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

delta_01 {LambertW} | R Documentation

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

### Description

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.

### 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 (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.

### 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

[Package LambertW version 0.6.7 Index]