gamma_Taylor  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)
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