The Lambert Way to Gaussianize Heavy-Tailed Data with the Inverse of Tukey’s h Transformation as a Special Case

Table 4

MLE fits to S&P 500 Tables 4(a), 4(b), 4(c), 4(d), and 4(e) and the Gaussianized data   Table  4(f).

(a) Double-tail Lambert    Gaussian = Tukey’s (S&P 500)

	Est.	se		Pr (>)
	0.06	0.015	3.66	0.00
	0.71	0.016	44.00	0.00
	0.19	0.021	8.99	0.00
	0.16	0.019	8.24	0.00

(b) Skew (S&P 500)

	Est.	se		Pr (>)
	0.10	0.061	1.65	0.10
	0.67	0.017	38.47	0.00
	−0.08	0.101	−0.77	0.44
	3.73	0.297	12.57	0.00

(c) Lambert    Gaussian = Tukey’s (S&P 500)

	Est.	se		Pr (>)
	0.06	0.015	3.65	0.000
	0.71	0.016	43.95	0.000
	0.17	0.016	11.05	0.000

(d) Student’s (S&P 500)

	Est.	se		Pr (>)
	0.06	0.015	3.65	0.00
	0.67	0.017	39.51	0.00
	3.72	0.295	12.61	0.00

(e) Gaussian (S&P 500)

	Est.	se		Pr (>)
	0.05	0.018	2.55	0.01
	0.95	0.013	74.57	0.00

(f) Gaussian ()

	Est.	se		Pr (>)
	0.05	0.013	3.81	0.00
	0.71	0.009	74.57	0.00