LambertW (version 0.6.9-1)

gamma_01: 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")

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.

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".

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)