Table 6 Bias, MSE and MRE under different estimation methods, when \(\tau = 2.5\)

From: Development of a novel extension of Rayleigh distribution with application to COVID-19 data

Sample size	Value	\(\Delta_{1}\)	\(\Delta_{2}\)	\(\Delta_{3}\)	\(\Delta_{4}\)	\(\Delta_{5}\)	\(\Delta_{6}\)	\(\Delta_{7}\)	\(\Delta_{8}\)	\(\Delta_{9}\)	\(\Delta_{10}\)
25	Bias [ranks]	0.17276 [1]	0.19399 [5]	0.20248 [6]	0.18638 [3]	0.20867 [8]	0.19188 [4]	0.17999 [2]	0.20865 [7]	0.22330 [10]	0.20874 [9]
	MSE [ranks]	0.04633 [1]	0.05708 [4]	0.06724 [6]	0.05395 [3]	0.06947 [8]	0.05870 [5]	0.05127 [2]	0.07086 [9]	0.07946 [10]	0.06920 [7]
	MRE [ranks]	0.06910 [1]	0.07760 [5]	0.08099 [6]	0.07455 [3]	0.08347 [8]	0.07675 [4]	0.07200 [2]	0.08346 [7]	0.08932 [10]	0.08349 [9]
	Sum ranks [ranks]	3 [1]	14 [5]	18 [6]	9 [3]	24 [8]	13 [4]	6 [2]	23 [7]	30 [10]	25 [9]
50	Bias [ranks]	0.12195 [2]	0.13244 [5]	0.14424 [6]	0.12136 [1]	0.14855 [8]	0.13232 [4]	0.12490 [3]	0.15680 [9]	0.16451 [10]	0.14620 [7]
	MSE [ranks]	0.02274 [1]	0.02730 [4]	0.03208 [6]	0.02345 [2]	0.03548 [8]	0.02757 [5]	0.02390 [3]	0.03859 [9]	0.04359 [10]	0.03504 [7]
	MRE [ranks]	0.04878 [2]	0.05297 [5]	0.05770 [6]	0.04854 [1]	0.05942 [8]	0.05293 [4]	0.04996 [3]	0.06272 [9]	0.06580 [10]	0.05848 [7]
	Sum ranks [ranks]	5 [2]	14 [5]	18 [6]	4 [1]	24 [8]	13 [4]	9 [3]	27 [9]	30 [10]	21 [7]
75	Bias [ranks]	0.10978 [6]	0.10784 [5]	0.11161 [7]	0.10116 [1]	0.10758 [4]	0.10329 [2]	0.10632 [3]	0.13049 [9]	0.13347 [10]	0.12322 [8]
	MSE [ranks]	0.01897 [6]	0.01800 [4]	0.01930 [7]	0.01569 [1]	0.01866 [5]	0.01676 [2]	0.01788 [3]	0.02634 [9]	0.02850 [10]	0.02331 [8]
	MRE [ranks]	0.04391 [6]	0.04314 [5]	0.04465 [7]	0.04046 [1]	0.04303 [4]	0.04131 [2]	0.04253 [3]	0.05220 [9]	0.05339 [10]	0.04929 [8]
	Sum ranks [ranks]	18 [6]	14 [5]	21 [7]	3 [1]	13 [4]	6 [2]	9 [3]	27 [9]	30 [10]	24 [8]
150	Bias [ranks]	0.07200 [1]	0.08106 [6]	0.08449 [7]	0.07341 [2]	0.08070 [5]	0.07889 [4]	0.07570 [3]	0.09603 [10]	0.09429 [9]	0.08582 [8]
	MSE [ranks]	0.00803 [1]	0.00993 [5]	0.01089 [7]	0.00843 [2]	0.01053 [6]	0.00928 [3]	0.00934 [4]	0.01395 [9]	0.01402 [10]	0.01201 [8]
	MRE [ranks]	0.02880 [1]	0.03242 [6]	0.03380 [7]	0.02937 [2]	0.03228 [5]	0.03156 [4]	0.03028 [3]	0.03841 [10]	0.03772 [9]	0.03433 [8]
	Sum ranks [ranks]	3 [1]	17 [6]	21 [7]	6 [2]	16 [5]	11 [4]	10 [3]	29 [10]	28 [9]	24 [8]
250	Bias [ranks]	0.05239 [1]	0.06149 [6]	0.06117 [4]	0.05690 [3]	0.06578 [8]	0.06122 [5]	0.05330 [2]	0.06756 [9]	0.07240 [10]	0.06480 [7]
	MSE [ranks]	0.00433 [1]	0.00585 [5]	0.00567 [4]	0.00521 [3]	0.00695 [8]	0.00612 [6]	0.00443 [2]	0.00714 [9]	0.00842 [10]	0.00675 [7]
	MRE [ranks]	0.02095 [1]	0.02459 [6]	0.02447 [4]	0.02276 [3]	0.02631 [8]	0.02449 [5]	0.02132 [2]	0.02703 [9]	0.02896 [10]	0.02592 [7]
	Sum ranks [ranks]	3 [1]	17 [6]	12 [4]	9 [3]	24 [8]	16 [5]	6 [2]	27 [9]	30 [10]	21 [7]
400	Bias [ranks]	0.04403 [2]	0.04686 [4]	0.04716 [5]	0.04218 [1]	0.04947 [7]	0.04826 [6]	0.04623 [3]	0.05786 [10]	0.05350 [9]	0.05224 [8]
	MSE [ranks]	0.00306 [2]	0.00338 [3]	0.00339 [5]	0.00277 [1]	0.00384 [7]	0.00362 [6]	0.00340 [4]	0.00524 [10]	0.00467 [9]	0.00434 [8]
	MRE [ranks]	0.01761 [2]	0.01874 [4]	0.01886 [5]	0.01687 [1]	0.01979 [7]	0.01930 [6]	0.01849 [3]	0.02314 [10]	0.02140 [9]	0.02089 [8]
	Sum ranks [ranks]	6 [2]	11 [4]	15 [5]	3 [1]	21 [7]	18 [6]	10 [3]	30 [10]	27 [9]	24 [8]
600	Bias [ranks]	0.03452 [2]	0.03654 [3]	0.03956 [6]	0.03451 [1]	0.04197 [8]	0.03853 [5]	0.03680 [4]	0.04521 [9]	0.04615 [10]	0.04196 [7]
	MSE [ranks]	0.00187 [2]	0.00213 [4]	0.00249 [6]	0.00185 [1]	0.00274 [7.5]	0.00228 [5]	0.00208 [3]	0.00325 [9]	0.00337 [10]	0.00274 [7.5]
	MRE [ranks]	0.01381 [2]	0.01462 [3]	0.01583 [6]	0.01380 [1]	0.01679 [8]	0.01541 [5]	0.01472 [4]	0.01808 [9]	0.01846 [10]	0.01678 [7]
	Sum ranks [ranks]	6 [2]	10 [3]	18 [6]	3 [1]	23.5 [8]	15 [5]	11 [4]	27 [9]	30 [10]	21.5 [7]

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Table 6 Bias, MSE and MRE under different estimation methods, when \(\tau = 2.5\)

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