Table 2 Bias, MSE and MRE under different estimation methods, when \(\tau = 0.5\)

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]	0.03416 [1]	0.03691 [4]	0.03867 [6]	0.03688 [3]	0.04097 [7]	0.03818 [5]	0.03457 [2]	0.04459 [10]	0.04405 [9]	0.04363 [8]
	MSE [ranks]	0.00180 [1]	0.00224 [4]	0.00232 [6]	0.00222 [3]	0.00256 [7]	0.00229 [5]	0.00191 [2]	0.00324 [10]	0.00321 [9]	0.00296 [8]
	MRE [ranks]	0.06831 [1]	0.07382 [4]	0.07734 [6]	0.07377 [3]	0.08194 [7]	0.07635 [5]	0.06914 [2]	0.08918 [10]	0.08810 [9]	0.08726 [8]
	Sum ranks [ranks]	3 [1]	12 [4]	18 [6]	9 [3]	21 [7]	15 [5]	6 [2]	30 [10]	27 [9]	24 [8]
50	Bias [ranks]	0.02475 [1]	0.02696 [2]	0.02816 [5]	0.02817 [6]	0.02812 [4]	0.02825 [7]	0.02764 [3]	0.03536 [10]	0.03412 [9]	0.03208 [8]
	MSE [ranks]	0.00098 [1]	0.00116 [2]	0.00128 [6]	0.00126 [4.5]	0.00126 [4.5]	0.00129 [7]	0.00118 [3]	0.00185 [9]	0.00195 [10]	0.00165 [8]
	MRE [ranks]	0.04950 [1]	0.05392 [2]	0.05633 [5]	0.05634 [6]	0.05624 [4]	0.05650 [7]	0.05528 [3]	0.07071 [10]	0.06824 [9]	0.06416 [8]
	Sum ranks [ranks]	3 [1]	6 [2]	16 [5]	16.5 [6]	13.5 [4]	21 [7]	9 [3]	29 [10]	28 [9]	24 [8]
75	Bias [ranks]	0.02242 [3]	0.02231 [2]	0.02499 [8]	0.02284 [4]	0.02473 [7]	0.02375 [5]	0.02128 [1]	0.02815 [10]	0.02783 [9]	0.02368 [6]
	MSE [ranks]	0.00078 [2]	0.00081 [4]	0.00099 [8]	0.00080 [3]	0.00098 [7]	0.00089 [5]	0.00072 [1]	0.00123 [9]	0.00125 [10]	0.00090 [6]
	MRE [ranks]	0.04484 [3]	0.04462 [2]	0.04998 [8]	0.04568 [4]	0.04946 [7]	0.04751 [6]	0.04256 [1]	0.05631 [10]	0.05566 [9]	0.04736 [5]
	Sum ranks [ranks]	8 [2.5]	8 [2.5]	24 [8]	11 [4]	21 [7]	16 [5]	3 [1]	29 [10]	28 [9]	17 [6]
150	Bias [ranks]	0.01492 [1]	0.01928 [8]	0.01748 [4]	0.01613 [3]	0.01923 [7]	0.01873 [6]	0.01545 [2]	0.02344 [10]	0.01986 [9]	0.01777 [5]
	MSE [ranks]	0.00035 [1]	0.00056 [7.5]	0.00051 [4.5]	0.00041 [3]	0.00056 [7.5]	0.00054 [6]	0.00036 [2]	0.00083 [10]	0.00063 [9]	0.00051 [4.5]
	MRE [ranks]	0.02984 [1]	0.03857 [8]	0.03497 [4]	0.03228 [3]	0.03846 [7]	0.03746 [6]	0.03090 [2]	0.04689 [10]	0.03973 [9]	0.03554 [5]
	Sum ranks [ranks]	3 [1]	23.5 [8]	13.5 [4]	9 [3]	21.5 [7]	18 [6]	6 [2]	30 [10]	27 [9]	14.5 [5]
250	Bias [ranks]	0.01219 [1]	0.01519 [6]	0.01458 [5]	0.01296 [3]	0.01524 [7]	0.01643 [9]	0.01275 [2]	0.02093 [10]	0.01554 [8]	0.01351 [4]
	MSE [ranks]	0.00023 [1]	0.00035 [6.5]	0.00032 [5]	0.00026 [3]	0.00035 [6.5]	0.00040 [9]	0.00025 [2]	0.00062 [10]	0.00038 [8]	0.00028 [4]
	MRE [ranks]	0.02437 [1]	0.03038 [6]	0.02916 [5]	0.02591 [3]	0.03047 [7]	0.03285 [9]	0.02549 [2]	0.04186 [10]	0.03109 [8]	0.02703 [4]
	Sum ranks [ranks]	3 [1]	18.5 [6]	15 [5]	9 [3]	20.5 [7]	27 [9]	6 [2]	30 [10]	24 [8]	12 [4]
400	Bias [ranks]	0.01019 [1]	0.01324 [6]	0.01339 [7]	0.01114 [3.5]	0.01356 [8]	0.01472 [9]	0.01114 [3.5]	0.01931 [10]	0.01188 [5]	0.01021 [2]
	MSE [ranks]	0.00016 [1]	0.00025 [6]	0.00026 [7]	0.00019 [3.5]	0.00028 [8]	0.00031 [9]	0.00019 [3.5]	0.00050 [10]	0.00022 [5]	0.00017 [2]
	MRE [ranks]	0.02037 [1]	0.02648 [6]	0.02679 [7]	0.02229 [4]	0.02711 [8]	0.02944 [9]	0.02228 [3]	0.03862 [10]	0.02375 [5]	0.02042 [2]
	Sum ranks [ranks]	3 [1]	18 [6]	21 [7]	11 [4]	24 [8]	27 [9]	10 [3]	30 [10]	15 [5]	6 [2]
600	Bias [ranks]	0.00877 [1]	0.01238 [6]	0.01250 [7]	0.00993 [3]	0.01276 [8]	0.01439 [9]	0.00995 [4]	0.01958 [10]	0.01018 [5]	0.00893 [2]
	MSE [ranks]	0.00011 [1]	0.00021 [6]	0.00023 [7.5]	0.00014 [3]	0.00023 [7.5]	0.00027 [9]	0.00015 [4]	0.00048 [10]	0.00016 [5]	0.00012 [2]
	MRE [ranks]	0.01754 [1]	0.02476 [6]	0.02500 [7]	0.01986 [3]	0.02551 [8]	0.02877 [9]	0.01989 [4]	0.03917 [10]	0.02035 [5]	0.01785 [2]
	Sum ranks [ranks]	3 [1]	18 [6]	21.5 [7]	9 [3]	23.5 [8]	27 [9]	12 [4]	30 [10]	15 [5]	6 [2]

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Table 2 Bias, MSE and MRE under different estimation methods, when \(\tau = 0.5\)

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