In this paper, an attempt is made to discover the distribution of COVID-19 spread in different countries such as; Saudi Arabia, Italy, Argentina and Angola by specifying an optimal statistical distribution for analyzing the mortality rate of COVID-19. A new generalization of the recently inverted Topp Leone distribution, called Kumaraswamy inverted Topp–Leone distribution, is proposed by combining the Kumaraswamy-G family and the inverted Topp–Leone distribution. We initially provide a linear representation of its density function. We give some of its structure properties, such as quantile function, median, moments, incomplete moments, Lorenz and Bonferroni curves, entropies measures and stress-strength reliability. Then, Bayesian and maximum likelihood estimators for parameters of the Kumaraswamy inverted Topp–Leone distribution under Type-II censored sample are considered. Bayesian estimator is regarded using symmetric and asymmetric loss functions. As analytical solution is too hard, behaviours of estimates have been done viz Monte Carlo simulation study and some reasonable comparisons have been presented. The outcomes of the simulation study confirmed the efficiencies of obtained estimates as well as yielded the superiority of Bayesian estimate under adequate priors compared to the maximum likelihood estimate. Application to COVID-19 in some countries showed that the new distribution is more appropriate than some other competitive models.
The inverted distributions are of great importance due to their applicability in many fields like; biological sciences, life testing problems, etc. The density and hazard rate shapes of inverted distributions exhibit dissimilar structure than matching the non-inverted distributions. Applications of inverted distributions have been discussed with various researchers, so the reader can refer to [
Recently, [
where,
Extensions and generalizations of probability distributions have been regarded by many researchers to enhance flexibility in modelling variety of data in many fields. A well-notable family of adding parameters is the Kumaraswamy-G (K-G) proposed in [
and,
where
In this work, we provide and study a generalization of ITL model, the so called Kumaraswamy inverted Topp–Leone (KITL) distribution. Using
where,
The KITL density function can exhibit different behavior for different parameters values (
The hazard rate function of KITL distribution is given as follows
Plots of the hazard rate function (hrf) of KITL distribution for specific values of parameters are shown in
We are motivated to suggest the KITL model according to: (a) Produce new useful form of ITL with three parameters; (b) discuss several statistical properties (c) introduce more flexible model with decreasing, increasing, and upside-down hazard rate shapes; (d) able to model the COVID-19 data, in Saudi Arabia, Italy, Argentina and Angola, than some other distributions. This article is addressed as follows. Section 2 deals with some important properties. Maximum likelihood (ML) and Bayesian estimators of parameters in presence of Type II censored (T2C) samples are given in Sections 3 and 4 respectively. Monte Carlo simulation is provided in Section 5. Analysis to COVID-19 data sets is carried in Section 6, and conclusions are presented in Section 7.
Here, some significant properties of KITL distribution, specifically, linear representation of the pdf, quantile function, moments, Rényi and
Here, an important mathematical formula of KITL distribution is provided. Consider the binomial theorem
in the pdf
Again, employ the binomial expansion in
where,
The KITL distribution is easily simulated by inverting
The
which gives
where,
Mean | Variance | Skewnss | Kurtosis | |
---|---|---|---|---|
(2,1,4) | 0.507 | 0.163 | 2.298 | 14.249 |
(3,2,4) | 0.726 | 0.15 | 1.43 | 7.072 |
(5,1,1) | 0.758 | 0.517 | 3.804 | 52.153 |
(5,2,3) | 0.56 | 0.08 | 1.268 | 6.103 |
(2,2,6) | 0.827 | 0.197 | 1.428 | 7.034 |
(2,4,6) | 1.553 | 0.457 | 1.281 | 6.363 |
(2,0.5,6) | 0.127 | 0.019 | 2.619 | 15.354 |
(1,3,5) | 3.031 | 5.036 | 3.51 | 45.836 |
The
where
The Lorenz and Bonferroni curves are useful applications of the first incomplete moment defined by
Here, we obtain Rényi and
where,
where,
Substituting
The
The
The stress-strength reliability (SSR) is defined as the probability that the system is strong frequently to beat the stress applied on it. Consider that
Using
Using the binomial expansion in last equation and after simplification we have
Here, the ML estimators of the model parameters are determined via T2C scheme. Let
The ML estimators of parameters are determined by solving the non-linear
Here, we discuss the Bayesian estimation of the parameters of the KITL distribution. The Bayesian estimator is considered under squared error (SE) loss function which can be defined as;
and linear exponential (LINEX) loss function which can be expressed as
where
Assuming that the prior distribution of
Based on the following likelihood function of the KITL distribution
and the joint prior density
Then the joint posterior can be written as
To obtain the Bayesian estimators, we can use the Markov Chain Monte Carlo (MCMC) approach. An important sub-class of the MCMC techniques is Gibbs sampling and more general Metropolis within Gibbs samplers. The Metropolis-Hastings (M-H) algorithm together with the Gibbs sampling are the two most popular example of a MCMC method. It’s similar to acceptance rejection sampling, the M-H algorithms consider that, to each iteration of the algorithm, a candidate value can be generated from the KITL distributions. We use the M-H within Gibbs sampling steps to generate random samples from conditional posterior densities of
and
The Bayesian estimates based on SE and LINEX loss functions are obtained in simulation section. For more information, please see as an example [
A simulation study for KITL model is conducted for samples of sizes
MLE | SE | LINEX (1.5) | LINEX (−1.5) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||
0.7 | 0.6 | 0.8 | 20 | 0.1057 | 0.1249 | 0.0292 | 0.0079 | 0.0095 | 0.0096 | 0.0550 | 0.0113 | |
0.1458 | 0.1702 | 0.0662 | 0.0230 | 0.0247 | 0.0146 | 0.1147 | 0.0391 | |||||
0.1235 | 0.2684 | 0.0979 | 0.0472 | 0.0091 | 0.0274 | 0.1747 | 0.0880 | |||||
50 | 0.0353 | 0.0309 | 0.0260 | 0.0071 | 0.0089 | 0.0067 | 0.0441 | 0.0101 | ||||
0.1085 | 0.1073 | 0.0461 | 0.0179 | 0.0101 | 0.0123 | 0.0888 | 0.0303 | |||||
0.0419 | 0.1439 | 0.0817 | 0.0427 | 0.0088 | 0.0243 | 0.1698 | 0.0852 | |||||
100 | 0.0097 | 0.0131 | 0.0154 | 0.0059 | 0.0040 | 0.0053 | 0.0272 | 0.0068 | ||||
0.0143 | 0.0263 | 0.0375 | 0.0159 | 0.0054 | 0.0112 | 0.0747 | 0.0255 | |||||
0.0537 | 0.0739 | 0.0756 | 0.0372 | 0.0080 | 0.0227 | 0.1550 | 0.0716 | |||||
0.7 | 0.6 | 1.5 | 20 | 0.1259 | 0.1180 | 0.0831 | 0.0347 | 0.0348 | 0.0226 | 0.1382 | 0.0572 | |
0.1692 | 0.1788 | 0.0872 | 0.0423 | 0.0332 | 0.0243 | 0.1513 | 0.0754 | |||||
0.0115 | 0.3471 | 0.1356 | 0.1279 | −0.0684 | 0.0760 | 0.4374 | 0.3878 | |||||
50 | 0.0492 | 0.0312 | 0.0522 | 0.0199 | 0.0256 | 0.0156 | 0.0807 | 0.0266 | ||||
0.1267 | 0.1159 | 0.0662 | 0.0352 | 0.0215 | 0.0212 | 0.1190 | 0.0600 | |||||
−0.0323 | 0.2608 | 0.1374 | 0.1143 | −0.0811 | 0.0673 | 0.4111 | 0.3783 | |||||
100 | 0.0222 | 0.0144 | 0.0274 | 0.0116 | 0.0123 | 0.0102 | 0.0432 | 0.0137 | ||||
0.0521 | 0.0315 | 0.0500 | 0.0266 | 0.0131 | 0.0179 | 0.0929 | 0.0422 | |||||
−0.0432 | 0.0800 | 0.1207 | 0.0796 | −0.1045 | 0.0596 | 0.3293 | 0.2629 | |||||
2.0 | 0.6 | 1.5 | 20 | 0.3617 | 0.8921 | 0.0437 | 0.0232 | −0.0515 | 0.0135 | 0.1518 | 0.0403 | |
0.3033 | 0.4602 | 0.0958 | 0.0530 | 0.0372 | 0.0314 | 0.1647 | 0.0928 | |||||
0.0182 | 0.4150 | 0.1050 | 0.0969 | −0.0735 | 0.0597 | 0.3573 | 0.3048 | |||||
50 | 0.1859 | 0.4189 | 0.0440 | 0.0184 | −0.0377 | 0.0155 | 0.1361 | 0.0389 | ||||
0.2641 | 0.4398 | 0.0822 | 0.0499 | 0.0322 | 0.0314 | 0.1401 | 0.0817 | |||||
0.0176 | 0.4635 | 0.0953 | 0.1005 | −0.0671 | 0.0629 | 0.3203 | 0.2736 | |||||
100 | 0.0482 | 0.1301 | 0.0395 | 0.0222 | −0.0286 | 0.0190 | 0.1153 | 0.0376 | ||||
0.1385 | 0.1392 | 0.0617 | 0.0325 | 0.0206 | 0.0214 | 0.1088 | 0.0516 | |||||
−0.0576 | 0.1657 | 0.0994 | 0.1023 | −0.0494 | 0.0627 | 0.2980 | 0.2515 | |||||
2.0 | 1.6 | 1.5 | 20 | 0.6651 | 1.9106 | 0.0495 | 0.0228 | −0.0389 | 0.0195 | 0.1496 | 0.0401 | |
0.2146 | 1.3451 | 0.0423 | 0.0325 | −0.0978 | 0.0320 | 0.2191 | 0.0931 | |||||
0.6411 | 2.0430 | 0.0964 | 0.0556 | −0.0244 | 0.0349 | 0.2501 | 0.1368 | |||||
50 | 0.1625 | 0.2665 | 0.0381 | 0.0208 | −0.0323 | 0.0179 | 0.1164 | 0.0368 | ||||
0.0013 | 0.5902 | 0.0369 | 0.0271 | −0.0970 | 0.0283 | 0.2054 | 0.0889 | |||||
0.3697 | 0.8672 | 0.0800 | 0.0356 | −0.0190 | 0.0228 | 0.2038 | 0.0863 | |||||
100 | 0.1726 | 0.1991 | 0.0375 | 0.0160 | −0.0162 | 0.0129 | 0.0959 | 0.0335 | ||||
0.0172 | 0.3438 | 0.0309 | 0.0242 | −0.0972 | 0.0261 | 0.1909 | 0.0888 | |||||
0.2026 | 0.4149 | 0.0806 | 0.0310 | −0.0081 | 0.0197 | 0.1899 | 0.0720 |
MLE | SE | LINEX 1.5 | LINEX −1.5 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | ||||
0.7 | 0.6 | 0.8 | 14 | 0.1810 | 0.2375 | 0.0835 | 0.0369 | 0.0319 | 0.0234 | 0.1431 | 0.0623 | |
0.1439 | 0.1645 | 0.0892 | 0.0619 | 0.0241 | 0.0363 | 0.1672 | 0.1115 | |||||
0.1675 | 0.2838 | 0.1989 | 0.1787 | 0.0562 | 0.0811 | 0.3691 | 0.3648 | |||||
35 | 0.0578 | 0.0447 | 0.0507 | 0.0215 | 0.0206 | 0.0165 | 0.0833 | 0.0294 | ||||
0.0952 | 0.1041 | 0.0760 | 0.0546 | 0.0201 | 0.0339 | 0.1421 | 0.0912 | |||||
0.0893 | 0.1741 | 0.1680 | 0.1576 | 0.0456 | 0.0790 | 0.3144 | 0.3087 | |||||
70 | 0.0286 | 0.0168 | 0.0306 | 0.0122 | 0.0132 | 0.0105 | 0.0489 | 0.0149 | ||||
0.0435 | 0.0316 | 0.0734 | 0.0518 | 0.0181 | 0.0316 | 0.1344 | 0.0894 | |||||
0.0264 | 0.0595 | 0.1559 | 0.1422 | 0.0428 | 0.0761 | 0.2918 | 0.2668 | |||||
0.7 | 0.6 | 1.5 | 14 | 0.1764 | 0.2303 | 0.0851 | 0.0358 | 0.0347 | 0.0228 | 0.1421 | 0.0593 | |
0.1929 | 0.2344 | 0.0888 | 0.0453 | 0.0325 | 0.0271 | 0.1554 | 0.0796 | |||||
0.0643 | 0.3551 | 0.1423 | 0.1111 | −0.0935 | 0.0608 | 0.4621 | 0.4064 | |||||
35 | 0.0524 | 0.0380 | 0.0508 | 0.0200 | 0.0220 | 0.0154 | 0.0822 | 0.0274 | ||||
0.1274 | 0.1061 | 0.0715 | 0.0361 | 0.0257 | 0.0226 | 0.1262 | 0.0612 | |||||
−0.0014 | 0.3023 | 0.1492 | 0.1016 | −0.0796 | 0.0601 | 0.4611 | 0.3616 | |||||
70 | 0.0278 | 0.0164 | 0.0330 | 0.0126 | 0.0159 | 0.0108 | 0.0509 | 0.0153 | ||||
0.0610 | 0.0416 | 0.0680 | 0.0299 | 0.0264 | 0.0188 | 0.1176 | 0.0517 | |||||
−0.0075 | 0.1233 | 0.1297 | 0.0913 | −0.0689 | 0.0574 | 0.4296 | 0.3411 | |||||
2.0 | 0.6 | 1.5 | 14 | 0.4853 | 1.2048 | 0.0473 | 0.0124 | −0.0506 | 0.0111 | 0.1582 | 0.0385 | |
0.3348 | 0.5642 | 0.0930 | 0.0552 | 0.0299 | 0.0326 | 0.1673 | 0.0982 | |||||
0.0911 | 0.5357 | 0.1068 | 0.0971 | −0.0769 | 0.0600 | 0.3687 | 0.3230 | |||||
35 | 0.2418 | 0.4747 | 0.0444 | 0.0118 | −0.0412 | 0.0105 | 0.1406 | 0.0366 | ||||
0.1924 | 0.3096 | 0.0929 | 0.0533 | 0.0361 | 0.0323 | 0.1589 | 0.0892 | |||||
0.1151 | 0.4940 | 0.0969 | 0.0961 | −0.0683 | 0.0596 | 0.3249 | 0.2777 | |||||
70 | 0.0866 | 0.1883 | 0.0382 | 0.0102 | −0.0330 | 0.0090 | 0.1174 | 0.0311 | ||||
0.1587 | 0.1902 | 0.0688 | 0.0392 | 0.0204 | 0.0246 | 0.1246 | 0.0654 | |||||
−0.0120 | 0.2408 | 0.1056 | 0.0896 | −0.0493 | 0.0544 | 0.3207 | 0.2555 | |||||
2.0 | 1.6 | 1.5 | 14 | 0.5410 | 1.3729 | 0.0422 | 0.0149 | −0.0476 | 0.0133 | 0.1434 | 0.0372 | |
0.2011 | 1.1437 | 0.0479 | 0.0259 | −0.0979 | 0.0265 | 0.2320 | 0.0977 | |||||
0.4969 | 1.2216 | 0.0926 | 0.0575 | −0.0369 | 0.0364 | 0.2555 | 0.1444 | |||||
35 | 0.2841 | 0.4206 | 0.0487 | 0.0129 | −0.0278 | 0.0125 | 0.1342 | 0.0359 | ||||
−0.0713 | 0.3989 | 0.0405 | 0.0228 | −0.0993 | 0.0258 | 0.2161 | 0.0935 | |||||
0.4033 | 0.7683 | 0.0871 | 0.0400 | −0.0178 | 0.0251 | 0.2166 | 0.0975 | |||||
70 | 0.1194 | 0.2157 | 0.0367 | 0.0224 | −0.0233 | 0.0191 | 0.1022 | 0.0349 | ||||
0.1170 | 0.5348 | 0.0365 | 0.0316 | −0.0984 | 0.0313 | 0.2048 | 0.0926 | |||||
0.2005 | 0.5432 | 0.0765 | 0.0296 | −0.0131 | 0.0196 | 0.1859 | 0.0691 |
MLE | SE | LINEX 1.5 | LINEX −1.5 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
r | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | ||||
0.7 | 0.6 | 0.8 | 18 | 0.1249 | 0.1486 | 0.0728 | 0.0322 | 0.0252 | 0.0213 | 0.1267 | 0.0524 | |
0.1274 | 0.1494 | 0.1016 | 0.0742 | 0.0330 | 0.0412 | 0.1826 | 0.1333 | |||||
0.1389 | 0.2865 | 0.1703 | 0.1601 | 0.0380 | 0.0814 | 0.3308 | 0.3193 | |||||
45 | 0.0493 | 0.0357 | 0.0510 | 0.0205 | 0.0231 | 0.0160 | 0.0811 | 0.0276 | ||||
0.0913 | 0.0958 | 0.0818 | 0.0739 | 0.0233 | 0.0454 | 0.1499 | 0.1204 | |||||
0.0616 | 0.1462 | 0.1669 | 0.1703 | 0.0469 | 0.0898 | 0.3114 | 0.3174 | |||||
90 | 0.0204 | 0.0129 | 0.0230 | 0.0100 | 0.0070 | 0.0088 | 0.0397 | 0.0120 | ||||
0.0269 | 0.0231 | 0.0502 | 0.0486 | 0.0037 | 0.0337 | 0.1042 | 0.0739 | |||||
0.0317 | 0.0485 | 0.1755 | 0.1504 | 0.0638 | 0.0784 | 0.3091 | 0.2868 | |||||
0.7 | 0.6 | 1.5 | 18 | 0.1142 | 0.1250 | 0.0714 | 0.0322 | 0.0242 | 0.0217 | 0.1247 | 0.0518 | |
0.1703 | 0.1849 | 0.0888 | 0.0445 | 0.0331 | 0.0260 | 0.1556 | 0.0818 | |||||
0.0196 | 0.3397 | 0.1340 | 0.1074 | −0.0925 | 0.0663 | 0.4414 | 0.3835 | |||||
45 | 0.0369 | 0.0297 | 0.0425 | 0.0179 | 0.0161 | 0.0143 | 0.0709 | 0.0236 | ||||
0.1340 | 0.1316 | 0.0660 | 0.0279 | 0.0214 | 0.0170 | 0.1199 | 0.0501 | |||||
−0.0204 | 0.3106 | 0.1240 | 0.1024 | −0.0926 | 0.0646 | 0.4178 | 0.3485 | |||||
90 | 0.0187 | 0.0120 | 0.0234 | 0.0101 | 0.0080 | 0.0089 | 0.0395 | 0.0119 | ||||
0.0634 | 0.0359 | 0.0557 | 0.0238 | 0.0189 | 0.0168 | 0.0982 | 0.0450 | |||||
−0.0479 | 0.0870 | 0.1304 | 0.0820 | −0.0741 | 0.0609 | 0.4054 | 0.3062 | |||||
2.0 | 0.6 | 1.5 | 18 | 0.4046 | 1.1029 | 0.0412 | 0.0137 | −0.0554 | 0.0131 | 0.1505 | 0.0383 | |
0.3207 | 0.5175 | 0.1006 | 0.0554 | 0.0391 | 0.0319 | 0.1728 | 0.0974 | |||||
0.0631 | 0.5414 | 0.0915 | 0.0880 | −0.0842 | 0.0588 | 0.3347 | 0.2638 | |||||
45 | 0.1902 | 0.4160 | 0.0386 | 0.0127 | −0.0438 | 0.0125 | 0.1306 | 0.0362 | ||||
0.2253 | 0.3705 | 0.0789 | 0.0412 | 0.0279 | 0.0253 | 0.1381 | 0.0693 | |||||
0.0697 | 0.4967 | 0.0906 | 0.0794 | −0.0683 | 0.0569 | 0.3249 | 0.2576 | |||||
90 | 0.0621 | 0.1382 | 0.0346 | 0.0120 | −0.0230 | 0.0116 | 0.1216 | 0.0356 | ||||
0.1926 | 0.1943 | 0.0748 | 0.0411 | 0.0269 | 0.0251 | 0.1267 | 0.0658 | |||||
−0.0872 | 0.2181 | 0.0861 | 0.0719 | −0.0607 | 0.0550 | 0.2845 | 0.2151 | |||||
2.0 | 1.6 | 1.5 | 18 | 0.5581 | 1.5919 | 0.0377 | 0.0150 | −0.0513 | 0.0140 | 0.1382 | 0.0364 | |
0.2539 | 1.3779 | 0.0484 | 0.0267 | −0.0946 | 0.0264 | 0.2288 | 0.0967 | |||||
0.5553 | 1.7396 | 0.0912 | 0.0504 | −0.0326 | 0.0324 | 0.2476 | 0.1289 | |||||
45 | 0.1938 | 0.3071 | 0.0403 | 0.0198 | −0.0321 | 0.0168 | 0.1208 | 0.0364 | ||||
−0.0282 | 0.2905 | 0.0295 | 0.0253 | −0.1039 | 0.0291 | 0.1974 | 0.0826 | |||||
0.2459 | 0.4593 | 0.0852 | 0.0367 | −0.0143 | 0.0232 | 0.2093 | 0.0882 | |||||
90 | 0.1914 | 0.2442 | 0.0410 | 0.0223 | −0.0148 | 0.0185 | 0.1017 | 0.0342 | ||||
0.0118 | 0.2615 | 0.0397 | 0.0277 | −0.0902 | 0.0279 | 0.2012 | 0.0848 | |||||
0.1376 | 0.3895 | 0.0710 | 0.0273 | −0.0164 | 0.0178 | 0.1798 | 0.0673 |
From the above tables, we conclude the following
As the sample size
As the sample size
As the value of
As the value of
As the value of
As the level of censoring increases, the bias and MSE decrease.
In this section, the KITL distribution is fitted to more famous fields of survival times of COVID-19 data with different country including Saudi Arabia, Italy, Argentina, Angola as well as March precipitation data. The data are available at https://covid19.who.int/. Reference [
Models | KS | |||||
---|---|---|---|---|---|---|
KIT | 20.5801 | 5.4540 | 0.7776 | 0.0720 | 0.0273 | 0.2147 |
17.5944 | 7.3773 | 0.4666 | ||||
IW | 1.525737 | 164.6467 | 0.0906 | 0.0420 | 0.3669 | |
0.118713 | 61.8164 | |||||
IL | 40.7961 | 0.7892 | 0.1941 | 0.0363 | 0.3145 | |
33.3320 | 0.6556 | |||||
IK | 1.4612 | 139.3747 | 0.0964 | 0.0339 | 0.3032 | |
0.1065 | 47.6837 | |||||
TLIK | 0.6682 | 187.1664 | 2.3385 | 0.0855 | 0.0447 | 0.3874 |
8.3490 | 91.9325 | 29.1325 |
Models | KS | |||||
---|---|---|---|---|---|---|
ITL | 0.1418 | 0.5815 | 0.122 | 0.867 | ||
0.0136 | ||||||
KIT | 190.4247 | 155.8897 | 0.5046 | 0.0735 | 0.097 | 0.700 |
159.5578 | 247.7791 | 0.1564 | ||||
IW | 0.722081 | 220.0379 | 0.3787 | 0.144 | 1.007 | |
0.031731 | 50.28962 | |||||
IL | 130.8241 | 16.1774 | 0.3559 | 0.162 | 1.124 | |
162.2328 | 20.1525 | |||||
IK | 0.7304 | 234.8483 | 0.3775 | 0.144 | 1.011 | |
0.0322 | 54.5390 | |||||
TLIK | 0.6131 | 1198.065 | 1.5354 | 0.3347 | 0.158 | 1.095 |
0.3507 | 309.7877 | 0.8783 |
Italy | KS | |||||
---|---|---|---|---|---|---|
ITL | 43.6078 | 0.1560 | 0.179 | 1.079 | ||
4.1391 | ||||||
KIT | 1.3430 | 20.4473 | 4.4464 | 0.0715 | 0.135 | 0.831 |
0.1180 | 28.9206 | 5.1612 | ||||
IW | 1.3507 | 0.0483 | 0.1907 | 1.324 | 7.127 | |
0.0818 | 0.0115 | |||||
IL | 17.7970 | 0.0069 | 0.2922 | 0.986 | 5.442 | |
7.1991 | 0.0029 | |||||
IK | 14.6443 | 4.8909 | 0.1202 | 0.398 | 2.330 | |
1.2584 | 0.7972 | |||||
TLIK | 30.0526 | 1.3699 | 1.8741 | 0.0740 | 0.142 | 0.870 |
8.8631 | 0.4298 | 0.2976 |
KS | ||||||
---|---|---|---|---|---|---|
ITL | 0.3683 | 0.4763 | 0.1179 | 0.7466 | ||
0.0709 | ||||||
KITL | 8.1278 | 59.2157 | 0.3181 | 0.1373 | 0.0745 | 0.4604 |
3.0705 | 111.8191 | 0.2076 | ||||
IW | 1.2946 | 48.5089 | 0.1879 | 0.2063 | 1.3152 | |
0.1642 | 22.5014 | |||||
IL | 20.3526 | 1.1276 | 0.2537 | 0.1481 | 0.9441 | |
34.6317 | 2.0001 | |||||
IK | 1.4000 | 71.6972 | 0.1798 | 0.1886 | 1.2027 | |
0.1848 | 38.4794 | |||||
TLIK | 1.7875 | 79.2851 | 0.7828 | 0.1820 | 0.1708 | 1.0897 |
0.9507 | 59.6138 | 0.3539 |
KS | ||||||
---|---|---|---|---|---|---|
ITL | 2.1281 | 0.2268 | 0.0327 | 0.2088 | ||
0.3885 | ||||||
KIT | 2.0695 | 22.6336 | 0.4720 | 0.0683 | 0.0186 | 0.1264 |
0.4407 | 54.9394 | 0.7279 | ||||
IW | 1.5496 | 1.0253 | 0.1523 | 0.1261 | 0.7722 | |
0.2027 | 0.1978 | |||||
IL | 30.3138 | 0.0381 | 0.2556 | 0.0796 | 0.4935 | |
34.2592 | 0.0441 | |||||
IK | 2.9879 | 8.5955 | 0.1143 | 0.0562 | 0.3506 | |
0.4732 | 3.1251 | |||||
TLIK | 1.9583 | 3.9593 | 1.4187 | 0.1098 | 0.0515 | 0.3227 |
1.8175 | 5.7236 | 1.0220 |
The following COVID-19 data represent the daily new deaths which belong to Argentina in 65 days recorded from 1 June to 4 August 2020: 20, 11, 19, 10, 18, 27, 27, 14, 14, 28, 19, 24, 31, 30, 17, 23, 20, 24, 43, 25, 25, 13, 24, 33, 36, 39, 43, 25, 25, 28, 38, 27, 53, 40, 50, 37, 33, 79, 52, 53, 42, 38, 31, 41, 67, 61, 85, 61,71, 42, 35, 145, 80, 111, 105, 125, 66, 43, 126, 118, 111, 155, 77, 69, and 55.
Furthermore, we plot the histogram, estimated pdf plots for all models for data of Argentina in
The following COVID-19 data belong to Saudi Arabia in 109 days recorded from 17 April to 4 August 2020 (data of daily new cases): 762, 1088, 1122, 1132, 1141, 1147, 1158, 1172, 1197, 1223, 1258, 1266, 1289, 1325, 1344, 1351, 1357, 1362, 1552, 1573, 1581, 1595, 1618, 1629, 1644, 1645, 1686, 1687, 1701, 1704, 1759, 1793, 1815, 1869, 1877, 1881, 1897, 1905, 1911, 1912, 1931, 1966, 1968, 1975, 1993, 2039, 2171, 2201, 2235, 2238, 2307, 2331, 2378, 2399, 2429, 2442, 2476, 2504, 2509, 2532, 2565, 2591, 2593, 2613, 2642, 2671, 2691, 2692, 2736, 2764, 2779, 2840 2852, 2994, 3036, 3045, 3121, 3123, 3139, 3159, 3183, 3288, 3366, 3369, 3372, 3379, 3383, 3392, 3393, 3402, 3580, 3717, 3733, 3921, 3927, 3938, 3941, 3943, 3989, 4128, 4193, 4207, 4233, 4267, 4301, 4387, 4507, 4757, 4919.
Furthermore, the histogram and estimated cdf plots for all models for data of Saudi Arabia are plotted in
The considered COVID-19 data belong to Italy of 111 days that are recorded from 1 April to 20 July 2020. This data formed of daily new deaths divided by daily new cases. The data are as follows: 0.2070, 0.1520, 0.1628, 0.1666, 0.1417, 0.1221, 0.1767, 0.1987, 0.1408, 0.1456, 0.1443, 0.1319, 0.1053, 0.1789, 0.2032, 0.2167, 0.1387, 0.1646, 0.1375, 0.1421, 0.2012, 0.1957, 0.1297, 0.1754, 0.1390, 0.1761, 0.1119, 0.1915, 0.1827, 0.1548, 0.1522, 0.1369, 0.2495, 0.1253, 0.1597, 0.2195, 0.2555, 0.1956, 0.1831, 0.1791, 0.2057, 0.2406, 0.1227, 0.2196, 0.2641, 0.3067, 0.1749, 0.2148, 0.2195, 0.1993, 0.2421, 0.2430, 0.1994, 0.1779, 0.0942, 0.3067, 0.1965, 0.2003, 0.1180, 0.1686, 0.2668, 0.2113, 0.3371, 0.1730, 0.2212, 0.4972, 0.1641, 0.2667, 0.2690, 0.2321, 0.2792, 0.3515, 0.1398, 0.3436, 0.2254, 0.1302, 0.0864, 0.1619, 0.1311, 0.1994, 0.3176, 0.1856, 0.1071, 0.1041, 0.1593, 0.0537, 0.1149, 0.1176, 0.0457, 0.1264, 0.0476, 0.1620, 0.1154, 0.1493, 0.0673, 0.0894, 0.0365, 0.0385, 0.2190, 0.0777, 0.0561, 0.0435, 0.0372, 0.0385, 0.0769, 0.1491, 0.0802, 0.0870, 0.0476, 0.0562, 0.0138.
Also, the histogram and estimated cdf plots for all models for data of Italy country are plotted in
The considered COVID19 data represent the daily new cases which are belonging to Angola of 27 days recorded from 8 July to 3 August 2020. The data are as follows: 33, 10, 62, 4, 21, 23, 19, 16, 35, 31, 31, 49, 18, 44, 30, 33, 39, 29, 36, 16, 18, 50, 78, 31, 39, 16, 116.
Reference [
This article formulates a generalization of inverted Topp–Leone distribution, named as Kumaraswamy inverted Topp–Leone distribution. Some statistical properties of the KITL distribution are provided. Bayesian and ML methods of estimation are considered. The Bayesian estimator is deduced under LINEX and SE loss functions. Monte Carlo simulation study is designed to assess the performance of estimates. Generally, we conclude that the Bayesian estimates are preferable than the corresponding other estimates in approximately most of the situations. Five real data of COVID-19 obtained from Saudi Arabia, Italy, Argentina, and Angola as well as March precipitation data are considered and they showed that KITL distribution is an adequate model for these data compared with other competitive distributions.