R: Estimate of delta by Taylor approximation

delta_Taylor {LambertW} | R Documentation

## Estimate of delta by Taylor approximation

### Description

Computes an initial estimate of δ based on the Taylor approximation of the kurtosis of Lambert W \times Gaussian RVs. See Details for the formula.

This is the initial estimate for `IGMM` and `delta_GMM`.

### Usage

    delta_Taylor(y, kurtosis.y = kurtosis(y), distname = "normal")

### Arguments

`y` | a numeric vector of data values.
`kurtosis.y` | kurtosis of y; default: empirical kurtosis of data `y`.
`distname` | string; name of the distribution. Currently only supports `"normal"`.

### Details

The second order Taylor approximation of the theoretical kurtosis of a heavy tail Lambert W x Gaussian RV around δ = 0 equals

γ_2(δ) = 3 + 12 δ + 66 δ^2 + \mathcal{O}(δ^3).

Ignoring higher order terms, using the empirical estimate on the left hand side, and solving for δ yields (positive root)

\widehat{δ}_{Taylor} = \frac{1}{66} \cdot ≤ft( √{66 \widehat{γ}_2(\mathbf{y}) - 162}-6 \right),

where \widehat{γ}_2(\mathbf{y}) is the empirical kurtosis of \mathbf{y}.

Since the kurtosis is finite only for δ < 1/4, `delta_Taylor` upper-bounds the returned estimate by 0.25.

### Value

scalar; estimated δ.

### See Also

`IGMM` to estimate all parameters jointly.

### Examples

    set.seed(2)
    # a little heavy-tailed (kurtosis does exist)
    y <- rLambertW(n = 1000, theta = list(beta = c(0, 1), delta = 0.2),
                   distname = "normal")
    # good initial estimate since true delta=0.2 close to 0, and
    # empirical kurtosis well-defined.
    delta_Taylor(y)
    delta_GMM(y) # iterative estimate

    y <- rLambertW(n = 1000, theta = list(beta = c(0, 1), delta = 1),
                   distname = "normal") # very heavy-tailed (like a Cauchy)
    delta_Taylor(y) # bounded by 1/4 (as otherwise kurtosis does not exist)
    delta_GMM(y) # iterative estimate

[Package LambertW version 0.6.7 Index]