gamma_Taylor: Estimate gamma by Taylor approximation

Description

Computes an initial estimate of $\gamma$ based on the Taylor approximation of the skewness of Lambert W $\times$ Gaussian RVs around $\gamma = 0$. See Details for the formula.

This is the initial estimate for IGMM and gamma_GMM.

Usage

gamma_Taylor(y, skewness.y = skewness(y), skewness.x = 0, degree = 3)

Value

Scalar; estimate of $\gamma$.

Arguments

y: a numeric vector of data values.
skewness.y: skewness of $y$; default: empirical skewness of data y.
skewness.x: skewness for input X; default: 0 (symmetric input).
degree: degree of the Taylor approximation; in Goerg (2011) it just uses the first order approximation ($6 \cdot \gamma$); a much better approximation is the third order ($6 \cdot \gamma + 8 \cdot \gamma^3$). By default it uses the better degree = 3 approximation.

Details

The first order Taylor approximation of the theoretical skewness $\gamma_1$ (not to be confused with the skewness parameter $\gamma$) of a Lambert W x Gaussian random variable around $\gamma = 0$ equals $$ \gamma_1(\gamma) = 6 \gamma + \mathcal{O}(\gamma^3). $$

Ignoring higher order terms, using the empirical estimate on the left hand side, and solving $\gamma$ yields a first order Taylor approximation estimate of $\gamma$ as $$ \widehat{\gamma}_{Taylor}^{(1)} = \frac{1}{6} \widehat{\gamma}_1(\mathbf{y}), $$ where $\widehat{\gamma}_1(\mathbf{y})$ is the empirical skewness of the data $\mathbf{y}$.

As the Taylor approximation is only good in a neighborhood of $\gamma = 0$, the output of gamma_Taylor is restricted to the interval $(-0.5, 0.5)$.

The solution of the third order Taylor approximation $$ \gamma_1(\gamma) = 6 \gamma + 8 \gamma^3 + \mathcal{O}(\gamma^5),$$ is also supported. See code for the solution to this third order polynomial.

Examples

Run this code


set.seed(2)
# a little skewness
yy <- rLambertW(n = 1000, theta = list(beta = c(0, 1), gamma = 0.1), 
                distname = "normal") 
# Taylor estimate is good because true gamma = 0.1 close to 0
gamma_Taylor(yy) 

# very highly negatively skewed
yy <- rLambertW(n = 1000, theta = list(beta = c(0, 1), gamma = -0.75), 
                distname = "normal") 
# Taylor estimate is bad since gamma = -0.75 is far from 0; 
# and gamma = -0.5 is the lower bound by default.
gamma_Taylor(yy)