W_gamma() R function from [LambertW]

Inverse transformation for skewed Lambert W RVs

Inverse transformation for skewed Lambert W RVs and its derivative.


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

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

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.

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

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(\gamma U) =: H_{\gamma}(U), \quad \gamma \in \mathbf{R},

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

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

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

If $\gamma = 0$ , then $z = u$ and the inverse also equals the identity.

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

W_{\gamma}(z) = \frac{1}{\gamma} W(\gamma z).

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

The derivative of $W_{\gamma}(z)$ with respect to $z$ simplifies to

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

deriv_W_gamma implements this derivative (for both branches).