W_gamma | R Documentation |

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 |

`gamma` |
skewness parameter; by default |

`branch` |
either |

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).

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

`H_gamma`

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