delta_Taylor: Estimate of delta by Taylor approximation

Description

Computes an initial estimate of $\delta$ 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")

Value

scalar; estimated $\delta$.

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 $\delta = 0$ equals

$$ \gamma_2(\delta) = 3 + 12 \delta + 66 \delta^2 + \mathcal{O}(\delta^3). $$

Ignoring higher order terms, using the empirical estimate on the left hand side, and solving for $\delta$ yields (positive root) $$\widehat{\delta}_{Taylor} = \frac{1}{66} \cdot \left( \sqrt{66 \widehat{\gamma}_2(\mathbf{y}) - 162}-6 \right), $$ where $\widehat{\gamma}_2(\mathbf{y})$ is the empirical kurtosis of $\mathbf{y}$.

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

Examples

Run this code


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