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

Table 6

Skew (𝛼₃) and Kurtosis (𝛼₄) results for the conventional moment procedure.

(a) 𝑛=25

Dist	Parameter	Estimate	95% Bootstrap C.I.	Standard error	Relative bias %
1	𝛼3=2	0.649	0.638, 0.659	0.0056	−67.6
	𝛼4=40	1.677	1.64, 1.72	0.0200	−95.8
2	𝛼3=1	0.305	0.293, 0.317	0.0060	−69.5
	𝛼4=15	1.73	1.69, 1.76	0.0175	−88.5
3	𝛼3=0.5	0.248	0.239, 0.257	0.0048	−50.4
	𝛼4=3	0.97	0.95, 1.00	0.0128	−67.7
4	𝛼3=0	0.0056	−0.0024, 0.0140	0.0042	—
	𝛼4=1.2	0.624	0.606, 0.643	0.0095	−48.0

(b) 𝑛=1000

Dist	Parameter	Estimate	95% Bootstrap C.I.	Standard error	Relative bias %
1	𝛼3=2	1.612	1.597, 1.627	0.0077	−19.4
	𝛼4=40	11.41	11.11, 11.72	0.1542	−71.5
2	𝛼3=1	0.834	0.823, 0.845	0.0057	−16.6
	𝛼4=15	7.68	7.50, 7.87	0.0962	−48.8
3	𝛼3=0.5	0.477	0.472, 0.481	0.0021	−4.6
	𝛼4=3	2.65	2.61, 2.70	0.0223	−11.7
4	𝛼3=0	−0.0003	−0.0023, 0.0018	0.0011	—
	𝛼4=1.2	1.166	1.158, 1.175	0.0046	−2.83