Extreme Value Distributions: An Overview of Estimation and Simulation

Table 5

Moment method estimator (MOM).

Parameter	Gumbel dist	2-Fréchet dist	3-Fréchet dist	2-Weibull dist	3-Weibull dist	GEV-distribution
N = 1,000	= 5,  = 9	= 2,  = 5	= 1,  = 5,  = 3,	= 3,  = 7	= 3,  = 1,  = 5,	= 1,  = 3,  = 0.01
Location	5.209467	—	0.3666667	—	1.2104030	1.182236950
95% CI	(3.974989, 6.443945)	—	(−1.245415, 1.978749)	—	(−0.589689, 3.01049)	(0.8251437, 1.53933)
C	0.923	—	0.409	—	0.119	0.890
Bias	0.209467	—	−0.6333333	—	−1.789597	0.18223695
MSE	0.044289216	—	0.401815014	—	3.203535133	0.033244846
Scale	8.590332	2.008617	7.5555556	2.994487	0.1442575	3.180989490
95% CI	(7.22174, 9.958924)	(1.734339, 2.282895)	(6.208391, 8.902721)	(2.854526, 3.134448)	(−1.141305, 1.42982)	(2.892358, 3.469621)
CP	0.881	0.929	0.551	0.957	0.393	0.948
Bias	−0.409668	0.008617	2.5555556	−0.005513	−0.8557425	0.18098949
MSE	0.655396	0.019656	7.0032859	0.0051295	1.1624990	0.05444299
Shape	—	5.465167	2.8888889	7.619065	3.7844660	0.018791750
95% CI	—	(4.856787, 6.073547)	(2.031479, 3.746299)	(6.967252, 8.270878)	(0.888693, 6.68024)	(0.0053161, 0.032267)
CP	—	0.779	0.864	0.742	0.286	0.931
Bias	—	0.465167	−0.1111111	0.619065	−1.215534	0.00879175
MSE	—	0.312727	0.2037117	0.493836	3.6603391	0.00012456
N = 1,000,000	= 3,  = 5	= 15,  = 30	= 0,  = 30,  = 10,	= 10,  = 10	= 0.5,  = 13,  = 10,	= 3,  = 5,  = 0.01
Location	2.966366	—	11.872800	—	0.50000009	3.006222137
95% CI	(2.577247, 3.355485)	—	(6.68255, 17.06305)	—	(0.455781, 0.544219)	(2.765454, 3.24699)
CP	0.859	—	0.110	—	0.979	0.961
Bias	−0.033634	—	11.8728	—	0.000000009	0.006222137
MSE	0.0405454	—	147.9757	—	0.000508984	0.015128564
Scale	5.011782	15.01847	1.0707010	9.999611	18.0001921	5.009725335
95% CI	(4.406102, 5.617462)	(14.76892, 15.26802)	(−14.83286, 16.97427)	(9.564931, 10.43429)	(14.57669, 21.42369)	(4.732609, 5.286841)
CP	0.917	0.941	0.019	0.935	0.367	0.897
Bias	0.011782	0.01847	−28.929299	−0.000389	5.0001921	0.009725335
MSE	0.0956324	0.0165518	902.74240	0.049184302	28.05282115	0.020084454
Shape	—	31.76106	10.274778	9.992710	22.0002100	0.008961094
95% CI	—	(30.50113, 33.02099)	(9.345474, 11.20408)	(9.599808, 10.38561)	(16.43889, 27.56153)	(−0.073961, 0.091883)
CP	—	0.687	0.762	0.899	0.153	0.951
Bias	—	1.76106	0.274778	−0.00729	12.00021	−0.001038906
MSE	—	3.101332	0.300305	0.0402370	152.0559	0.001790970