delta_01 {LambertW}

Input parameters to get zero mean, unit variance output given delta

Description

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

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

Usage

delta_01(delta, mu.y = 0, sigma.y = 1, distname = "normal")

Arguments

delta

scalar; heavy-tail parameter.

mu.y

output mean; default: 0.

sigma.y

output standard deviation; default: 1.

distname

string; distribution name. Currently this function only supports "normal".

Value

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

Examples

delta_01(0) # for delta = 0, input == output, therefore (0,1,0,0,1)
# delta > 0 (heavy-tails): 
#   Y is symmetric for all delta: 
#   mean = 0; however, sd must be smaller 
delta_01(0.1) 
delta_01(1/3)  # only moments up to order 2 exist
delta_01(1)  # neither mean nor variance exist, thus NA