Table 4 Bias, MSE and MRE under different estimation methods, when \(\tau = 1.25\).

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]	0.08344 [1]	0.09385 [4]	0.10478 [8]	0.08977 [2]	0.09612 [5]	0.10131 [6]	0.09123 [3]	0.10394 [7]	0.11235 [10]	0.10642 [9]
	MSE [ranks]	0.01134 [1]	0.01387 [4]	0.01732 [7]	0.01259 [2]	0.01454 [5]	0.01572 [6]	0.01307 [3]	0.01795 [8]	0.02006 [10]	0.01878 [9]
	MRE [ranks]	0.06676 [1]	0.07508 [4]	0.08382 [8]	0.07182 [2]	0.07690 [5]	0.08105 [6]	0.07299 [3]	0.08315 [7]	0.08988 [10]	0.08514 [9]
	Sum ranks [ranks]	3 [1]	12 [4]	23 [8]	6 [2]	15 [5]	18 [6]	9 [3]	22 [7]	30 [10]	27 [9]
50	Bias [ranks]	0.05943 [1]	0.06287 [3]	0.06685 [5]	0.06233 [2]	0.06879 [6]	0.07019 [7]	0.06570 [4]	0.08163 [10]	0.07670 [9]	0.07503 [8]
	MSE [ranks]	0.00544 [1]	0.00626 [3]	0.00721 [5]	0.00620 [2]	0.00776 [6]	0.00788 [7]	0.00656 [4]	0.01041 [10]	0.00938 [9]	0.00919 [8]
	MRE [ranks]	0.04754 [1]	0.05030 [3]	0.05348 [5]	0.04986 [2]	0.05503 [6]	0.05615 [7]	0.05256 [4]	0.06530 [10]	0.06136 [9]	0.06002 [8]
	Sum ranks [ranks]	3 [1]	9 [3]	15 [5]	6 [2]	18 [6]	21 [7]	12 [4]	30 [10]	27 [9]	24 [8]
75	Bias [ranks]	0.05115 [2]	0.05614 [5]	0.05967 [7]	0.05094 [1]	0.05645 [6]	0.05604 [4]	0.05142 [3]	0.06385 [9]	0.06652 [10]	0.06004 [8]
	MSE [ranks]	0.00403 [1]	0.00500 [4]	0.00555 [7]	0.00420 [2]	0.00509 [6]	0.00505 [5]	0.00425 [3]	0.00614 [9]	0.00706 [10]	0.00567 [8]
	MRE [ranks]	0.04092 [2]	0.04491 [5]	0.04774 [7]	0.04075 [1]	0.04516 [6]	0.04483 [4]	0.04114 [3]	0.05108 [9]	0.05322 [10]	0.04803 [8]
	Sum ranks [ranks]	5 [2]	14 [5]	21 [7]	3 [1]	18 [6]	13 [4]	9 [3]	27 [9]	30 [10]	24 [8]
150	Bias [ranks]	0.03517 [1]	0.03711 [3]	0.04090 [6]	0.03632 [2]	0.04151 [7]	0.03846 [5]	0.03820 [4]	0.04334 [9]	0.04989 [10]	0.04217 [8]
	MSE [ranks]	0.00198 [1]	0.00223 [3]	0.00264 [6]	0.00205 [2]	0.00269 [7]	0.00229 [4]	0.00237 [5]	0.00289 [9]	0.00404 [10]	0.00276 [8]
	MRE [ranks]	0.02813 [2]	0.02969 [4]	0.03272 [7]	0.02905 [3]	0.02365 [1]	0.03076 [6]	0.03056 [5]	0.03467 [9]	0.03992 [10]	0.03374 [8]
	Sum ranks [ranks]	4 [1]	10 [3]	19 [7]	7 [2]	15 [5.5]	15 [5.5]	14 [4]	27 [9]	30 [10]	24 [8]
250	Bias [ranks]	0.02725 [1]	0.03013 [6]	0.03199 [7]	0.02748 [2]	0.02957 [5]	0.02912 [4]	0.02822 [3]	0.03508 [9]	0.03588 [10]	0.03358 [8]
	MSE [ranks]	0.00116 [1]	0.00142 [6]	0.00166 [7]	0.00120 [2]	0.00134 [5]	0.00131 [4]	0.00129 [3]	0.00196 [9]	0.00205 [10]	0.00181 [8]
	MRE [ranks]	0.02180 [1]	0.02411 [5]	0.02559 [6]	0.02198 [2]	0.03321 [10]	0.02329 [4]	0.02257 [3]	0.02806 [8]	0.02870 [9]	0.02686 [7]
	Sum ranks [ranks]	3 [1]	17 [5]	20 [6.5]	6 [2]	20 [6.5]	12 [4]	9 [3]	26 [9]	29 [10]	25 [8]
400	Bias [ranks]	0.02094 [1]	0.02451 [5]	0.02462 [6]	0.02224 [2]	0.02494 [7]	0.02322 [4]	0.02254 [3]	0.02821 [10]	0.02779 [8]	0.02791 [9]
	MSE [ranks]	0.00071 [1]	0.00094 [6]	0.00092 [5]	0.00080 [2]	0.00098 [7]	0.00088 [4]	0.00082 [3]	0.00124 [10]	0.00121 [8.5]	0.00121 [8.5]
	MRE [ranks]	0.01675 [1]	0.01961 [5]	0.01970 [6]	0.01779 [2]	0.01995 [7]	0.01857 [4]	0.01803 [3]	0.02257 [10]	0.02223 [8]	0.02233 [9]
	Sum ranks [ranks]	3 [1]	16 [5]	17 [6]	6 [2]	21 [7]	12 [4]	9 [3]	30 [10]	24.5 [8]	26.5 [9]
600	Bias [ranks]	0.01760 [2]	0.01881 [4]	0.02148 [8]	0.01734 [1]	0.01938 [6]	0.01918 [5]	0.01771 [3]	0.02129 [7]	0.02244 [10]	0.02175 [9]
	MSE [ranks]	0.00048 [2]	0.00056 [4]	0.00070 [7.5]	0.00046 [1]	0.00059 [5.5]	0.00059 [5.5]	0.00050 [3]	0.00070 [7.5]	0.00080 [10]	0.00071 [9]
	MRE [ranks]	0.01408 [2]	0.01505 [4]	0.01718 [8]	0.01387 [1]	0.01550 [6]	0.01534 [5]	0.01417 [3]	0.01703 [7]	0.01795 [10]	0.01740 [9]
	Sum ranks [ranks]	6 [2]	12 [4]	23.5 [8]	3 [1]	17.5 [6]	15.5 [5]	9 [3]	21.5 [7]	30 [10]	27 [9]

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Table 4 Bias, MSE and MRE under different estimation methods, when \(\tau = 1.25\).

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