Table 5 Bias, MSE and MRE under different estimation methods, when \(\tau = 1.75\).

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]	0.12543 [2]	0.13819 [6]	0.13450 [4]	0.13037 [3]	0.14345 [7]	0.13655 [5]	0.12524 [1]	0.16243 [10]	0.15019 [8]	0.15121 [9]
	MSE [ranks]	0.02416 [1]	0.02871 [4]	0.02876 [5]	0.02648 [3]	0.03221 [7]	0.03029 [6]	0.02475 [2]	0.04184 [10]	0.03584 [9]	0.03545 [8]
	MRE [ranks]	0.07167 [2]	0.07896 [6]	0.07686 [4]	0.07450 [3]	0.08197 [7]	0.07803 [5]	0.07156 [1]	0.09281 [10]	0.08582 [8]	0.08641 [9]
	Sum ranks [ranks]	5 [2]	16 [5.5]	13 [4]	9 [3]	21 [7]	16 [5.5]	4 [1]	30 [10]	25 [8]	26 [9]
50	Bias [ranks]	0.08819 [1]	0.08971 [2]	0.09966 [7]	0.09112 [3]	0.09857 [6]	0.09228 [4]	0.09277 [5]	0.11457 [10]	0.11280 [9]	0.10296 [8]
	MSE [ranks]	0.01231 [1]	0.01265 [2]	0.01532 [6]	0.01296 [3]	0.01663 [7]	0.01318 [4]	0.01335 [5]	0.02012 [9]	0.02060 [10]	0.01665 [8]
	MRE [ranks]	0.05039 [1]	0.05126 [2]	0.05695 [7]	0.05207 [3]	0.05633 [6]	0.05273 [4]	0.05301 [5]	0.06547 [10]	0.06445 [9]	0.05884 [8]
	Sum ranks [ranks]	3 [1]	6 [2]	20 [7]	9 [3]	19 [6]	12 [4]	15 [5]	29 [10]	28 [9]	24 [8]
75	Bias [ranks]	0.07145 [1]	0.07927 [5]	0.08026 [7]	0.07761 [4]	0.07998 [6]	0.07470 [2]	0.07539 [3]	0.08304 [8]	0.09226 [10]	0.09115 [9]
	MSE [ranks]	0.00784 [1]	0.00995 [6]	0.00974 [5]	0.00928 [4]	0.01067 [7]	0.00884 [3]	0.00874 [2]	0.01101 [8]	0.01354 [10]	0.01322 [9]
	MRE [ranks]	0.04083 [1]	0.04530 [5]	0.04586 [7]	0.04435 [4]	0.04570 [6]	0.04269 [2]	0.04308 [3]	0.04745 [8]	0.05272 [10]	0.05208 [9]
	Sum ranks [ranks]	3 [1]	16 [5]	19 [6.5]	12 [4]	19 [6.5]	7 [2]	8 [3]	24 [8]	30 [10]	27 [9]
150	Bias [ranks]	0.05258 [1]	0.05348 [4]	0.05834 [7]	0.05283 [2]	0.05750 [6]	0.05506 [5]	0.05327 [3]	0.06823 [10]	0.06393 [9]	0.06190 [8]
	MSE [ranks]	0.00423 [1]	0.00442 [2]	0.00537 [7]	0.00449 [3]	0.00528 [6]	0.00471 [5]	0.00459 [4]	0.00733 [10]	0.00644 [9]	0.00598 [8]
	MRE [ranks]	0.03005 [1]	0.03056 [4]	0.03334 [7]	0.03019 [2]	0.03286 [6]	0.03146 [5]	0.03044 [3]	0.03899 [10]	0.03653 [9]	0.03537 [8]
	Sum ranks [ranks]	3 [1]	10 [3.5]	21 [7]	7 [2]	18 [6]	15 [5]	10 [3.5]	30 [10]	27 [9]	24 [8]
250	Bias [ranks]	0.03911 [1]	0.04157 [3]	0.04418 [7]	0.04188 [4]	0.04308 [5]	0.04313 [6]	0.04086 [2]	0.04886 [9]	0.05016 [10]	0.04573 [8]
	MSE [ranks]	0.00236 [1]	0.00267 [3]	0.00314 [7]	0.00268 [4]	0.00294 [5.5]	0.00294 [5.5]	0.00266 [2]	0.00372 [9]	0.00428 [10]	0.00331 [8]
	MRE [ranks]	0.02235 [1]	0.02375 [3]	0.02524 [7]	0.02393 [4]	0.02461 [5]	0.02465 [6]	0.02335 [2]	0.02792 [9]	0.02866 [10]	0.02613 [8]
	Sum ranks [ranks]	3 [1]	9 [3]	21 [7]	12 [4]	15.5 [5]	17.5 [6]	6 [2]	27 [9]	30 [10]	24 [8]
400	Bias [ranks]	0.03012 [1]	0.03157 [2]	0.03630 [7.5]	0.03283 [4]	0.03630 [7.5]	0.03363 [5]	0.03197 [3]	0.03964 [9]	0.04107 [10]	0.03429 [6]
	MSE [ranks]	0.00143 [1]	0.00158 [3]	0.00215 [8]	0.00163 [4]	0.00205 [7]	0.00177 [5]	0.00153 [3]	0.00244 [9]	0.00261 [10]	0.00187 [6]
	MRE [ranks]	0.01721 [1]	0.01804 [2]	0.02074 [7.5]	0.01876 [4]	0.02074 [7.5]	0.01922 [5]	0.01827 [3]	0.02265 [9]	0.02347 [10]	0.01960 [6]
	Sum ranks [ranks]	3 [1]	7 [2]	23 [8]	12 [4]	22 [7]	15 [5]	9 [3]	27 [9]	30 [10]	18 [6]
600	Bias [ranks]	0.02625 [2]	0.02816 [5]	0.02991 [8]	0.02605 [1]	0.02939 [7]	0.02700 [4]	0.02639 [3]	0.03147 [9]	0.03177 [10]	0.02893 [6]
	MSE [ranks]	0.00106 [1]	0.00123 [5]	0.00135 [7]	0.00108 [2]	0.00129 [6]	0.00113 [4]	0.00110 [3]	0.00154 [9]	0.00155 [10]	0.00136 [8]
	MRE [ranks]	0.01500 [2]	0.01609 [5]	0.01709 [8]	0.01489 [1]	0.01680 [7]	0.01543 [4]	0.01508 [3]	0.01798 [9]	0.01816 [10]	0.01653 [6]
	Sum ranks [ranks]	5 [2]	15 [5]	23 [8]	4 [1]	20 [6.5]	12 [4]	9 [3]	27 [9]	30 [10]	20 [6.5]

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Table 5 Bias, MSE and MRE under different estimation methods, when \(\tau = 1.75\).

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