Inference about the Tail of a Distribution: Improvementon the Hill Estimator

Table 1

	𝑃	𝑁rand	𝐷𝐿	𝐷𝑅	𝐿	𝑅	𝜎	input 𝜇	𝜇Hill	𝜇^[5]
(1)	𝑥^−5	1000	3.0	150.0	3.0004	14.37	1.339	5	5.157	5.115
(2)	𝑥^−5	1000	3.0	4.0	3.0015	3.962	1.208	5	10.175	5.784
(3)	𝑥^−5	5000	3.0	4.0	3.0002	3.999	1.214	5	9.634	5.142
(4)	1/√𝑥	1000	3.0	150.0	3.017	149.53	3.642	0.5	1.394	0.511
(5)	1/√𝑥	1000	3.0	1500.0	3.058	1494.7	5.585	0.5	1.223	0.508
(6)	1/√𝑥	1000	3.0	15000.0	3.115	14945	7.682	0.5	1.153	0.518
(7)	1/(1+𝑝2𝑥^2+𝑝4𝑥^4)	1000	1.0	2.0	1.0001	1.992	0.234	≈4	5.269	3.977
(8)	1/(1+𝑝2𝑥^2+𝑝4𝑥^4)	1000	1.0	5.0	1.0001	4.686	0.320	≈4	4.126	3.980
(9)	ln(𝑥)/𝑥	1000	100.0	400.0	100.07	399.11	5.322	≈1	2.397	0.849
(10)	ln(𝑥)/𝑥	1000	2000.0	5000.0	2001.7	9973.2	8.423	≈1	2.217	0.914
(11)	1/ln(𝑥)𝑥	5000	8000.0	10000.0	8000.8	9999.0	9.098	≈1	10.054	1.247
(12)	1/ln(𝑥)𝑥	5000	3000.0	6000.0	3000.9	5998.1	8.346	≈1	3.944	1.164
(13)	𝑥^3.5	1000	3.0	10000.0	1828.3	9995.9	8.989	−3.5	1.677	−3.503