# On the Analysis of New COVID-19 Cases in Pakistan Using an Exponentiated Version of the M Family of Distributions

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## Abstract

**:**

## 1. Introduction

## 2. The EM Family

#### 2.1. Definition

- if $\gamma \le 1$, we have $F(x;{\xi}_{1},{\xi}_{2})\le F(x;{\xi}_{1},{\xi}_{2},\gamma )$,
- if $\gamma >1$, we have $F(x;{\xi}_{1},{\xi}_{2},\gamma )\le F(x;{\xi}_{1},{\xi}_{2})$,

#### 2.2. Reliability Functions

#### 2.3. Properties

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Corollary**

**1.**

## 3. On a Special EM Distribution

#### 3.1. Definition and Shapes’ Analysis

- ${F}_{1}(x;\alpha )={e}^{-\alpha /x}$, $x>0$, corresponding to the cdf of the inverse exponential distribution with parameter $\alpha $ (see [20]),
- ${F}_{2}(x;\beta )=1-{e}^{-\beta x}$, $x>0$, corresponding to the cdf of the standard exponential distribution with parameter $\beta $.

#### 3.2. On Different Measures

- the mean of X defined by ${\mu}_{1}^{\prime}$, remaining the central parameter of the distribution,
- the variance of X given as $\mathrm{Var}={\mu}_{2}^{\prime}-{({\mu}_{1}^{\prime})}^{2}$, providing a dispersion parameter,
- the standard deviation of X defined as $\sigma ={\mathrm{Var}}^{1/2}$, corresponding to a dispersion parameter with the same unit as the mean,
- the skewness of X given by $\mathrm{SK}=[{\mu}_{3}^{\prime}-3{\mu}_{1}^{\prime}{\mu}_{2}^{\prime}+2{({\mu}_{1}^{\prime})}^{3}]/{\sigma}^{3}$, measuring the lack of symmetry of tails of the EMIEE distribution (about the ${\mu}_{1}^{\prime}$),
- the kurtosis of X specified by $\mathrm{KU}=[{\mu}_{4}^{\prime}-4{\mu}_{1}^{\prime}{\mu}_{3}^{\prime}+6{({\mu}_{1}^{\prime})}^{2}{\mu}_{2}^{\prime}-3{({\mu}_{1}^{\prime})}^{4}]/{\sigma}^{4}$, measuring how heavily the tails of the EMIEE distribution differ from those of a normal distribution,
- the coefficient of variation of X defined as $CV=\sigma /{\mu}_{1}^{\prime}$, providing a dispersion parameter that can serve as a benchmark for comparison.

## 4. Parameter Estimation

## 5. Application to a COVID-19 Dataset

- (i)
- Provide a precise estimation for some measures of interest related to COVID-19 cases in Pakistan (mean of cases, probability to have a certain number of cases, and so on),
- (ii)
- Compare the repartitions of the number of COVID-19 cases in Pakistan with those in other countries,
- (iii)
- Propose an efficient strategy for fitting data on COVID-19 cases in other countries,
- (iv)
- In a more challenging way, model the distribution of the number of cases for any pandemic with similar features and under a similar environment (with comparable populations, comparable climate, sanitary system, etc.).

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Plots for the hazard rate function (hrf) of the EMIEE distribution, for several values of the parameters.

**Table 1.**Numerical values of some moments, variance, skewness (SK), kurtosis (KU), and the coefficient of variation (CV) of the EMIEE distribution for some selected values of $\gamma $ and at $\alpha =0.5$ and $\beta =0.5$.

Measure | $\mathit{\gamma}=1.5$ | $\mathit{\gamma}=2.0$ | $\mathit{\gamma}=2.5$ | $\mathit{\gamma}=3.0$ | $\mathit{\gamma}=3.5$ | $\mathit{\gamma}=4.0$ | $\mathit{\gamma}=4.5$ |
---|---|---|---|---|---|---|---|

${\mu}_{1}^{\prime}$ | 1.709 | 1.921 | 2.051 | 2.344 | 2.601 | 2.830 | 3.037 |

${\mu}_{2}^{\prime}$ | 6.381 | 7.465 | 8.162 | 9.823 | 11.382 | 12.853 | 14.247 |

${\mu}_{3}^{\prime}$ | 37.486 | 44.483 | 49.068 | 60.272 | 71.130 | 81.674 | 91.927 |

${\mu}_{4}^{\prime}$ | 297.27 | 354.846 | 392.912 | 487.007 | 579.644 | 670.901 | 760.85 |

Var | 3.459 | 3.775 | 3.956 | 4.329 | 4.616 | 4.842 | 5.024 |

SK | 2.294 | 2.132 | 2.046 | 1.882 | 1.766 | 1.679 | 1.611 |

KU | 10.633 | 9.646 | 9.155 | 8.281 | 7.711 | 7.315 | 7.026 |

CV | 1.088 | 1.012 | 0.970 | 0.888 | 0.826 | 0.777 | 0.738 |

**Table 2.**Numerical values of some moments, variance, SK, KU, and CV of the EMIEE distribution for some selected values of $\alpha $ and at $\gamma =2.0$ and $\beta =0.5$.

Measure | $\mathit{\alpha}=0.1$ | $\mathit{\alpha}=0.2$ | $\mathit{\alpha}=0.3$ | $\mathit{\alpha}=0.4$ | $\mathit{\alpha}=0.6$ | $\mathit{\alpha}=0.7$ | $\mathit{\alpha}=0.75$ |
---|---|---|---|---|---|---|---|

${\mu}_{1}^{\prime}$ | 1.846 | 1.910 | 1.963 | 2.009 | 2.088 | 2.123 | 2.139 |

${\mu}_{2}^{\prime}$ | 7.645 | 7.782 | 7.913 | 8.040 | 8.280 | 8.395 | 8.451 |

${\mu}_{3}^{\prime}$ | 47.023 | 47.542 | 48.055 | 48.564 | 49.567 | 50.061 | 50.306 |

${\mu}_{4}^{\prime}$ | 380.997 | 383.988 | 386.972 | 389.947 | 395.867 | 398.811 | 400.279 |

Var | 4.238 | 4.134 | 4.060 | 4.002 | 3.919 | 3.887 | 3.874 |

SK | 1.980 | 2.009 | 2.027 | 2.039 | 2.051 | 2.053 | 2.053 |

KU | 8.646 | 8.846 | 8.982 | 9.081 | 9.209 | 9.249 | 9.264 |

CV | 1.115 | 1.065 | 1.026 | 0.996 | 0.948 | 0.929 | 0.920 |

**Table 3.**Numerical values of some moments, variance, SK, KU, and CV of the EMIEE distribution for some selected values of $\beta $ and at $\alpha =0.5$ and $\gamma =2.0$.

Measure | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.4$ | $\mathit{\beta}=0.7$ | $\mathit{\beta}=1.0$ | $\mathit{\beta}=1.2$ | $\mathit{\beta}=1.5$ | $\mathit{\beta}=1.8$ |
---|---|---|---|---|---|---|---|

${\mu}_{1}^{\prime}$ | 9.229 | 2.512 | 1.516 | 1.107 | 0.944 | 0.777 | 0.664 |

${\mu}_{2}^{\prime}$ | 191.116 | 12.562 | 4.283 | 2.180 | 1.549 | 1.023 | 0.730 |

${\mu}_{3}^{\prime}$ | 5878 | 94.852 | 18.244 | 6.439 | 3.794 | 1.993 | 1.182 |

${\mu}_{4}^{\prime}$ | 238,100 | 952.018 | 103.814 | 25.473 | 12.458 | 5.209 | 2.562 |

Var | 105.938 | 6.254 | 1.983 | 0.955 | 0.658 | 0.419 | 0.290 |

SK | 1.980 | 2.039 | 2.053 | 2.049 | 2.040 | 2.023 | 2.001 |

KU | 8.646 | 9.081 | 9.249 | 9.304 | 9.304 | 9.266 | 9.198 |

CV | 1.115 | 0.996 | 0.929 | 0.883 | 0.860 | 0.832 | 0.811 |

**Table 4.**Numerical values of the Rényi entropy of the EMIEE distribution at different values of $\alpha $, $\beta $, and $\gamma $.

Parameters | Rényi Entropy | ||||
---|---|---|---|---|---|

$\mathbf{\alpha}$ | $\mathbf{\beta}$ | $\mathbf{\gamma}$ | $\mathbf{\delta}=\mathbf{1.2}$ | $\mathbf{\delta}=\mathbf{2}$ | $\mathbf{\delta}=\mathbf{3}$ |

0.5 | 0.5 | 1 | 0.451 | 0.285 | 0.182 |

0.5 | 0.5 | 1.8 | 0.650 | 0.531 | 0.449 |

0.5 | 0.5 | 3 | 0.782 | 0.705 | 0.656 |

0.5 | 0.5 | 4 | 0.834 | 0.770 | 0.730 |

0.5 | 0.5 | 5 | 0.865 | 0.805 | 0.768 |

1.5 | 0.5 | 3 | 0.788 | 0.714 | 0.666 |

2 | 0.5 | 3 | 0.793 | 0.720 | 0.674 |

2.5 | 0.5 | 3 | 0.799 | 0.727 | 0.682 |

3 | 0.5 | 3 | 0.805 | 0.734 | 0.690 |

0.5 | 1.5 | 3 | 0.311 | 0.236 | 0.190 |

0.5 | 2 | 3 | 0.191 | 0.118 | 0.072 |

0.5 | 2.5 | 3 | 0.100 | 0.028 | −0.017 |

0.5 | 3 | 3 | 0.026 | −0.044 | −0.088 |

**Table 5.**Simulation study for the EMIEE model: MLEs and MSEs with the following sets of parameters: $\mathrm{Y}1$($\alpha =1.2,\beta =1.5,\gamma =2.0$), $\mathrm{Y}2$($\alpha =1.8,\beta =1.5,\gamma =2.0$), and $\mathrm{Y}3$($\alpha =2.5,\beta =1.5,\gamma =2.0$).

n | $\mathrm{Y}1$ | $\mathrm{Y}2$ | $\mathrm{Y}3$ | |||
---|---|---|---|---|---|---|

MLE | MLE | MLE | MLE | MLE | MLE | |

50 | 1.517 | 1.476 | 1.962 | 0.920 | 2.784 | 4.450 |

1.712 | 0.176 | 1.541 | 0.057 | 1.622 | 0.112 | |

2.498 | 1.907 | 2.051 | 0.181 | 2.423 | 1.019 | |

100 | 1.259 | 0.257 | 1.880 | 0.831 | 1.866 | 1.769 |

1.568 | 0.069 | 1.628 | 0.045 | 1.483 | 0.019 | |

2.124 | 0.130 | 2.241 | 0.104 | 2.380 | 0.436 | |

200 | 1.363 | 0.095 | 2.081 | 0.613 | 2.843 | 1.709 |

1.583 | 0.014 | 1.550 | 0.026 | 1.497 | 0.012 | |

2.056 | 0.017 | 1.992 | 0.023 | 2.013 | 0.027 | |

500 | 1.224 | 0.059 | 1.697 | 0.333 | 2.779 | 1.121 |

1.532 | 0.013 | 1.463 | 0.008 | 1.504 | 0.010 | |

2.083 | 0.045 | 2.009 | 0.011 | 2.031 | 0.025 | |

1000 | 1.131 | 0.029 | 1.825 | 0.262 | 2.552 | 0.461 |

1.484 | 0.005 | 1.484 | 0.006 | 1.487 | 0.002 | |

2.003 | 0.016 | 2.007 | 0.010 | 1.978 | 0.003 |

**Table 6.**Simulation study for the EMIEE model: MLEs and MSEs with the following sets of parameters: $\mathrm{Y}4$($\alpha =1.2,\beta =2.0,\gamma =2.0$), $\mathrm{Y}5$($\alpha =1.8,\beta =1.5,\gamma =4.0$), and $\mathrm{Y}6$($\alpha =3.0,\beta =1.5,\gamma =4.0$).

n | $\mathrm{Y}4$ | $\mathrm{Y}5$ | $\mathrm{Y}6$ | |||
---|---|---|---|---|---|---|

MLE | MLE | MLE | MLE | MLE | MLE | |

50 | 1.118 | 0.567 | 2.060 | 1.969 | 3.239 | 0.315 |

2.016 | 0.134 | 1.774 | 0.165 | 2.082 | 0.521 | |

2.446 | 1.021 | 5.169 | 3.634 | 5.419 | 4.331 | |

100 | 1.180 | 0.198 | 1.452 | 0.181 | 2.895 | 0.198 |

1.931 | 0.127 | 1.517 | 0.034 | 1.800 | 0.172 | |

2.056 | 0.153 | 4.171 | 0.475 | 5.045 | 3.873 | |

200 | 1.226 | 0.147 | 1.394 | 0.177 | 2.812 | 0.052 |

1.990 | 0.042 | 1.550 | 0.020 | 1.722 | 0.086 | |

2.028 | 0.043 | 4.353 | 0.335 | 4.660 | 0.649 | |

500 | 1.299 | 0.141 | 1.400 | 0.173 | 2.774 | 0.074 |

2.018 | 0.014 | 1.509 | 0.012 | 1.736 | 0.069 | |

2.113 | 0.032 | 4.197 | 0.206 | 4.434 | 0.300 | |

1000 | 1.177 | 0.050 | 1.669 | 0.093 | 2.781 | 0.056 |

2.029 | 0.016 | 1.511 | 0.004 | 1.732 | 0.055 | |

2.044 | 0.009 | 4.200 | 0.077 | 4.289 | 0.222 |

**Table 7.**MLEs and standard errors (SEs) (under parentheses) of the model parameters for the COVID-19 dataset: Weibull-exponential (WE) model, the Lomax-exponential (LE) model, the gamma-exponentiated exponential (GaE) model, the beta Weibull (BW) model, the Kumaraswamy exponential (KE) model, the Burr X-exponential (BXE) model, the exponentiated exponential (EE) model, the CStransformation of exponential (CE) model, the standard exponential (E) model, the alpha-power inverse Weibull (AIW) model, the Gompertz inverse exponential (GomIE) model, the Weibull-inverse exponential (WIE) model, the inverse Weibull-inverse exponential (IWIE) model, the inverse exponential (IE) model.

Model | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\gamma}$ | $\mathit{\lambda}$ | $\mathit{\theta}$ | $\mathit{\eta}$ |
---|---|---|---|---|---|---|

EMIEE | 1.5577 | 0.1184 | 2.4646 | - | - | - |

(0.5860) | (0.0188) | (0.5511) | - | - | - | |

WE | - | - | - | 1.2850 | 0.9664 | 0.0528 |

- | - | - | (1.3391) | (0.1893) | (0.0387) | |

LE | 44.7999 | 0.0477 | 67.3943 | - | - | - |

(13.9974) | (0.0199) | (1.2461) | - | - | ||

GaE | - | - | - | 0.0822 | 0.7317 | 1.2936 |

- | - | - | (0.0450) | (0.3352) | (0.5946) | |

BW | - | - | 1.6505 | 0.9444 | 3.4276 | 0.0867 |

- | - | (0.0083) | (0.0069) | (1.2777) | (0.0109) | |

KE | - | - | - | 1.1891 | 2.9329 | 0.0982 |

- | - | - | (0.0191) | (0.1555) | (0.0744) | |

BXE | 0.0391 | 0.3782 | - | - | - | - |

(0.0027) | (0.0489) | - | - | - | - | |

EE | 1.3633 | 0.1284 | - | - | - | - |

(0.2239) | (0.0184) | - | - | - | - | |

CE | 16.9399 | 16.6595 | 8.1926 | - | - | - |

(2.27821) | (3.1211) | (2.116) | - | - | - | |

E | - | 0.1061 | - | - | - | - |

- | (0.0126) | - | - | - | - | |

AIW | 7.3531 | 2.6650 | 1.1892 | - | - | - |

(1.3953) | (0.8931) | (0.1124) | - | - | - | |

GomIE | 0.2035 | 1.0786 | 1.9361 | - | - | - |

(0.1355) | (0.1925) | (0.8576) | - | - | - | |

WIE | 0.2472 | 0.9974 | 2.0841 | - | - | - |

(0.2197) | (0.1553) | (1.2271) | - | - | - | |

IWIE | 0.3554 | 0.9506 | 9.7728 | - | - | - |

(0.5157) | (0.1172) | (5.2750) | - | - | - | |

IE | 3.7629 | - | - | - | - | - |

(0.4497) | - | - | - | - | - | |

MIEE | 4.3601 | 0.0741 | - | - | - | - |

(1.11734) | (0.0123) | - | - | - | - |

**Table 8.**Values of $-\widehat{\ell}$, AIC, BIC, W, Anderson–Darling (A) criterion, KS, and KS p-value of the 16 considered models for the COVID-19 dataset.

Model | $-\widehat{\mathit{\ell}}$ | AIC | BIC | W | A | KS | KS p-Value |
---|---|---|---|---|---|---|---|

EMIEE | 221.3346 | 448.6692 | 455.4147 | 0.1228 | 0.8148 | 0.0991 | 0.5679 |

WE | 223.8891 | 453.7781 | 460.5236 | 0.1393 | 0.9449 | 0.1036 | 0.4122 |

LE | 223.9463 | 453.8925 | 460.6380 | 0.1399 | 0.9494 | 0.1060 | 0.3838 |

GaE | 223.8191 | 453.6381 | 460.3836 | 0.1396 | 0.9447 | 0.1042 | 0.4050 |

BW | 223.8808 | 455.7615 | 464.7555 | 0.1629 | 1.0745 | 0.1172 | 0.2695 |

KE | 224.0199 | 454.0399 | 460.7854 | 0.1577 | 1.0497 | 0.1130 | 0.3088 |

BXE | 224.5807 | 453.1613 | 457.6583 | 0.1552 | 1.0425 | 0.1040 | 0.4074 |

EE | 225.2817 | 454.5633 | 459.0603 | 0.1632 | 1.1044 | 0.1085 | 0.3563 |

CE | 228.1471 | 462.2942 | 469.0396 | 0.1240 | 0.8424 | 0.1392 | 0.1203 |

E | 226.9804 | 455.9608 | 458.2093 | 0.1608 | 1.0887 | 0.1048 | 0.3979 |

AIW | 231.7256 | 469.4511 | 476.1966 | 0.3406 | 2.0919 | 0.12049 | 0.2413 |

GomIE | 223.1276 | 452.2553 | 459.0007 | 0.1454 | 0.9659 | 0.1107 | 0.3327 |

WIE | 223.2084 | 452.4168 | 459.1623 | 0.1518 | 1.0024 | 0.1134 | 0.3049 |

IWIE | 233.9508 | 473.9016 | 480.6471 | 0.4037 | 2.4377 | 0.1367 | 0.1327 |

IE | 233.4393 | 468.8786 | 471.1271 | 0.3844 | 2.3318 | 0.1330 | 0.1532 |

MIEE | 228.6196 | 461.2392 | 465.7362 | 0.2013 | 1.2916 | 0.1577 | 0.0548 |

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**MDPI and ACS Style**

Bantan, R.A.R.; Chesneau, C.; Jamal, F.; Elgarhy, M.
On the Analysis of New COVID-19 Cases in Pakistan Using an Exponentiated Version of the M Family of Distributions. *Mathematics* **2020**, *8*, 953.
https://doi.org/10.3390/math8060953

**AMA Style**

Bantan RAR, Chesneau C, Jamal F, Elgarhy M.
On the Analysis of New COVID-19 Cases in Pakistan Using an Exponentiated Version of the M Family of Distributions. *Mathematics*. 2020; 8(6):953.
https://doi.org/10.3390/math8060953

**Chicago/Turabian Style**

Bantan, Rashad A. R., Christophe Chesneau, Farrukh Jamal, and Mohammed Elgarhy.
2020. "On the Analysis of New COVID-19 Cases in Pakistan Using an Exponentiated Version of the M Family of Distributions" *Mathematics* 8, no. 6: 953.
https://doi.org/10.3390/math8060953