Table 3 Bias, MSE and MRE under different estimation methods, when \(\tau = 0.75\).

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]	0.05293 [2]	0.05485 [3]	0.06101 [7]	0.05633 [5]	0.06264 [8]	0.05539 [4]	0.05141 [1]	0.06663 [9]	0.06980 [10]	0.05985 [6]
	MSE [ranks]	0.00426 [2]	0.00478 [3]	0.00598 [7]	0.00510 [5]	0.00617 [8]	0.00505 [4]	0.00418 [1]	0.00695 [9]	0.00786 [10]	0.00570 [6]
	MRE [ranks]	0.07057 [2]	0.07313 [3]	0.08135 [7]	0.07510 [5]	0.08352 [8]	0.07385 [4]	0.06855 [1]	0.08884 [9]	0.09307 [10]	0.07980 [6]
	Sum ranks [ranks]	6 [2]	9 [3]	21 [7]	15 [5]	24 [8]	12 [4]	3 [1]	27 [9]	30 [10]	18 [6]
50	Bias [ranks]	0.03584 [1]	0.04009 [4]	0.04434 [6]	0.03837 [2]	0.04482 [7]	0.04138 [5]	0.03947 [3]	0.04666 [9]	0.04743 [10]	0.04507 [8]
	MSE [ranks]	0.00200 [1]	0.00259 [4]	0.00310 [6]	0.00230 [2]	0.00313 [7]	0.00272 [5]	0.00236 [3]	0.00356 [9]	0.00360 [10]	0.00316 [8]
	MRE [ranks]	0.04779 [1]	0.05346 [4]	0.05912 [6]	0.05116 [2]	0.05976 [7]	0.05517 [5]	0.05262 [3]	0.06221 [9]	0.06324 [10]	0.06009 [8]
	Sum ranks [ranks]	3 [1]	12 [4]	18 [6]	6 [2]	21 [7]	15 [5]	9 [3]	27 [9]	30 [10]	24 [8]
75	Bias [ranks]	0.03133 [1]	0.03298 [5]	0.03438 [6]	0.03270 [4]	0.03462 [8]	0.03226 [3]	0.03216 [2]	0.03776 [9]	0.04031 [10]	0.03439 [7]
	MSE [ranks]	0.00151 [1]	0.00169 [5]	0.00191 [8]	0.00167 [3]	0.00190 [7]	0.00168 [4]	0.00164 [2]	0.00230 [9]	0.00264 [10]	0.00189 [6]
	MRE [ranks]	0.04178 [1]	0.04398 [5]	0.04584 [6]	0.04360 [4]	0.04616 [8]	0.04302 [3]	0.04288 [2]	0.05035 [9]	0.05374 [10]	0.04586 [7]
	Sum ranks [ranks]	3 [1]	15 [5]	20 [6.5]	11 [4]	23 [8]	10 [3]	6 [2]	27 [9]	30 [10]	20 [6.5]
150	Bias [ranks]	0.02079 [1]	0.02186 [2]	0.02386 [6]	0.02198 [3]	0.02445 [7]	0.02371 [5]	0.02200 [4]	0.02746 [9]	0.02941 [10]	0.02513 [8]
	MSE [ranks]	0.00069 [1]	0.00078 [3]	0.00090 [6]	0.00079 [4]	0.00094 [7]	0.00088 [5]	0.00074 [2]	0.00116 [9]	0.00139 [10]	0.00103 [8]
	MRE [ranks]	0.02772 [1]	0.02915 [2]	0.03181 [6]	0.02931 [3]	0.03259 [7]	0.03161 [5]	0.02934 [4]	0.03661 [9]	0.03922 [10]	0.03351 [8]
	Sum ranks [ranks]	3 [1]	7 [2]	18 [6]	10 [3]	21 [7]	15 [5]	10 [4]	27 [9]	30 [10]	24 [8]
250	Bias [ranks]	0.01605 [1]	0.01810 [5]	0.01861 [7]	0.01695 [4]	0.01900 [8]	0.01670 [3]	0.01667 [2]	0.02098 [9]	0.02127 [10]	0.01817 [6]
	MSE [ranks]	0.00041 [1]	0.00049 [5]	0.00054 [7]	0.00046 [4]	0.00056 [8]	0.00044 [2.5]	0.00044 [2.5]	0.00070 [9]	0.00072 [10]	0.00051 [6]
	MRE [ranks]	0.02140 [1]	0.02413 [5]	0.02482 [7]	0.02260 [4]	0.02534 [8]	0.02226 [3]	0.02223 [2]	0.02797 [9]	0.02836 [10]	0.02423 [6]
	Sum ranks [ranks]	3 [1]	15 [5]	21 [7]	12 [4]	24 [8]	8.5 [3]	6.5 [2]	27 [9]	30 [10]	18 [6]
400	Bias [ranks]	0.01262 [1]	0.01271 [2]	0.01476 [5]	0.01360 [3]	0.01510 [6]	0.01530 [7]	0.01409 [4]	0.01641 [10]	0.01611 [8]	0.01619 [9]
	MSE [ranks]	0.00025 [1]	0.00026 [2]	0.00034 [5]	0.00028 [3]	0.00036 [7]	0.00035 [6]	0.00032 [4]	0.00042 [10]	0.00041 [9]	0.00040 [8]
	MRE [ranks]	0.01682 [1]	0.01695 [2]	0.01968 [5]	0.01814 [3]	0.02013 [6]	0.02040 [7]	0.01879 [4]	0.02188 [10]	0.02148 [8]	0.02159 [9]
	Sum ranks [ranks]	3 [1]	6 [2]	15 [5]	9 [3]	19 [6]	20 [7]	12 [4]	30 [10]	25 [8]	26 [9]
600	Bias [ranks]	0.01008 [1]	0.01203 [6]	0.01141 [5]	0.01124 [3]	0.01243 [8]	0.01126 [4]	0.01076 [2]	0.01385 [10]	0.01357 [9]	0.01229 [7]
	MSE [ranks]	0.00016 [1]	0.00022 [6]	0.00021 [5]	0.00020 [3.5]	0.00024 [7]	0.00020 [3.5]	0.00018 [2]	0.00030 [10]	0.00029 [9]	0.00025 [8]
	MRE [ranks]	0.01345 [1]	0.01604 [6]	0.01521 [5]	0.01499 [3]	0.01657 [8]	0.01502 [4]	0.01434 [2]	0.01847 [10]	0.01809 [9]	0.01639 [7]
	Sum ranks [ranks]	3 [1]	18 [6]	15 [5]	9.5 [3]	23 [8]	11.5 [4]	6 [2]	30 [10]	27 [9]	22 [7]

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Table 3 Bias, MSE and MRE under different estimation methods, when \(\tau = 0.75\).

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