| gamma_Taylor {LambertW} | R Documentation |
Computes an initial estimate of γ based on the Taylor approximation of the skewness of Lambert W \times Gaussian RVs around γ = 0. See Details for the formula.
This is the initial estimate for IGMM and
gamma_GMM.
gamma_Taylor(y, skewness.y = skewness(y), skewness.x = 0, degree = 3)
y |
a numeric vector of data values. |
skewness.y |
skewness of y; default: empirical skewness of data
|
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 γ); a much better
approximation is the third order (6 \cdot γ + 8 \cdot
γ^3). By default it uses the better |
The first order Taylor approximation of the theoretical skewness γ_1 (not to be confused with the skewness parameter γ) of a Lambert W x Gaussian random variable around γ = 0 equals
γ_1(γ) = 6 γ + \mathcal{O}(γ^3).
Ignoring higher order terms, using the empirical estimate on the left hand side, and solving γ yields a first order Taylor approximation estimate of γ as
\widehat{γ}_{Taylor}^{(1)} = \frac{1}{6} \widehat{γ}_1(\mathbf{y}),
where \widehat{γ}_1(\mathbf{y}) is the empirical skewness of the data \mathbf{y}.
As the Taylor approximation is only good in a neighborhood of γ =
0, the output of gamma_Taylor is restricted to the interval
(-0.5, 0.5).
The solution of the third order Taylor approximation
γ_1(γ) = 6 γ + 8 γ^3 + \mathcal{O}(γ^5),
is also supported. See code for the solution to this third order polynomial.
Scalar; estimate of γ.
IGMM to estimate all parameters jointly.
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)