Statistical Inference for the Kavya–Manoharan Kumaraswamy Model under Ranked Set Sampling with Applications
<p>Plots of pdf for the KMKu model.</p> "> Figure 2
<p>Plots of hrf for the KMKu model.</p> "> Figure 2 Cont.
<p>Plots of hrf for the KMKu model.</p> "> Figure 3
<p>MSE of parameters based on SRS with different sample sizes.</p> "> Figure 4
<p>Heat-map of MSE for parameters based on RSS with different sample sizes.</p> "> Figure 5
<p>Estimated cdf, pdf and P-P plot for KMKu: COVID-19 data of the United Kingdom.</p> "> Figure 6
<p>Profile MLE for KMKu parameters: COVID-19 data of the United Kingdom.</p> "> Figure 7
<p>Estimated cdf, pdf and P-P plot for KMKu: COVID-19 data of Turkey.</p> "> Figure 8
<p>Profile MLE for KMKu parameters: COVID-19 data of Turkey.</p> "> Figure 9
<p>Estimated cdf, pdf and P–P plot for KMKu: data III.</p> "> Figure 10
<p>Profile MLE for KMKu parameters: data III.</p> ">
Abstract
:1. Introduction
- The new KMKu is more flexible than the Ku model, and they have the same number of parameters.
- The curves of the pdf for the KMKu model are similar to the Ku model, and it can be asymmetric, such as (i) bathtub when , , (ii) decreasing when , , (iii) increasing when , , (iv) unimodal when .
- The KMKu model has a closed form for the quantile function, making it simple to generate random numbers from the KMKu proposed model.
- Several general statistical features of the KMku model were investigated.
- The maximum likelihood estimation technique was employed to calculate the parameters of the KMKu model, employing simple and ranked set sampling.
- The KMKu model gives more fit than the Ku model and numerous other well-known models for modeling real-world data sets in different fields, and we recommended that in the application section.
2. The Construction of The Suggested Model
3. Useful Expansions
4. General Statistical Properties
4.1. Quantile Function
4.2. Moments and Incomplete Moments
4.3. Probability Weighted Moments
4.4. Order Statistics
5. Different Types of Entropy
5.1. Rényi Entropy
5.2. Havrda and Charvat Entropy
5.3. Tsallis Entropy
5.4. Arimoto Entropy
6. Sampling Techniques
7. Maximum Likelihood Estimation
7.1. MLE Based on SRS
7.2. MLE Based on RSS
7.3. Asymptotic Confidence Interval
8. Simulation
9. Application
10. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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var | CV | ||||||||
---|---|---|---|---|---|---|---|---|---|
0.6 | 0.6 | 0.4230 | 0.2870 | 0.2240 | 0.1870 | 0.1080 | 0.3180 | 1.7130 | 0.7760 |
0.8 | 0.3480 | 0.2140 | 0.1560 | 0.1240 | 0.0920 | 0.6210 | 2.1050 | 0.8730 | |
1.0 | 0.2930 | 0.1640 | 0.1130 | 0.0860 | 0.0780 | 0.8650 | 2.5950 | 0.9540 | |
1.2 | 0.2510 | 0.1290 | 0.0840 | 0.0610 | 0.0660 | 1.0720 | 3.1370 | 1.0220 | |
1.4 | 0.2180 | 0.1030 | 0.0630 | 0.0440 | 0.0560 | 1.2520 | 3.7110 | 1.0810 | |
1.6 | 0.1920 | 0.0840 | 0.0490 | 0.0330 | 0.0470 | 1.4140 | 4.3040 | 1.1330 | |
1.8 | 0.1700 | 0.0690 | 0.0380 | 0.0240 | 0.0400 | 1.5600 | 4.9080 | 1.1790 | |
2.0 | 0.1520 | 0.0580 | 0.0300 | 0.0190 | 0.0350 | 1.6930 | 5.5200 | 1.2210 | |
0.8 | 0.6 | 0.4880 | 0.3400 | 0.2680 | 0.2240 | 0.1020 | 0.0830 | 1.6660 | 0.6540 |
0.8 | 0.4160 | 0.2640 | 0.1960 | 0.1560 | 0.0910 | 0.3530 | 1.8630 | 0.7270 | |
1.0 | 0.3610 | 0.2110 | 0.1470 | 0.1130 | 0.0810 | 0.5630 | 2.1490 | 0.7860 | |
1.2 | 0.3190 | 0.1720 | 0.1140 | 0.0840 | 0.0710 | 0.7350 | 2.4730 | 0.8350 | |
1.4 | 0.2840 | 0.1430 | 0.0900 | 0.0630 | 0.0620 | 0.8820 | 2.8140 | 0.8770 | |
1.6 | 0.2570 | 0.1210 | 0.0720 | 0.0490 | 0.0550 | 1.0090 | 3.1620 | 0.9130 | |
1.8 | 0.2330 | 0.1030 | 0.0590 | 0.0380 | 0.0480 | 1.1210 | 3.5100 | 0.9440 | |
2.0 | 0.2130 | 0.0890 | 0.0480 | 0.0300 | 0.0430 | 1.2210 | 3.8550 | 0.9710 | |
1 | 0.6 | 0.5390 | 0.3840 | 0.3060 | 0.2570 | 0.0940 | −0.0960 | 1.7240 | 0.5680 |
0.8 | 0.4710 | 0.3090 | 0.2310 | 0.1860 | 0.0870 | 0.1550 | 1.7990 | 0.6270 | |
1.0 | 0.4180 | 0.2540 | 0.1800 | 0.1390 | 0.0790 | 0.3450 | 1.9650 | 0.6740 | |
1.2 | 0.3760 | 0.2130 | 0.1440 | 0.1060 | 0.0720 | 0.4970 | 2.1700 | 0.7120 | |
1.4 | 0.3420 | 0.1820 | 0.1170 | 0.0830 | 0.0650 | 0.6240 | 2.3910 | 0.7430 | |
1.6 | 0.3140 | 0.1570 | 0.0960 | 0.0660 | 0.0580 | 0.7320 | 2.6170 | 0.7700 | |
1.8 | 0.2900 | 0.1370 | 0.0800 | 0.0530 | 0.0530 | 0.8260 | 2.8430 | 0.7930 | |
2.0 | 0.2690 | 0.1200 | 0.0680 | 0.0430 | 0.0480 | 0.9090 | 3.0640 | 0.8140 | |
1.2 | 0.6 | 0.5810 | 0.4230 | 0.3400 | 0.2870 | 0.0860 | −0.2390 | 1.8310 | 0.5040 |
0.8 | 0.5160 | 0.3480 | 0.2640 | 0.2140 | 0.0820 | −0.0012 | 1.8190 | 0.5530 | |
1.0 | 0.4660 | 0.2930 | 0.2110 | 0.1640 | 0.0760 | 0.1760 | 1.9060 | 0.5920 | |
1.2 | 0.4250 | 0.2510 | 0.1720 | 0.1290 | 0.0700 | 0.3150 | 2.0360 | 0.6230 | |
1.4 | 0.3920 | 0.2180 | 0.1430 | 0.1030 | 0.0650 | 0.4300 | 2.1840 | 0.6480 | |
1.6 | 0.3640 | 0.1920 | 0.1210 | 0.0840 | 0.0590 | 0.5260 | 2.3390 | 0.6690 | |
1.8 | 0.3400 | 0.1700 | 0.1030 | 0.0690 | 0.0550 | 0.6090 | 2.4940 | 0.6880 | |
2.0 | 0.3190 | 0.1520 | 0.0890 | 0.0580 | 0.0500 | 0.6810 | 2.6470 | 0.7030 | |
1.4 | 0.6 | 0.6160 | 0.4580 | 0.3700 | 0.3150 | 0.0780 | −0.3590 | 1.9640 | 0.4540 |
0.8 | 0.5550 | 0.3840 | 0.2940 | 0.2400 | 0.0760 | −0.1290 | 1.8840 | 0.4960 | |
1.0 | 0.5070 | 0.3290 | 0.2400 | 0.1880 | 0.0720 | 0.0390 | 1.9140 | 0.5290 | |
1.2 | 0.4680 | 0.2860 | 0.2000 | 0.1510 | 0.0670 | 0.1700 | 1.9910 | 0.5550 | |
1.4 | 0.4350 | 0.2520 | 0.1690 | 0.1230 | 0.0630 | 0.2760 | 2.0900 | 0.5760 | |
1.6 | 0.4080 | 0.2250 | 0.1450 | 0.1020 | 0.0590 | 0.3640 | 2.1980 | 0.5940 | |
1.8 | 0.3840 | 0.2020 | 0.1260 | 0.0860 | 0.0550 | 0.4390 | 2.3080 | 0.6090 | |
2.0 | 0.3640 | 0.1830 | 0.1100 | 0.0730 | 0.0510 | 0.5040 | 2.4180 | 0.6220 |
RE | HCE | TE | AE | RE | HCE | TE | AE | ||
---|---|---|---|---|---|---|---|---|---|
0.6 | 0.6 | −0.0260 | −0.1400 | −0.0590 | −0.0580 | −0.0470 | −0.1780 | −0.1060 | −0.1060 |
0.8 | −0.0420 | −0.2240 | −0.0950 | −0.0930 | −0.0760 | −0.2870 | −0.1720 | −0.1710 | |
1.0 | −0.0690 | −0.3540 | −0.1530 | −0.1470 | −0.1210 | −0.4510 | −0.2700 | −0.2680 | |
1.2 | −0.1000 | −0.4960 | −0.2170 | −0.2050 | −0.1700 | −0.6290 | −0.3780 | −0.3740 | |
1.4 | −0.1330 | −0.6350 | −0.2830 | −0.2630 | −0.2210 | −0.8030 | −0.4840 | −0.4780 | |
1.6 | −0.1660 | −0.7650 | −0.3470 | −0.3170 | −0.2700 | −0.9690 | −0.5850 | −0.5770 | |
1.8 | −0.1980 | −0.8850 | −0.4080 | −0.3670 | −0.3180 | −1.1250 | −0.6810 | −0.6690 | |
2.0 | −0.2310 | −0.9950 | −0.4660 | −0.4120 | −0.3630 | −1.2700 | −0.7710 | −0.7550 | |
0.8 | 0.6 | −0.0091 | −0.0500 | −0.0210 | −0.0210 | −0.0160 | −0.0620 | −0.0370 | −0.0370 |
0.8 | −0.0130 | −0.0690 | −0.0290 | −0.0290 | −0.0210 | −0.0810 | −0.0480 | −0.0480 | |
1.0 | −0.0270 | −0.1480 | −0.0620 | −0.0610 | −0.0450 | −0.1730 | −0.1030 | −0.1030 | |
1.2 | −0.0470 | −0.2500 | −0.1060 | −0.1040 | −0.0760 | −0.2890 | −0.1730 | −0.1720 | |
1.4 | −0.0700 | −0.3590 | −0.1550 | −0.1490 | −0.1100 | −0.4130 | −0.2470 | −0.2450 | |
1.6 | −0.0930 | −0.4670 | −0.2040 | −0.1930 | −0.1440 | −0.5350 | −0.3210 | −0.3180 | |
1.8 | −0.1170 | −0.5710 | −0.2520 | −0.2360 | −0.1770 | −0.6520 | −0.3920 | −0.3880 | |
2.0 | −0.1410 | −0.6680 | −0.2990 | −0.2770 | −0.2100 | −0.7640 | −0.4600 | −0.4550 | |
1 | 0.6 | −0.0071 | −0.0390 | −0.0160 | −0.0160 | −0.0130 | −0.0500 | −0.0300 | −0.0300 |
0.8 | −0.0020 | −0.0110 | −0.0045 | −0.0045 | −0.0032 | −0.0120 | −0.0073 | −0.0073 | |
1.0 | −0.0089 | −0.0490 | −0.0200 | −0.0200 | −0.0140 | −0.0550 | −0.0330 | −0.0330 | |
1.2 | −0.0220 | −0.1180 | −0.0490 | −0.0490 | −0.0340 | −0.1290 | −0.0770 | −0.0770 | |
1.4 | −0.0370 | −0.1990 | −0.0840 | −0.0830 | −0.0570 | −0.2160 | −0.1290 | −0.1290 | |
1.6 | −0.0550 | −0.2850 | −0.1220 | −0.1180 | −0.0810 | −0.3070 | −0.1830 | −0.1830 | |
1.8 | −0.0720 | −0.3710 | −0.1600 | −0.1530 | −0.1060 | −0.3970 | −0.2380 | −0.2360 | |
2.0 | −0.0900 | −0.4530 | −0.1980 | −0.1880 | −0.1300 | −0.4850 | −0.2900 | −0.2880 | |
1.2 | 0.6 | −0.0130 | −0.0710 | −0.0300 | −0.0290 | −0.0220 | −0.0860 | −0.0520 | −0.0510 |
0.8 | −0.0016 | −0.0087 | −0.0036 | −0.0036 | −0.0025 | −0.0097 | −0.0058 | −0.0058 | |
1.0 | −0.0029 | −0.0160 | −0.0066 | −0.0066 | −0.0045 | −0.0180 | −0.0100 | −0.0100 | |
1.2 | −0.0100 | −0.0580 | −0.0240 | −0.0240 | −0.0160 | −0.0620 | −0.0370 | −0.0370 | |
1.4 | −0.0210 | −0.1160 | −0.0490 | −0.0480 | −0.0320 | −0.1230 | −0.0730 | −0.0730 | |
1.6 | −0.0340 | −0.1820 | −0.0770 | −0.0750 | −0.0500 | −0.1900 | −0.1140 | −0.1130 | |
1.8 | −0.0480 | −0.2510 | −0.1070 | −0.1040 | −0.0680 | −0.2600 | −0.1550 | −0.1550 | |
2.0 | −0.0620 | −0.3200 | −0.1370 | −0.1320 | −0.0870 | −0.3290 | −0.1970 | −0.1960 | |
1.4 | 0.6 | −0.0230 | −0.1250 | −0.0530 | −0.0520 | −0.0380 | −0.1460 | −0.0870 | −0.0870 |
0.8 | −0.0071 | −0.0390 | −0.0160 | −0.0160 | −0.0110 | −0.0410 | −0.0250 | −0.0250 | |
1.0 | −0.0041 | −0.0230 | −0.0095 | −0.0094 | −0.0062 | −0.0240 | −0.0140 | −0.0140 | |
1.2 | −0.0078 | −0.0430 | −0.0180 | −0.0180 | −0.0120 | −0.0460 | −0.0270 | −0.0270 | |
1.4 | −0.0150 | −0.0830 | −0.0350 | −0.0340 | −0.0230 | −0.0870 | −0.0520 | −0.0520 | |
1.6 | −0.0250 | −0.1330 | −0.0560 | −0.0550 | −0.0360 | −0.1370 | −0.0820 | −0.0820 | |
1.8 | −0.0350 | −0.1870 | −0.0790 | −0.0770 | −0.0500 | −0.1910 | −0.1140 | −0.1140 | |
2.0 | −0.0460 | −0.2430 | −0.1030 | −0.1010 | −0.0650 | −0.2460 | −0.1470 | −0.1460 |
SRS | RSS r = 1 | RSS r = 3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
m | Bias | MSE | Bias | MSE | Bias | MSE | RE1 | RE2 | RE3 | ||
0.2, 0.75 | 4 | 0.1651 | 0.1474 | 0.0890 | 0.0346 | 0.0813 | 0.0296 | 426% | 499% | 117% | |
0.9047 | 3.1293 | 0.5787 | 1.3401 | 0.4815 | 1.0431 | 234% | 300% | 128% | |||
7 | 0.0717 | 0.0349 | 0.0366 | 0.0087 | 0.0309 | 0.0084 | 403% | 416% | 103% | ||
0.4008 | 0.8798 | 0.2021 | 0.2457 | 0.1974 | 0.2407 | 358% | 366% | 102% | |||
10 | 0.0519 | 0.0147 | 0.0190 | 0.0036 | 0.0172 | 0.0033 | 408% | 445% | 109% | ||
0.2775 | 0.3929 | 0.1120 | 0.0847 | 0.0962 | 0.0743 | 464% | 529% | 114% | |||
15 | 0.0386 | 0.0093 | 0.0078 | 0.0015 | 0.0076 | 0.0013 | 634% | 686% | 108% | ||
0.2331 | 0.3179 | 0.0512 | 0.0339 | 0.0511 | 0.0316 | 938% | 1005% | 107% | |||
0.2, 1.5 | 4 | 0.0712 | 0.0328 | 0.0340 | 0.0084 | 0.0305 | 0.0076 | 391% | 432% | 110% | |
0.8320 | 3.7420 | 0.6290 | 7.1765 | 0.3595 | 0.7037 | 52% | 532% | 1020% | |||
7 | 0.0318 | 0.0093 | 0.0095 | 0.0023 | 0.0055 | 0.0021 | 408% | 442% | 109% | ||
0.3885 | 0.8633 | 0.0996 | 0.1745 | 0.0900 | 0.1672 | 495% | 516% | 104% | |||
10 | 0.0273 | 0.0060 | 0.0061 | 0.0013 | 0.0058 | 0.0011 | 474% | 527% | 111% | ||
0.3359 | 0.6755 | 0.1135 | 0.1803 | 0.0940 | 0.1531 | 375% | 441% | 118% | |||
15 | 0.0188 | 0.0040 | 0.0003 | 0.0004 | 0.0002 | 0.0003 | 1071% | 1148% | 107% | ||
0.2567 | 0.4604 | 0.0175 | 0.0289 | 0.0131 | 0.0241 | 1595% | 1910% | 120% | |||
0.2, 3 | 4 | 0.0221 | 0.0077 | 0.0072 | 0.0022 | 0.0068 | 0.0020 | 354% | 380% | 107% | |
0.2639 | 0.8307 | 0.2024 | 0.8127 | 0.0940 | 0.3292 | 102% | 252% | 247% | |||
7 | 0.0102 | 0.0028 | 0.0030 | 0.0006 | 0.0010 | 0.0006 | 469% | 466% | 99% | ||
0.2096 | 1.3086 | 0.0326 | 0.0649 | 0.0269 | 0.0505 | 2016% | 2591% | 129% | |||
10 | 0.0081 | 0.0019 | 0.0016 | 0.0005 | 0.0039 | 0.0005 | 393% | 387% | 99% | ||
0.0895 | 0.4816 | 0.0851 | 0.4519 | 0.0818 | 0.3829 | 107% | 126% | 118% | |||
15 | 0.0060 | 0.0014 | −0.0003 | 0.0001 | −0.0002 | 0.0001 | 1051% | 1227% | 117% | ||
0.1332 | 0.5084 | 0.0139 | 0.0326 | 0.0077 | 0.0291 | 1558% | 1746% | 112% | |||
0.5, 0.75 | 4 | 0.4500 | 0.9750 | 0.2569 | 0.2633 | 0.2410 | 0.2403 | 370% | 406% | 110% | |
1.3133 | 6.3410 | 0.8140 | 2.7293 | 0.7887 | 2.6856 | 232% | 236% | 102% | |||
7 | 0.1936 | 0.2332 | 0.0982 | 0.0600 | 0.0845 | 0.0587 | 389% | 397% | 102% | ||
0.5368 | 1.7774 | 0.2345 | 0.3502 | 0.2327 | 0.3494 | 507% | 509% | 100% | |||
10 | 0.1380 | 0.0989 | 0.0533 | 0.0264 | 0.0576 | 0.0261 | 375% | 379% | 101% | ||
0.3433 | 0.7597 | 0.1324 | 0.1237 | 0.1111 | 0.0991 | 614% | 767% | 125% | |||
15 | 0.0996 | 0.0604 | 0.0199 | 0.0094 | 0.0215 | 0.0090 | 644% | 668% | 104% | ||
0.2526 | 0.3891 | 0.0527 | 0.0364 | 0.0523 | 0.0328 | 1067% | 1185% | 111% | |||
0.5, 1.5 | 4 | 0.2394 | 0.2725 | 0.1415 | 0.0892 | 0.1333 | 0.0847 | 306% | 322% | 105% | |
1.6074 | 7.5157 | 1.1325 | 4.4234 | 0.9699 | 3.9644 | 170% | 190% | 112% | |||
7 | 0.1170 | 0.0834 | 0.0564 | 0.0261 | 0.0440 | 0.0244 | 319% | 341% | 107% | ||
0.8210 | 2.8512 | 0.3655 | 0.7510 | 0.3359 | 0.6951 | 380% | 410% | 108% | |||
10 | 0.0980 | 0.0532 | 0.0356 | 0.0145 | 0.0341 | 0.0142 | 367% | 375% | 102% | ||
0.6881 | 2.4032 | 0.2709 | 0.4835 | 0.2478 | 0.4554 | 497% | 528% | 106% | |||
15 | 0.0733 | 0.0360 | 0.0114 | 0.0056 | 0.0119 | 0.0056 | 643% | 642% | 100% | ||
0.5825 | 3.0070 | 0.0978 | 0.1478 | 0.0937 | 0.1348 | 2034% | 2231% | 110% | |||
0.5, 3 | 4 | 0.0923 | 0.0755 | 0.0420 | 0.0231 | 0.0424 | 0.0224 | 327% | 337% | 103% | |
1.1344 | 5.2244 | 0.7868 | 4.8975 | 0.6287 | 2.6572 | 107% | 197% | 184% | |||
7 | 0.0478 | 0.0290 | 0.0101 | 0.0069 | 0.0034 | 0.0067 | 419% | 432% | 103% | ||
0.7166 | 2.9233 | 0.1408 | 0.5447 | 0.1252 | 0.5316 | 537% | 550% | 102% | |||
10 | 0.0461 | 0.0216 | 0.0108 | 0.0051 | 0.0106 | 0.0051 | 422% | 425% | 101% | ||
0.7557 | 3.3001 | 0.2461 | 0.8321 | 0.2375 | 0.7869 | 877% | 928% | 106% | |||
15 | 0.0255 | 0.0142 | −0.0067 | 0.0013 | 0.0010 | 0.0012 | 1102% | 1153% | 105% | ||
0.5203 | 2.6514 | 0.0383 | 0.1276 | 0.0345 | 0.1237 | 11482% | 11847% | 103% |
SRS | RSS r = 1 | RSS r = 3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
m | Lower | Upper | CV | Lower | Upper | CV | Lower | Upper | CV | ||
0.2, 0.75 | 4 | 0.002 | 0.772 | 97.0% | 0.026 | 0.487 | 96.3% | 0.014 | 0.504 | 95.0% | |
0.106 | 3.388 | 93.3% | 0.191 | 2.388 | 94.7% | 0.233 | 2.626 | 94.3% | |||
7 | 0.012 | 0.568 | 95.0% | 0.077 | 0.396 | 96.0% | 0.058 | 0.389 | 96.0% | ||
0.165 | 3.126 | 97.0% | 0.029 | 1.920 | 97.0% | 3.029 | 1.870 | 95.3% | |||
10 | 0.030 | 0.474 | 95.7% | 0.108 | 0.334 | 93.3% | 0.113 | 0.327 | 97.3% | ||
0.030 | 2.093 | 95.3% | 0.362 | 1.375 | 94.3% | 0.350 | 1.372 | 95.0% | |||
15 | 0.082 | 0.392 | 96.7% | 0.134 | 0.286 | 96.0% | 0.135 | 0.279 | 95.7% | ||
0.126 | 1.768 | 96.3% | 0.468 | 1.144 | 95.7% | 0.472 | 1.119 | 94.7% | |||
0.2, 1.5 | 4 | 0.028 | 0.557 | 96.0% | 0.070 | 0.397 | 96.0% | 0.066 | 0.395 | 94.0% | |
0.339 | 4.062 | 95.3% | 0.377 | 4.050 | 98.3% | 0.305 | 3.203 | 96.3% | |||
7 | 0.045 | 0.450 | 95.7% | 0.114 | 0.316 | 94.7% | 0.097 | 0.316 | 95.0% | ||
0.002 | 3.387 | 97.7% | 0.483 | 2.248 | 96.7% | 0.469 | 2.131 | 96.7% | |||
10 | 0.067 | 0.407 | 96.0% | 0.128 | 0.296 | 93.7% | 0.133 | 0.288 | 95.7% | ||
0.046 | 3.159 | 94.7% | 0.639 | 2.039 | 95.7% | 0.659 | 1.983 | 97.0% | |||
15 | 0.099 | 0.356 | 96.3% | 0.151 | 0.257 | 95.7% | 0.150 | 0.254 | 95.3% | ||
0.338 | 2.623 | 94.7% | 0.867 | 1.619 | 95.3% | 0.835 | 1.636 | 96.0% | |||
0.2, 3 | 4 | 0.084 | 0.344 | 95.7% | 0.127 | 0.282 | 94.7% | 0.124 | 0.281 | 95.0% | |
1.613 | 4.815 | 95.0% | 1.632 | 4.560 | 95.7% | 1.718 | 3.399 | 99.0% | |||
7 | 0.112 | 0.312 | 95.3% | 0.158 | 0.246 | 94.7% | 0.153 | 0.248 | 95.7% | ||
1.395 | 4.209 | 95.7% | 2.581 | 3.477 | 95.3% | 2.486 | 3.508 | 96.0% | |||
10 | 0.127 | 0.289 | 96.3% | 0.162 | 0.240 | 95.0% | 0.162 | 0.244 | 96.7% | ||
1.768 | 4.049 | 94.3% | 2.408 | 3.470 | 94.7% | 2.106 | 3.122 | 95.3% | |||
15 | 0.137 | 0.277 | 95.0% | 0.173 | 0.228 | 96.7% | 0.174 | 0.227 | 96.7% | ||
2.232 | 3.964 | 95.0% | 2.553 | 3.250 | 97.0% | 2.552 | 3.053 | 96.0% | |||
0.5, 0.75 | 4 | 0.0395 | 1.9725 | 0.9399 | 0.0310 | 1.2898 | 0.9533 | 0.0219 | 1.3072 | 0.9400 | |
0.2752 | 4.4734 | 0.9333 | 0.1702 | 3.1423 | 0.9500 | 0.2963 | 3.1355 | 0.9433 | |||
7 | 0.0520 | 1.4757 | 0.9500 | 0.1898 | 1.0052 | 0.9600 | 0.1261 | 1.0117 | 0.9600 | ||
0.2907 | 3.9506 | 0.9567 | 0.0536 | 2.0574 | 0.9633 | 0.3216 | 2.2705 | 0.9700 | |||
10 | 0.0681 | 1.2080 | 0.9467 | 0.2446 | 0.8759 | 0.9400 | 0.2751 | 0.8347 | 0.9600 | ||
0.0534 | 2.5092 | 0.9667 | 0.2412 | 1.5481 | 0.9567 | 0.2790 | 1.4747 | 0.9567 | |||
15 | 0.2056 | 0.9832 | 0.9633 | 0.3339 | 0.7175 | 0.9600 | 0.3365 | 0.6988 | 0.9567 | ||
0.1256 | 1.7818 | 0.9567 | 0.4521 | 1.1662 | 0.9633 | 0.4691 | 1.1243 | 0.9467 | |||
0.5, 1.5 | 4 | 0.0105 | 1.3514 | 0.9600 | 0.1852 | 1.0026 | 0.9600 | 0.1675 | 1.0249 | 0.9567 | |
0.3629 | 6.5529 | 0.9600 | 0.4315 | 4.9020 | 0.9400 | 0.2164 | 5.2904 | 0.9500 | |||
7 | 0.1532 | 1.1011 | 0.9500 | 0.2781 | 0.8357 | 0.9533 | 0.2407 | 0.8216 | 0.9667 | ||
0.3760 | 5.2429 | 0.9567 | 0.2514 | 3.5801 | 0.9667 | 0.2006 | 3.3864 | 0.9533 | |||
10 | 0.1800 | 1.0181 | 0.9600 | 0.3095 | 0.7711 | 0.9367 | 0.3212 | 0.7557 | 0.9633 | ||
0.4251 | 4.8780 | 0.9400 | 0.5150 | 3.0744 | 0.9400 | 0.5552 | 2.9837 | 0.9633 | |||
15 | 0.2595 | 0.8825 | 0.9667 | 0.3661 | 0.6651 | 0.9500 | 0.3695 | 0.6479 | 0.9567 | ||
0.1953 | 3.7385 | 0.9533 | 0.8980 | 2.3224 | 0.9567 | 0.9243 | 2.2329 | 0.9633 | |||
0.5, 3 | 4 | 0.1642 | 0.9694 | 0.9500 | 0.2860 | 0.7555 | 0.9433 | 0.2767 | 0.7550 | 0.9367 | |
0.6024 | 8.7044 | 0.9733 | 1.1364 | 5.5158 | 0.9600 | 0.7555 | 5.8615 | 0.9867 | |||
7 | 0.2484 | 0.8595 | 0.9567 | 0.3547 | 0.6613 | 0.9467 | 0.3280 | 0.6623 | 0.9433 | ||
0.7034 | 7.0543 | 0.9733 | 1.6482 | 4.6778 | 0.9500 | 1.4562 | 4.6331 | 0.9600 | |||
10 | 0.2761 | 0.8144 | 0.9600 | 0.3773 | 0.6553 | 0.9567 | 0.3776 | 0.6505 | 0.9500 | ||
1.2968 | 6.0141 | 0.9900 | 1.6491 | 4.0989 | 0.9400 | 1.6483 | 4.8747 | 0.9467 | |||
15 | 0.3068 | 0.7480 | 0.9533 | 0.4253 | 0.5784 | 0.9533 | 0.4256 | 0.5736 | 0.9467 | ||
0.6290 | 5.1354 | 0.9867 | 2.3825 | 3.7132 | 0.9600 | 2.3523 | 3.7115 | 0.9633 |
Models | Estimates | SE | KSS | PVKS | AIC | BIC | CAIC | HQIC | WS | AS | |
---|---|---|---|---|---|---|---|---|---|---|---|
KMKu | 1.3680 | 0.1123 | 0.0531 | 0.9749 | −386.4656 | −381.6522 | −386.3137 | −384.5331 | 0.0385 | 0.2875 | |
62.0471 | 22.1058 | ||||||||||
UW | 0.0024 | 0.0003 | 0.0737 | 0.7639 | −381.6038 | −376.7903 | −381.4519 | −379.6713 | 0.0988 | 0.7080 | |
4.3158 | 0.1099 | ||||||||||
Ku | 1.2399 | 0.1055 | 0.0597 | 0.9322 | −384.6698 | −379.8564 | −384.5179 | −382.7373 | 0.0601 | 0.4228 | |
55.7476 | 18.3042 | ||||||||||
UG | 0.0181 | 0.0071 | 0.1082 | 0.2924 | −363.6988 | −358.8853 | −363.5469 | −361.7662 | 0.2941 | 1.9827 | |
0.9759 | 0.0803 | ||||||||||
UEHL | 1.2515 | 0.1030 | 0.0574 | 0.9496 | −385.0542 | −380.2408 | −384.9023 | −383.1217 | 0.0555 | 0.3931 | |
29.3838 | 9.3197 |
Data | Observation | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | i = 1 | i = 2 | i = 3 | i = 4 | i = 5 | i = 6 | i = 7 | i = 8 | i = 9 | i = 10 |
0.0023 | 0.0065 | 0.0069 | 0.0067 | 0.0069 | 0.0093 | 0.0162 | 0.0093 | 0.0116 | 0.0116 | |
i = 11 | i = 12 | i = 13 | i = 14 | i = 15 | i = 16 | i = 17 | i = 18 | i = 19 | i = 20 | |
0.0159 | 0.0162 | 0.0161 | 0.0115 | 0.0162 | 0.0161 | 0.0187 | 0.0162 | 0.0239 | 0.023 | |
i = 21 | i = 22 | i = 23 | i = 24 | i = 25 | i = 26 | i = 27 | i = 28 | i = 29 | i = 30 | |
0.0207 | 0.0138 | 0.0312 | 0.03 | 0.03 | 0.0255 | 0.023 | 0.0346 | 0.0297 | 0.0314 | |
i = 31 | i = 32 | i = 33 | i = 34 | i = 35 | i = 36 | i = 37 | i = 38 | i = 39 | i = 40 | |
0.0346 | 0.0312 | 0.0394 | 0.0379 | 0.0588 | 0.0468 | 0.0468 | 0.0419 | 0.0379 | 0.0521 | |
i = 41 | i = 42 | i = 43 | i = 44 | i = 45 | i = 46 | i = 47 | i = 48 | i = 49 | i = 50 | |
0.0425 | 0.0471 | 0.0588 | 0.0715 | 0.0679 | 0.0942 | 0.0628 | 0.0766 | 0.1343 | 0.1343 | |
2 | i = 1 | i = 2 | i = 3 | i = 4 | i = 5 | i = 6 | i = 7 | i = 8 | i = 9 | i = 10 |
0.01133 | 0.01796 | 0.02288 | 0.04829 | 0.02288 | 0.0385 | 0.0439 | 0.0507 | 0.0605 | 0.0605 | |
i = 11 | i = 12 | i = 13 | i = 14 | i = 15 | ||||||
0.0955 | 0.0818 | 0.1476 | 0.1099 | 0.1476 | ||||||
3 | i = 1 | i = 2 | i = 3 | i = 4 | i = 5 | i = 6 | i = 7 | i = 8 | i = 9 | i = 10 |
0.032 | 0.023 | 0.032 | 0.188 | 0.169 | 0.105 | 0.216 | 0.361 | 0.361 | 0.463 | |
i = 11 | i = 12 | i = 13 | i = 14 | i = 15 | ||||||
0.642 | 0.674 | 0.823 | 0.823 | 0.926 |
Data | Cycle | Observation | ||||
---|---|---|---|---|---|---|
i = 1 | i = 2 | i = 3 | i = 4 | i = 5 | ||
1 | s = 1 | 0.0093 | 0.0093 | 0.0464 | 0.0384 | 0.096 |
s = 2 | 0.0065 | 0.0202 | 0.0225 | 0.0715 | 0.1223 | |
s = 2 | 0.0093 | 0.0093 | 0.0355 | 0.0255 | 0.096 | |
s = 3 | 0.0115 | 0.0187 | 0.0116 | 0.0464 | 0.1781 | |
s = 3 | 0.0115 | 0.0384 | 0.0187 | 0.0715 | 0.0942 | |
s = 4 | 0.0116 | 0.0115 | 0.0255 | 0.0314 | 0.1781 | |
s = 4 | 0.0115 | 0.0314 | 0.0202 | 0.0501 | 0.0942 | |
s = 5 | 0.0093 | 0.0161 | 0.0255 | 0.0384 | 0.0715 | |
s = 5 | 0.0161 | 0.0355 | 0.0255 | 0.0115 | 0.0501 | |
s = 6 | 0.0065 | 0.0115 | 0.0255 | 0.0501 | 0.1223 | |
2 | i = 1 | i = 2 | i = 3 | i = 4 | i = 5 | |
s = 1 | 0.0180 | 0.0385 | 0.0095 | 0.0231 | 0.1099 | |
s = 2 | 0.0074 | 0.0180 | 0.0180 | 0.0515 | 0.1099 | |
s = 3 | 0.0074 | 0.0212 | 0.0212 | 0.0385 | 0.1388 | |
3 | i = 1 | i = 2 | i = 3 | i = 4 | i = 5 | |
s = 1 | 0.032 | 0.105 | 0.463 | 0.169 | 0.752 | |
s = 2 | 0.023 | 0.395 | 0.169 | 0.311 | 0.823 | |
s = 3 | 0.023 | 0.032 | 0.127 | 0.255 | 0.823 |
Estimates | SE | Lower | Upper | ||
---|---|---|---|---|---|
SRS | 1.3751 | 0.0201 | 1.3357 | 1.4146 | |
56.2365 | 27.5437 | 2.2509 | 110.2221 | ||
RSS s = 1 | 1.4676 | 0.0029 | 1.4619 | 1.4733 | |
91.3018 | 13.2753 | 65.2823 | 117.3213 | ||
RSS s = 10 | 1.2903 | 0.0100 | 1.2706 | 1.3100 | |
43.7122 | 9.5155 | 25.0618 | 62.3627 |
Models | Estimates | SE | KSS | PVKS | AIC | BIC | CAIC | HQIC | WS | AS | |
---|---|---|---|---|---|---|---|---|---|---|---|
KMKu | 1.5517 | 0.2452 | 0.1025 | 0.9315 | −94.8552 | −92.4174 | −94.3097 | −94.1791 | 0.0261 | 0.1989 | |
55.3232 | 37.2191 | ||||||||||
UW | 0.0054 | 0.0031 | 0.1362 | 0.6923 | −93.1603 | −90.7226 | −92.6149 | −92.4842 | 0.0652 | 0.3860 | |
4.1597 | 0.4182 | ||||||||||
Ku | 1.4164 | 0.2303 | 0.1022 | 0.9329 | −94.6877 | −92.2499 | −94.1422 | −94.0115 | 0.0278 | 0.2151 | |
50.9406 | 31.3225 | ||||||||||
B | 1.7485 | 0.4553 | 0.1031 | 0.9284 | −94.8924 | −92.4547 | −94.3470 | −94.2163 | 0.0258 | 0.1944 | |
29.5605 | 8.8186 | ||||||||||
MOKu | 0.4536 | 0.6246 | 0.1047 | 0.9205 | −92.9571 | −89.3005 | −91.8143 | −91.9429 | 0.0262 | 0.1994 | |
1.6814 | 0.4880 | ||||||||||
66.1138 | 51.4394 | ||||||||||
UG | 0.0167 | 0.0125 | 0.1601 | 0.4936 | −89.1731 | −86.7354 | −88.6277 | −88.4970 | 0.1241 | 0.7255 | |
1.1446 | 0.1796 | ||||||||||
MOETL | 0.0062 | 0.0046 | 0.1058 | 0.9147 | −92.2698 | −89.8320 | −91.7243 | −91.5936 | 0.0559 | 0.3477 | |
2.0660 | 0.2976 | ||||||||||
UEHL | 1.4306 | 0.2250 | 0.1026 | 0.9311 | −94.6713 | −92.2336 | −94.1259 | −93.9952 | 0.0274 | 0.2127 | |
26.8822 | 15.9639 |
Estimates | SE | Lower | Upper | ||
---|---|---|---|---|---|
SRS | 1.6497 | 0.1006 | 1.4525 | 1.8468 | |
111.3630 | 47.0753 | 19.0954 | 203.6306 | ||
RSS s = 1 | 1.6652 | 0.0322 | 1.6022 | 1.7282 | |
59.8835 | 21.3712 | 17.9959 | 101.7711 | ||
RSS s = 3 | 1.2537 | 0.0313 | 1.1898 | 1.3176 | |
39.7230 | 19.7096 | 1.0922 | 78.3539 |
Models | Estimates | SE | KSS | PVKS | AIC | BIC | CAIC | HQIC | WS | AS | |
---|---|---|---|---|---|---|---|---|---|---|---|
KMKu | 1.0826 | 0.2138 | 0.0569 | 0.9999 | −3.0417 | −0.2393 | −2.5973 | −2.1452 | 0.0151 | 0.1215 | |
1.3797 | 0.3935 | ||||||||||
Ku | 0.9627 | 0.2017 | 0.0650 | 0.9987 | −2.6221 | 0.1803 | −2.1776 | −1.7256 | 0.0183 | 0.1551 | |
1.6084 | 0.4137 | ||||||||||
B | 0.9667 | 0.2238 | 0.0669 | 0.9979 | −2.6101 | 0.1923 | −2.1657 | −1.7136 | 0.0184 | 0.1559 | |
1.6205 | 0.4107 | ||||||||||
MOKu | 0.4365 | 0.4732 | 0.0628 | 0.9992 | −1.2087 | 2.9949 | −0.2856 | 0.1361 | 0.0152 | 0.1224 | |
1.1872 | 0.3472 | ||||||||||
1.2585 | 0.6458 | ||||||||||
MOETL | 1.0929 | 0.7021 | 0.0672 | 0.9978 | −1.8272 | 0.9752 | −1.3828 | −0.9307 | 0.0203 | 0.1710 | |
1.0628 | 0.3883 |
Estimates | SE | Lower | Upper | ||
---|---|---|---|---|---|
SRS | 0.8111 | 0.0552 | 0.7030 | 0.9192 | |
1.1550 | 0.2076 | 0.7481 | 1.5619 | ||
RSS s = 1 | 1.1444 | 0.0279 | 1.0897 | 1.1992 | |
1.6753 | 0.1506 | 1.3801 | 1.9705 | ||
RSS s = 2 | 0.8179 | 0.0288 | 0.7614 | 0.8744 | |
1.4021 | 0.1488 | 1.0333 | 1.7709 |
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Alotaibi, N.; Elbatal, I.; Shrahili, M.; Al-Moisheer, A.S.; Elgarhy, M.; Almetwally, E.M. Statistical Inference for the Kavya–Manoharan Kumaraswamy Model under Ranked Set Sampling with Applications. Symmetry 2023, 15, 587. https://doi.org/10.3390/sym15030587
Alotaibi N, Elbatal I, Shrahili M, Al-Moisheer AS, Elgarhy M, Almetwally EM. Statistical Inference for the Kavya–Manoharan Kumaraswamy Model under Ranked Set Sampling with Applications. Symmetry. 2023; 15(3):587. https://doi.org/10.3390/sym15030587
Chicago/Turabian StyleAlotaibi, Naif, Ibrahim Elbatal, Mansour Shrahili, A. S. Al-Moisheer, Mohammed Elgarhy, and Ehab M. Almetwally. 2023. "Statistical Inference for the Kavya–Manoharan Kumaraswamy Model under Ranked Set Sampling with Applications" Symmetry 15, no. 3: 587. https://doi.org/10.3390/sym15030587