| delta_Taylor {LambertW} | R Documentation |
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")
y |
a numeric vector of data values. |
kurtosis.y |
kurtosis of y; default: empirical kurtosis of data |
distname |
string; name of the distribution. Currently only supports |
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.
IGMM to estimate all parameters jointly.
set.seed(2)
# 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_Taylor(y)
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