W_delta function

Inverse transformation for heavy-tail Lambert W RVs

Inverse transformation W_delta_alpha for heavy-tail Lambert W RVs and its derivative. This is the inverse of Tukey's h transformation as a special case of alpha = 1.

W_delta(z, delta = 0)

W_delta_alpha(z, delta = 0, alpha = 1)

W_2delta(z, delta = c(0, 1/5))

W_2delta_2alpha(z, delta = c(0, 0), alpha = c(1, 1))

deriv_W_delta(z, delta = 0)

deriv_W_delta_alpha(z, delta = 1, alpha = 1)

Arguments

• z: a numeric vector of real values.
• delta: heavy-tail parameter(s); by default delta = 0, which implies W_delta(z) = z. If a vector of length 2 is supplied, then delta[1] on the left and delta[2] on the right (of the center) will be used.
• alpha: heavy-tail exponent(s) in $(u^2)^{\alpha}$; default: alpha = 1.

Returns

Computes sgnc("$(z) \\left(\\frac{1}{\\alpha \\delta} W(\\alpha \\delta (z^2)^{\\alpha})\n$", "$\\right)^{1/2 \\alpha}$"). If $z$ is a vector, so is the output.

Examples

G_delta(0)
W_delta(0)

# W_delta is the inverse of G_delta
u.v <- -2:2
W_delta(G_delta(u.v, delta = 0.3), delta = 0.3)

# with alpha too
G_delta_alpha(u.v, delta = 1, alpha = 0.33)
W_delta_alpha(G_delta_alpha(u.v, delta = 1, alpha = 0.33), 
              delta = 1, alpha = 0.33) # the inverse