delta_Taylor: Estimate of delta by Taylor approximation

View source: R/delta_Taylor.R

delta_TaylorR Documentation

Estimate of delta by Taylor approximation


Computes an initial estimate of δ 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.


delta_Taylor(y, kurtosis.y = kurtosis(y), distname = "normal")



a numeric vector of data values.


kurtosis of y; default: empirical kurtosis of data y.


string; name of the distribution. Currently only supports "normal".


The second order Taylor approximation of the theoretical kurtosis of a heavy tail Lambert W x Gaussian RV around δ = 0 equals

γ_2(δ) = 3 + 12 δ + 66 δ^2 + \mathcal{O}(δ^3).

Ignoring higher order terms, using the empirical estimate on the left hand side, and solving for δ yields (positive root)

\widehat{δ}_{Taylor} = \frac{1}{66} \cdot ≤ft( √{66 \widehat{γ}_2(\mathbf{y}) - 162}-6 \right),

where \widehat{γ}_2(\mathbf{y}) is the empirical kurtosis of \mathbf{y}.

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


scalar; estimated δ.

See Also

IGMM to estimate all parameters jointly.


# 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_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

LambertW documentation built on Sept. 22, 2022, 5:07 p.m.