LambertW (version 0.6.4)

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)

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.

Value

Scalar; estimate of $\gamma$.

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 $${\hat{\gamma }}_{Taylor}^{(1)}=\frac{1}{6}{\hat{\gamma }}_1(\mathbf{y}),$$ where ${\hat{\gamma }}_1(\mathbf{y})$ 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.

See Also

IGMM to estimate all parameters jointly.

Examples

# NOT RUN {
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) 

# }