Table 7 Bias, MSE and MRE under different estimation methods, when \(\tau = 4.0\)

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]	1.02902 [10]	0.34500 [4]	0.32686 [1]	0.75053 [8]	0.34155 [3]	0.32757 [2]	0.44777 [7]	0.36026 [5]	0.39297 [6]	0.87676 [9]
	MSE [ranks]	1.64617 [9]	0.20585 [5]	0.17419 [2]	1.40051 [8]	0.18148 [3]	0.16494 [1]	0.58860 [7]	0.20479 [4]	0.23302 [6]	1.84185 [10]
	MRE [ranks]	0.25725 [10]	0.08625 [4]	0.08172 [1]	0.18763 [8]	0.08539 [3]	0.08189 [2]	0.11194 [7]	0.09007 [5]	0.09824 [6]	0.21919 [9]
	Sum ranks [ranks]	29 [10]	13 [4]	4 [1]	24 [8]	9 [3]	5 [2]	21 [7]	14 [5]	18 [6]	28 [9]
50	Bias [ranks]	0.87099 [10]	0.25112 [2]	0.27264 [5]	0.58757 [8]	0.24499 [1]	0.25984 [3]	0.27144 [4]	0.27575 [6]	0.38328 [7]	0.76460 [9]
	MSE [ranks]	1.19157 [9]	0.09916 [2]	0.11030 [4]	0.84105 [8]	0.09622 [1]	0.10079 [3]	0.12163 [6]	0.12154 [5]	0.19901 [7]	1.48645 [10]
	MRE [ranks]	0.21775 [10]	0.06278 [2]	0.06816 [5]	0.14689 [8]	0.06125 [1]	0.06496 [3]	0.06786 [4]	0.06894 [6]	0.09582 [7]	0.19115 [9]
	Sum ranks [ranks]	29 [10]	6 [2]	14 [4.5]	24 [8]	3 [1]	9 [3]	14 [4.5]	17 [6]	21 [7]	28 [9]
75	Bias [ranks]	0.81805 [10]	0.22158 [1]	0.23166 [3]	0.55375 [8]	0.22599 [2]	0.24395 [5]	0.24137 [4]	0.24563 [6]	0.38934 [7]	0.79866 [9]
	MSE [ranks]	1.05099 [9]	0.08078 [3]	0.07960 [2]	0.69288 [8]	0.07549 [1]	0.08803 [4]	0.09998 [6]	0.09082 [5]	0.18896 [7]	1.40101 [10]
	MRE [ranks]	0.20451 [10]	0.05540 [1]	0.05792 [3]	0.13844 [8]	0.05650 [2]	0.06099 [5]	0.06034 [4]	0.06141 [6]	0.09734 [7]	0.19967 [9]
	Sum ranks [ranks]	29 [10]	5 [1.5]	8 [3]	24 [8]	5 [1.5]	14 [4.5]	14 [4.5]	17 [6]	21 [7]	28 [9]
150	Bias [ranks]	0.78184 [10]	0.16814 [1]	0.19707 [3]	0.52600 [8]	0.21034 [5]	0.21433 [6]	0.16924 [2]	0.20203 [4]	0.38831 [7]	0.79456 [9]
	MSE [ranks]	0.79335 [9]	0.04408 [1]	0.05352 [3]	0.55874 [8]	0.06130 [5]	0.06251 [6]	0.04503 [2]	0.06059 [4]	0.17203 [7]	1.21483 [10]
	MRE [ranks]	0.19546 [9]	0.04204 [1]	0.04927 [3]	0.13150 [8]	0.05258 [5]	0.05358 [6]	0.04231 [2]	0.05051 [4]	0.09708 [7]	0.19864 [10]
	Sum ranks [ranks]	28 [9]	3 [1]	9 [3]	24 [8]	15 [5]	18 [6]	6 [2]	12 [4]	21 [7]	29 [10]
250	Bias [ranks]	0.76747 [9]	0.15749 [2]	0.20141 [6]	0.42780 [8]	0.19724 [4]	0.19889 [5]	0.15027 [1]	0.19703 [3]	0.39409 [7]	0.81945 [10]
	MSE [ranks]	0.72031 [9]	0.03562 [2]	0.05211 [6]	0.37341 [8]	0.05029 [3]	0.05101 [4]	0.03488 [1]	0.05141 [5]	0.16837 [7]	1.17555 [10]
	MRE [ranks]	0.19187 [9]	0.03937 [2]	0.05035 [6]	0.10695 [8]	0.04931 [4]	0.04972 [5]	0.03757 [1]	0.04926 [3]	0.09852 [7]	0.20486 [10]
	Sum ranks [ranks]	27 [9]	6 [2]	18 [6]	24 [8]	11 [3.5]	14 [5]	3 [1]	11 [3.5]	21 [7]	30 [10]
400	Bias [ranks]	0.77947 [10]	0.15336 [2]	0.19569 [4]	0.42958 [8]	0.20145 [5]	0.20222 [6]	0.13795 [1]	0.18055 [3]	0.38918 [7]	0.71606 [9]
	MSE [ranks]	0.69235 [9]	0.03204 [2]	0.04598 [4]	0.37133 [8]	0.04870 [5]	0.04896 [6]	0.02953 [1]	0.04127 [3]	0.15990 [7]	0.98755 [10]
	MRE [ranks]	0.19487 [10]	0.03834 [2]	0.05035 [4]	0.10740 [8]	0.05036 [5]	0.05055 [6]	0.03449 [1]	0.04514 [3]	0.09729 [7]	0.17901 [9]
	Sum ranks [ranks]	29 [10]	6 [2]	12 [4]	24 [8]	15 [5]	18 [6]	3 [1]	9 [3]	21 [7]	28 [9]
600	Bias [ranks]	0.75989 [10]	0.14187 [2]	0.19454 [5]	0.40699 [8]	0.19436 [4]	0.19852 [6]	0.12127 [1]	0.18194 [3]	0.38821 [7]	0.67150 [9]
	MSE [ranks]	0.62454 [9]	0.02594 [2]	0.04892 [6]	0.34579 [8]	0.04374 [4]	0.04429 [5]	0.02184 [1]	0.03958 [3]	0.15664 [7]	0.89907 [10]
	MRE [ranks]	0.18997 [10]	0.03547 [2]	0.04864 [5]	0.10175 [8]	0.04859 [4]	0.04963 [6]	0.03032 [1]	0.04549 [3]	0.09705 [7]	0.16787 [9]
	Sum ranks [ranks]	29 [10]	6 [2]	16 [5]	24 [8]	12 [4]	17 [6]	3 [1]	9 [3]	21 [7]	28 [9]

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Table 7 Bias, MSE and MRE under different estimation methods, when \(\tau = 4.0\)

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