gamma_01 {LambertW}

Input parameters to get a zero mean, unit variance output for a given gamma

Description

Computes the input mean \mu_x(\gamma) and standard deviation \sigma_x(\gamma) for input X \sim F(x \mid \boldsymbol \beta) such that the resulting skewed Lambert W x F RV Y with \gamma has zero-mean and unit-variance. So far works only for Gaussian input and scalar \gamma.

The function works for any output mean and standard deviation, but \mu_y = 0 and \sigma_y = 1 are set as default as they are the most useful, e.g., to generate a standardized Lambert W white noise sequence.

Usage

gamma_01(gamma, mu.y = 0, sigma.y = 1, distname = "normal")

Arguments

gamma

skewness parameter

mu.y

output mean; default: 0.

sigma.y

output standard deviation; default: 1.

distname

string; name of distribution. Currently only supports "normal".

Value

A 5-dimensional vector (\mu_x(\gamma), \sigma_x(\gamma), \gamma, 0, 1), where \delta = 0 and \alpha = 1 are set for the sake of compatiblity with other functions.

Examples

gamma_01(0) # for gamma = 0, input == output, therefore (0,1,0,0,1)
# input mean must be slightly negative to get a zero-mean output
gamma_01(0.1) # gamma = 0.1 means it is positively skewed
gamma_01(1)