Characterizing Tukey ℎ and ℎℎ-Distributions through 𝐿-Moments and the 𝐿-Correlation

Table 7

𝐿-skew (𝜏₃) and 𝐿-kurtosis (𝜏₄) results.

(a) 𝑛=25

Dist	Parameter	Estimate	95% Bootstrap C.I.	Stand. error	Relative bias %
1	𝜏3=0.110610	0.09724	0.09574, 0.09897	0.00082	−12.1
	𝜏4=0.195758	0.1882	0.1869, 0.1893	0.00061	−3.9
2	𝜏3=0.050040	0. 04361	0.04193, 0.04534	0.00086	−12.8
	𝜏4=0.210839	0.2038	0.2026, 0.2050	0.00060	−3.3
3	𝜏3=0.041025	0.03726	0.03592, 0.03883	0.00074	−9.2
	𝜏4=0.175084	0.1723	0.1713, 0.1734	0.00054	−1.6
4	𝜏3=0	0.0008	−0.0007, 0.0021	0.00070	—
	𝜏4=0.160173	0.1591	0.1582, 0.1602	0.00050	—

(b) 𝑛=1000

Dist	Parameter	Estimate	95% Bootstrap C.I.	Stand. error	Relative bias %
1	𝜏3=0.110610	0.1101	0.1098, 0.1104	0.00014	−0.46
	𝜏4=0.195758	0. 1955	0. 1953, 0.1958	0.00010	—
2	𝜏3=0.050040	0.0498	0.04946, 0.050041	0.00015	—
	𝜏4=0.210839	0.2106	0.2104, 0.2109	0.00010	—
3	𝜏3=0.041025	0.04086	0.04062, 0.04109	0.00012	—
	𝜏4=0.175084	0.1750	0.1748, 0.1751	0.00010	—
4	𝜏3=0	−0.00003	−0.00027, 0.00015	0.00011	—
	𝜏4=0.160173	0.1602	0.1600, 0.1603	0.00010	—