $ | 0.3057 | 0.1908 | 1.2240 | 0.0922 | 0.0843 | 1.0799 | 0.0098 | 0.0148 | 0.4700 | |

$\beta $ | 0.0074 | 0.0021 | 0.1795 | 0.0175 | 0.0028 | 0.1954 | 0.0163 | 0.0020 | 0.1788 | |||

$\lambda $ | 0.0091 | 0.0397 | 0.7803 | −0.0018 | 0.0000 | 0.0206 | 0.0052 | 0.0143 | 0.4714 | |||

80 | $\alpha $ | 0.2302 | 0.0835 | 0.6854 | 0.0813 | 0.0449 | 0.6496 | 0.0095 | 0.0075 | 0.3238 | ||

$\beta $ | 0.0007 | 0.0007 | 0.1072 | 0.0093 | 0.0010 | 0.1189 | 0.0119 | 0.0005 | 0.0912 | |||

$\lambda $ | 0.0142 | 0.0114 | 0.4159 | −0.0009 | 0.0000 | 0.0125 | 0.0029 | 0.0071 | 0.3150 | |||

150 | $\alpha $ | 0.2164 | 0.0630 | 0.4985 | 0.0816 | 0.0395 | 0.4831 | 0.0091 | 0.0026 | 0.1930 | ||

$\beta $ | 0.0001 | 0.0004 | 0.0760 | 0.0052 | 0.0005 | 0.0823 | 0.0085 | 0.0004 | 0.0759 | |||

$\lambda $ | 0.0174 | 0.0066 | 0.3124 | −0.0004 | 0.0000 | 0.0086 | 0.0019 | 0.0028 | 0.2043 |

**Table 17.**${M}_{L}E$, $MP{S}_{p}$ and Bayesian estimation methods for parameters of UEPD when $\beta =2$.

$\mathit{\beta}=2$ | ${\mathit{M}}_{\mathit{L}}\mathit{E}$ | ${\mathbf{MPS}}_{\mathit{p}}$ | Bayesian | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\beta}$ | $\mathit{\lambda}$ | n | Bias | AVSE | L.CI | Bias | AVSE | L.CI | Bias | AVSE | L.CI | |

0.5 | 0.6 | 30 | $\alpha $ | 0.0776 | 0.0133 | 0.3347 | 0.0409 | 0.0064 | 0.2993 | 0.0285 | 0.0059 | 0.2787 |

$\beta $ | 0.0338 | 0.0053 | 0.2518 | 0.0315 | 0.0031 | 0.1958 | 0.0096 | 0.0049 | 0.2354 | |||

$\lambda $ | 0.0595 | 0.0128 | 0.3768 | −0.0528 | 0.0094 | 0.3399 | 0.0170 | 0.0099 | 0.3707 | |||

80 | $\alpha $ | 0.0583 | 0.0054 | 0.1738 | 0.0337 | 0.0029 | 0.1651 | 0.0235 | 0.0023 | 0.1611 | ||

$\beta $ | 0.0425 | 0.0036 | 0.1640 | 0.0297 | 0.0018 | 0.1185 | 0.0091 | 0.0035 | 0.1528 | |||

$\lambda $ | 0.0542 | 0.0061 | 0.2191 | −0.0510 | 0.0054 | 0.2056 | 0.0153 | 0.0049 | 0.2672 | |||

150 | $\alpha $ | 0.0559 | 0.0042 | 0.1284 | 0.0245 | 0.0027 | 0.1245 | 0.0187 | 0.0013 | 0.1189 | ||

$\beta $ | 0.0427 | 0.0031 | 0.1376 | 0.0243 | 0.0016 | 0.0940 | 0.0041 | 0.0023 | 0.1486 | |||

$\lambda $ | 0.0529 | 0.0046 | 0.1672 | −0.0429 | 0.0044 | 0.1596 | 0.0135 | 0.0021 | 0.1634 | |||

2.4 | 30 | $\alpha $ | 0.0763 | 0.0122 | 0.3123 | 0.0424 | 0.0057 | 0.2799 | 0.0392 | 0.0063 | 0.2646 | |

$\beta $ | 0.0222 | 0.0235 | 0.5955 | 0.0833 | 0.0392 | 0.7041 | 0.0049 | 0.0077 | 0.3312 | |||

$\lambda $ | 0.0676 | 0.0260 | 0.5744 | −0.0449 | 0.0099 | 0.3483 | 0.0066 | 0.0159 | 0.4865 | |||

80 | $\alpha $ | 0.0599 | 0.0057 | 0.1799 | 0.0414 | 0.0031 | 0.1707 | 0.0369 | 0.0033 | 0.1749 | ||

$\beta $ | 0.0257 | 0.0101 | 0.3805 | 0.0352 | 0.0135 | 0.4350 | 0.0042 | 0.0048 | 0.2683 | |||

$\lambda $ | 0.0863 | 0.0181 | 0.4044 | −0.0176 | 0.0026 | 0.1866 | 0.0063 | 0.0069 | 0.3350 | |||

150 | $\alpha $ | 0.0550 | 0.0041 | 0.1293 | 0.0401 | 0.0026 | 0.1254 | 0.0293 | 0.0018 | 0.1179 | ||

$\beta $ | 0.0298 | 0.0057 | 0.2722 | 0.0209 | 0.0064 | 0.3033 | 0.0035 | 0.0022 | 0.1819 | |||

$\lambda $ | 0.0897 | 0.0143 | 0.3112 | −0.0103 | 0.0011 | 0.1256 | 0.0018 | 0.0027 | 0.2055 | |||

2 | 0.6 | 30 | $\alpha $ | 0.0751 | 0.0123 | 0.3204 | 0.0400 | 0.0058 | 0.2855 | 0.0284 | 0.0055 | 0.2557 |

$\beta $ | 0.1086 | 0.0257 | 0.4621 | 0.0106 | 0.0003 | 0.0665 | 0.0006 | 0.0093 | 0.3750 | |||

$\lambda $ | 0.0569 | 0.0158 | 0.4393 | −0.0707 | 0.0150 | 0.4395 | 0.0167 | 0.0087 | 0.3506 | |||

80 | $\alpha $ | 0.0616 | 0.0060 | 0.1822 | 0.0370 | 0.0033 | 0.1732 | 0.0273 | 0.0026 | 0.1651 | ||

$\beta $ | 0.1259 | 0.0241 | 0.3559 | 0.0099 | 0.0002 | 0.0413 | 0.0012 | 0.0061 | 0.2933 | |||

$\lambda $ | 0.0610 | 0.0083 | 0.2662 | −0.0667 | 0.0094 | 0.2761 | 0.0188 | 0.0051 | 0.2646 | |||

150 | $\alpha $ | 0.0549 | 0.0041 | 0.1271 | 0.0224 | 0.0026 | 0.1233 | 0.0196 | 0.0012 | 0.1125 | ||

$\beta $ | 0.1335 | 0.0244 | 0.3190 | 0.0070 | 0.0002 | 0.0291 | 0.0002 | 0.0026 | 0.2020 | |||

$\lambda $ | 0.0600 | 0.0058 | 0.1855 | −0.0495 | 0.0075 | 0.1963 | 0.0158 | 0.0022 | 0.1754 | |||

2.4 | 30 | $\alpha $ | 0.0792 | 0.0129 | 0.3181 | 0.0262 | 0.0060 | 0.2848 | 0.0374 | 0.0062 | 0.2586 | |

$\beta $ | 0.0378 | 0.1281 | 1.3957 | 0.0375 | 0.0491 | 0.8566 | 0.0048 | 0.0102 | 0.3813 | |||

$\lambda $ | 0.0566 | 0.0816 | 1.0984 | −0.1018 | 0.1248 | 1.3268 | 0.0020 | 0.0141 | 0.4586 | |||

80 | $\alpha $ | 0.0579 | 0.0053 | 0.1754 | 0.0333 | 0.0029 | 0.1665 | 0.0372 | 0.0034 | 0.1757 | ||

$\beta $ | 0.0213 | 0.0513 | 0.8842 | 0.0247 | 0.0187 | 0.5276 | 0.0028 | 0.0058 | 0.2932 | |||

$\lambda $ | 0.0482 | 0.0335 | 0.6920 | −0.0547 | 0.0480 | 0.8318 | 0.0001 | 0.0067 | 0.3178 | |||

150 | $\alpha $ | 0.0551 | 0.0041 | 0.1291 | 0.0402 | 0.0026 | 0.1255 | 0.0290 | 0.0018 | 0.1203 | ||

$\beta $ | 0.0411 | 0.0297 | 0.6561 | 0.0202 | 0.0099 | 0.3819 | 0.0030 | 0.0025 | 0.1934 | |||

$\lambda $ | 0.0389 | 0.0169 | 0.4867 | −0.0394 | 0.0243 | 0.5916 | 0.0001 | 0.0029 | 0.2091 |

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## Share and Cite

**MDPI and ACS Style**

Haj Ahmad, H.; Almetwally, E.M.; Elgarhy, M.; Ramadan, D.A.
On Unit Exponential Pareto Distribution for Modeling the Recovery Rate of COVID-19. *Processes* **2023**, *11*, 232.
https://doi.org/10.3390/pr11010232

**AMA Style**

Haj Ahmad H, Almetwally EM, Elgarhy M, Ramadan DA.
On Unit Exponential Pareto Distribution for Modeling the Recovery Rate of COVID-19. *Processes*. 2023; 11(1):232.
https://doi.org/10.3390/pr11010232

**Chicago/Turabian Style**

Haj Ahmad, Hanan, Ehab M. Almetwally, Mohammed Elgarhy, and Dina A. Ramadan.
2023. "On Unit Exponential Pareto Distribution for Modeling the Recovery Rate of COVID-19" *Processes* 11, no. 1: 232.
https://doi.org/10.3390/pr11010232