A New Inverted Topp-Leone Distribution: Applications to the COVID-19 Mortality Rate in Two Different Countries
Abstract
:1. Introduction
2. MKITL Distribution
3. All Statistical Properties of the Proposed Distribution of MKITL
3.1. Linear Representation for the MKITL Distribution
3.2. Quantile for the MKITL Distribution
3.3. Moments for the MKITL Distribution
4. Methods of Estimation for the Distribution Parameters
4.1. Maximum Likelihood Estimators
4.2. Least Squares and Weighted Least Squares Methods
4.3. Maxzimum Product of Spacings Method
4.4. Cramr-von-Mises Method
4.5. Anderson-Darling Method
5. Simulation Results
Concluded Observation on the Simulation Results
- When , and as the value of increases the MSE of the two parameters increases in most cases, while the mean values of both parameters tends to the initial values.
- For a fixed value of , and , by increasing the sample size, the mean values of the parameters tend to the initial values, and MSE decreases.
- In most cases, the MPS was the best method for estimating the parameter referring to the mean values of the parameters and MSE as in Table 1, Table 2 and Table 3, while the AD was the best efficient method for estimating the parameters referring to the mean values of the parameters and MSE in Table 3 when .
- All estimators perform very well and provides very small MSE and the mean value of estimates for all estimators tend to the initial value of the parameters.
- The differences between all estimators values are very small, referring to the MSE values and the mean value of the parameters.
6. Application of Real Data Analysis
6.1. Application 1
6.2. Application 2
6.3. Concluding Remarks on the Two Application
- Referring to data set one we can see that MKITL provides the highest P-value, and the lowest W*, A* and lowest KS distance.
- From Figure 3 , we can deduce that MKITL was the best model for fitting the first data set.
- Referring to data set two, we can see that MKITL provides the highest P-value, and the lowest A*, W* and KS distance.
- From Figure 4, we can deduce that MKITL was the best model for fitting the second data set.
- Referring to Table 4 we can see that IE, IR, ITL and MOEx distribution provides poor fitting for the first data set.
- Referring to Table 5 we can see that IE, MOEx distribution provides poor fitting for the second data set.
- We can conclude that from both applications that the MKITL distribution provides the best fitting among all its competitive distributions, which gives it superiority in fitting this kind of mortality rate for COVID-19 data.
7. Summary
Author Contributions
Funding
Conflicts of Interest
References
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MLE | LS | MPS | WLS | CVM | AD | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | |||
0.6 | 25 | 0.5693 | 0.0167 | 0.6021 | 0.0180 | 0.5692 | 0.0131 | 0.6075 | 0.0164 | 0.6376 | 0.0223 | 0.6120 | 0.0152 | |
0.6114 | 0.0208 | 0.6191 | 0.0280 | 0.6113 | 0.0189 | 0.6194 | 0.0252 | 0.6331 | 0.0310 | 0.6196 | 0.0226 | |||
70 | 0.5839 | 0.0047 | 0.6024 | 0.0057 | 0.5839 | 0.0044 | 0.6055 | 0.0049 | 0.6145 | 0.0062 | 0.6058 | 0.0048 | ||
0.5990 | 0.0065 | 0.6046 | 0.0083 | 0.5990 | 0.0062 | 0.6046 | 0.0073 | 0.6091 | 0.0086 | 0.6050 | 0.0072 | |||
150 | 0.5892 | 0.0022 | 0.6008 | 0.0029 | 0.5891 | 0.0021 | 0.6028 | 0.0024 | 0.6064 | 0.0030 | 0.6023 | 0.0024 | ||
0.5988 | 0.0028 | 0.6022 | 0.0038 | 0.5987 | 0.0028 | 0.6027 | 0.0034 | 0.6043 | 0.0039 | 0.6026 | 0.0033 | |||
1.7 | 25 | 0.5777 | 0.0175 | 0.6178 | 0.0203 | 0.5779 | 0.0125 | 0.6227 | 0.0188 | 0.6542 | 0.0260 | 0.6248 | 0.0170 | |
1.7236 | 0.1568 | 1.7546 | 0.2115 | 1.7229 | 0.1391 | 1.7522 | 0.1883 | 1.7938 | 0.2350 | 1.7525 | 0.1714 | |||
70 | 0.5809 | 0.0051 | 0.6027 | 0.0062 | 0.5809 | 0.0048 | 0.6064 | 0.0056 | 0.6149 | 0.0068 | 0.6050 | 0.0052 | ||
1.7125 | 0.0550 | 1.7287 | 0.0781 | 1.7130 | 0.0514 | 1.7294 | 0.0678 | 1.7418 | 0.0813 | 1.7276 | 0.0646 | |||
150 | 0.5897 | 0.0022 | 0.6012 | 0.0027 | 0.5897 | 0.0022 | 0.6031 | 0.0024 | 0.6067 | 0.0028 | 0.6028 | 0.0023 | ||
1.7102 | 0.0226 | 1.7238 | 0.0304 | 1.7102 | 0.0215 | 1.7239 | 0.0265 | 1.7298 | 0.0311 | 1.7240 | 0.0264 | |||
3 | 25 | 0.5731 | 0.0180 | 0.6062 | 0.0196 | 0.5731 | 0.0135 | 0.6108 | 0.0180 | 0.6424 | 0.0245 | 0.6164 | 0.0168 | |
3.0467 | 0.4346 | 3.1018 | 0.6014 | 3.0466 | 0.3497 | 3.1171 | 0.6154 | 3.1728 | 0.6821 | 3.1198 | 0.5300 | |||
70 | 0.5811 | 0.0048 | 0.5982 | 0.0061 | 0.5811 | 0.0045 | 0.6018 | 0.0053 | 0.6102 | 0.0066 | 0.6018 | 0.0051 | ||
3.0047 | 0.1554 | 3.0384 | 0.2108 | 3.0047 | 0.1346 | 3.0384 | 0.1850 | 3.0616 | 0.2194 | 3.0396 | 0.1788 | |||
150 | 0.5924 | 0.0022 | 0.6016 | 0.0029 | 0.5923 | 0.0021 | 0.6045 | 0.0025 | 0.6072 | 0.0030 | 0.6040 | 0.0024 | ||
2.9988 | 0.0687 | 3.0243 | 0.0892 | 2.9989 | 0.0650 | 3.0249 | 0.0783 | 3.0348 | 0.0909 | 3.0238 | 0.0771 |
MLE | LS | MPS | WLS | CVM | AD | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | |||
0.6 | 25 | 1.5856 | 0.0862 | 1.6591 | 0.0890 | 1.5859 | 0.0791 | 1.6831 | 0.0833 | 1.7591 | 0.1119 | 1.7054 | 0.0786 | |
0.6037 | 0.0029 | 0.6047 | 0.0032 | 0.6036 | 0.0028 | 0.6059 | 0.0030 | 0.6094 | 0.0034 | 0.6072 | 0.0030 | |||
70 | 1.6483 | 0.0328 | 1.6898 | 0.0338 | 1.6482 | 0.0306 | 1.7088 | 0.0330 | 1.7220 | 0.0351 | 1.7100 | 0.0320 | ||
0.6004 | 0.0010 | 0.6012 | 0.0012 | 0.6003 | 0.0010 | 0.6017 | 0.0011 | 0.6028 | 0.0012 | 0.6019 | 0.0011 | |||
150 | 1.6706 | 0.0146 | 1.7023 | 0.0182 | 1.6706 | 0.0144 | 1.7098 | 0.0154 | 1.7185 | 0.0189 | 1.7101 | 0.0158 | ||
0.5998 | 0.0005 | 0.6005 | 0.0006 | 0.5998 | 0.0005 | 0.6007 | 0.0005 | 0.6012 | 0.0006 | 0.6008 | 0.0005 | |||
1.7 | 25 | 1.6116 | 0.1145 | 1.7011 | 0.1276 | 1.6117 | 0.0896 | 1.7188 | 0.1152 | 1.8055 | 0.1582 | 1.7370 | 0.1032 | |
1.7086 | 0.0237 | 1.7105 | 0.0252 | 1.7086 | 0.0227 | 1.7139 | 0.0240 | 1.7238 | 0.0266 | 1.7173 | 0.0239 | |||
70 | 1.6371 | 0.0307 | 1.6867 | 0.0385 | 1.6373 | 0.0308 | 1.7001 | 0.0337 | 1.7221 | 0.0407 | 1.7017 | 0.0324 | ||
1.7028 | 0.0075 | 1.7042 | 0.0086 | 1.7027 | 0.0074 | 1.7058 | 0.0080 | 1.7089 | 0.0088 | 1.7067 | 0.0080 | |||
150 | 1.6671 | 0.0140 | 1.7032 | 0.0195 | 1.6670 | 0.0140 | 1.7092 | 0.0161 | 1.7198 | 0.0204 | 1.7083 | 0.0156 | ||
1.7003 | 0.0035 | 1.7019 | 0.0040 | 1.7002 | 0.0034 | 1.7026 | 0.0038 | 1.7040 | 0.0041 | 1.7028 | 0.0037 | |||
3 | 25 | 1.6110 | 0.1158 | 1.7090 | 0.1431 | 1.6113 | 0.0897 | 1.7256 | 0.1267 | 1.8146 | 0.1798 | 1.7391 | 0.1109 | |
2.9987 | 0.0649 | 3.0010 | 0.0731 | 2.9985 | 0.0623 | 3.0065 | 0.0690 | 3.0247 | 0.0766 | 3.0119 | 0.0677 | |||
70 | 1.6407 | 0.0306 | 1.7040 | 0.0445 | 1.6407 | 0.0301 | 1.7128 | 0.0364 | 1.7400 | 0.0485 | 1.7116 | 0.0338 | ||
3.0027 | 0.0264 | 3.0074 | 0.0300 | 3.0025 | 0.0259 | 3.0099 | 0.0282 | 3.0156 | 0.0306 | 3.0104 | 0.0280 | |||
150 | 1.6627 | 0.0137 | 1.6982 | 0.0187 | 1.6628 | 0.0142 | 1.7040 | 0.0159 | 1.7147 | 0.0194 | 1.7033 | 0.0152 | ||
3.0005 | 0.0118 | 3.0028 | 0.0130 | 3.0003 | 0.0117 | 3.0040 | 0.0122 | 3.0066 | 0.0131 | 3.0042 | 0.0122 |
MLE | LS | MPS | WLS | CVM | AD | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | |||
0.6 | 25 | 2.8337 | 0.1408 | 2.8969 | 0.1058 | 2.8339 | 0.1606 | 2.9309 | 0.0985 | 2.9607 | 0.0714 | 2.9618 | 0.0585 | |
0.5988 | 0.0009 | 0.5991 | 0.0011 | 0.5988 | 0.0009 | 0.5999 | 0.0010 | 0.6010 | 0.0011 | 0.6005 | 0.0010 | |||
70 | 2.9155 | 0.0496 | 2.9782 | 0.0295 | 2.9156 | 0.0506 | 2.9962 | 0.0259 | 3.0149 | 0.0344 | 2.9865 | 0.0424 | ||
0.5982 | 0.0003 | 0.5991 | 0.0004 | 0.5982 | 0.0003 | 0.5994 | 0.0003 | 0.5999 | 0.0004 | 0.5996 | 0.0003 | |||
150 | 2.9443 | 0.0275 | 2.9853 | 0.0082 | 2.9440 | 0.0238 | 3.0012 | 0.0129 | 2.9954 | 0.0083 | 2.9949 | 0.0090 | ||
0.5994 | 0.0001 | 0.6001 | 0.0002 | 0.5994 | 0.0001 | 0.6002 | 0.0002 | 0.6004 | 0.0002 | 0.6003 | 0.0002 | |||
1.7 | 25 | 2.8113 | 0.3529 | 2.9196 | 0.2694 | 2.8113 | 0.2987 | 2.9733 | 0.2791 | 3.0759 | 0.2844 | 3.0107 | 0.2637 | |
1.7034 | 0.0080 | 1.7038 | 0.0085 | 1.7034 | 0.0077 | 1.7057 | 0.0081 | 1.7113 | 0.0088 | 1.7085 | 0.0080 | |||
70 | 2.8962 | 0.0967 | 2.9926 | 0.1139 | 2.8966 | 0.0970 | 3.0082 | 0.0987 | 3.0529 | 0.1203 | 3.0167 | 0.1010 | ||
1.7005 | 0.0027 | 1.7022 | 0.0030 | 1.7005 | 0.0027 | 1.7028 | 0.0029 | 1.7048 | 0.0031 | 1.7032 | 0.0028 | |||
150 | 2.9470 | 0.0445 | 2.9955 | 0.0457 | 2.9468 | 0.0435 | 3.0217 | 0.0452 | 3.0213 | 0.0454 | 3.0175 | 0.0429 | ||
1.6974 | 0.0013 | 1.6970 | 0.0014 | 1.6974 | 0.0013 | 1.6981 | 0.0013 | 1.6981 | 0.0014 | 1.6982 | 0.0013 | |||
3 | 25 | 2.8351 | 0.3245 | 2.9918 | 0.3290 | 2.8347 | 0.2596 | 3.0273 | 0.3148 | 3.1730 | 0.4019 | 3.0626 | 0.2887 | |
2.9931 | 0.0229 | 2.9967 | 0.0251 | 2.9933 | 0.0224 | 2.9996 | 0.0239 | 3.0099 | 0.0258 | 3.0031 | 0.0236 | |||
70 | 2.9073 | 0.0946 | 3.0050 | 0.1201 | 2.9072 | 0.0896 | 3.0279 | 0.1075 | 3.0684 | 0.1303 | 3.0275 | 0.1001 | ||
3.0017 | 0.0079 | 3.0027 | 0.0086 | 3.0018 | 0.0078 | 3.0045 | 0.0081 | 3.0074 | 0.0088 | 3.0051 | 0.0080 | |||
150 | 2.9351 | 0.0390 | 2.9925 | 0.0530 | 2.9349 | 0.0405 | 3.0076 | 0.0447 | 3.0218 | 0.0546 | 3.0055 | 0.0435 | ||
2.9967 | 0.0040 | 2.9981 | 0.0044 | 2.9968 | 0.0040 | 2.9991 | 0.0041 | 3.0003 | 0.0044 | 2.9990 | 0.0041 |
KS | p-Value | W * | A * | ||||
---|---|---|---|---|---|---|---|
MKITL | MLE | 3.3555 | 0.7286 | 0.1454 | 0.4323 | 0.1087 | 0.6473 |
SE | 0.4046 | 0.0275 | |||||
MKEx | MLE | 2.2855 | 0.1864 | 0.1695 | 0.2523 | 0.2299 | 1.3306 |
SE | 0.2705 | 0.0103 | |||||
IW | MLE | 3.1691 | 23.4053 | 0.1737 | 0.2274 | 0.2548 | 1.5284 |
SE | 0.3656 | 8.1037 | |||||
IR | MLE | 8.2339 | 0.2790 | 0.0074 | 0.1770 | 1.0691 | |
SE | 1.3723 | ||||||
IE | MLE | 3.0078 | 0.4284 | 0.0000 | 0.1289 | 0.7751 | |
SE | 0.5013 | ||||||
ITL | MLE | 1.1523 | 0.4390 | 0.0000 | 0.1505 | 0.8600 | |
SE | 0.1920 | ||||||
MOEx | MLE | 83.0836 | 1.4411 | 0.9890 | 0.0000 | 0.3894 | 2.2258 |
SE | 39.1521 | 0.1530 |
KS | p-Value | W* | A* | ||||
---|---|---|---|---|---|---|---|
MKITL | MLE | 1.3599 | 12.7931 | 0.0936 | 0.9715 | 0.0244 | 0.1762 |
SE | 0.2319 | 1.4051 | |||||
MKEX | MLE | 2.1755 | 2.3241 | 0.0956 | 0.9658 | 0.0254 | 0.1831 |
SE | 0.3598 | 0.1636 | |||||
ITL | MLE | 21.7398 | 0.2371 | 0.1136 | 0.0403 | 0.2776 | |
SE | 4.4376 | ||||||
IR | MLE | 0.0399 | 0.1666 | 0.4685 | 0.1541 | 0.9476 | |
SE | 0.0081 | ||||||
IW | MLE | 2.3035 | 0.0229 | 0.1869 | 0.3292 | 0.1698 | 1.0330 |
SE | 0.3350 | 0.0149 | |||||
IE | MLE | 0.2200 | 0.3752 | 0.0015 | 0.1052 | 0.6731 | |
SE | 0.0449 | ||||||
MOEX | MLE | 50.8938 | 15.5659 | 0.9817 | 0.0000 | 0.1196 | 0.7700 |
SE | 36.2538 | 2.5549 |
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Almetwally, E.M.; Alharbi, R.; Alnagar, D.; Hafez, E.H. A New Inverted Topp-Leone Distribution: Applications to the COVID-19 Mortality Rate in Two Different Countries. Axioms 2021, 10, 25. https://doi.org/10.3390/axioms10010025
Almetwally EM, Alharbi R, Alnagar D, Hafez EH. A New Inverted Topp-Leone Distribution: Applications to the COVID-19 Mortality Rate in Two Different Countries. Axioms. 2021; 10(1):25. https://doi.org/10.3390/axioms10010025
Chicago/Turabian StyleAlmetwally, Ehab M., Randa Alharbi, Dalia Alnagar, and Eslam Hossam Hafez. 2021. "A New Inverted Topp-Leone Distribution: Applications to the COVID-19 Mortality Rate in Two Different Countries" Axioms 10, no. 1: 25. https://doi.org/10.3390/axioms10010025
APA StyleAlmetwally, E. M., Alharbi, R., Alnagar, D., & Hafez, E. H. (2021). A New Inverted Topp-Leone Distribution: Applications to the COVID-19 Mortality Rate in Two Different Countries. Axioms, 10(1), 25. https://doi.org/10.3390/axioms10010025