R: Inverse transformation for skewed Lambert W RVs

W_gamma {LambertW} | R Documentation

## Inverse transformation for skewed Lambert W RVs

### Description

Inverse transformation for skewed Lambert W RVs and its derivative.

### Usage

    W_gamma(z, gamma = 0, branch = 0)

    deriv_W_gamma(z, gamma = 0, branch = 0)

### Arguments

`z` | a numeric vector of real values; note that `W(Inf, branch = 0) = Inf`.
`gamma` | skewness parameter; by default `gamma = 0`, which implies `W_gamma(z) = z`.
`branch` | either `0` or `-1` for the principal or non-principal branch solution.

### Details

A skewed Lambert W\times F RV Z (for simplicity assume zero mean, unit variance input) is defined by the transformation (see `H_gamma`)

z = U \exp(γ U) =: H_{γ}(U), \quad γ \in \mathbf{R},

where U is a zero-mean and/or unit-variance version of the distribution F.

The inverse transformation is W_{γ}(z) := \frac{W(γ z)}{γ}, where W is the Lambert W function.

`W_gamma(z, gamma, branch = 0)` (and `W_gamma(z, gamma, branch = -1)`) implement this inverse.

If γ = 0, then z = u and the inverse also equals the identity.

If γ \neq 0, the inverse transformation can be computed by

W_{γ}(z) = \frac{1}{γ} W(γ z).

Same holds for `W_gamma(z, gamma, branch = -1)`.

The derivative of W_{γ}(z) with respect to z simplifies to

\frac{d}{dz} W_{γ}(z) = \frac{1}{γ} \cdot W'(γ z) \cdot γ = W'(γ z)

`deriv_W_gamma` implements this derivative (for both branches).

### Value

numeric; if z is a vector, so is the output.

### See Also

`H_gamma`

[Package LambertW version 0.6.7 Index]