R: Inverse transformation for heavy-tail Lambert W RVs

W_delta {LambertW} | R Documentation

## Inverse transformation for heavy-tail Lambert W RVs

### Description

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`.

### Usage

    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)^{α}; default: `alpha = 1`.

### Value

Computes sgn(z) ≤ft(\frac{1}{α δ} W(α δ (z^2)^{α}) \right)^{1/2 α}. 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

[Package LambertW version 0.6.7 Index]