delta_01: Input parameters to get zero mean, unit variance output given...

View source: R/delta_01.R

delta_01R Documentation

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


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

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


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



scalar; heavy-tail parameter.


output mean; default: 0.


output standard deviation; default: 1.


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


5-dimensional vector (μ_x(δ), σ_x(δ), 0, δ, 1), where γ = 0 and α = 1 are set for the sake of compatiblity with other functions.


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(1/3)  # only moments up to order 2 exist
delta_01(1)  # neither mean nor variance exist, thus NA

LambertW documentation built on Sept. 22, 2022, 5:07 p.m.