delta_Taylor {LambertW}R Documentation

Estimate of delta by Taylor approximation

Description

Computes an initial estimate of δ\delta 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.

Usage

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

Arguments

y

a numeric vector of data values.

kurtosis.y

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

distname

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

Details

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

γ2(δ)=3+12δ+66δ2+O(δ3). \gamma_2(\delta) = 3 + 12 \delta + 66 \delta^2 + \mathcal{O}(\delta^3).

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

δ^Taylor=166(66γ^2(y)1626),\widehat{\delta}_{Taylor} = \frac{1}{66} \cdot \left( \sqrt{66 \widehat{\gamma}_2(\mathbf{y}) - 162}-6 \right),

where γ^2(y)\widehat{\gamma}_2(\mathbf{y}) is the empirical kurtosis of y\mathbf{y}.

Since the kurtosis is finite only for δ<1/4\delta < 1/4, delta_Taylor upper-bounds the returned estimate by 0.250.25.

Value

scalar; estimated δ\delta.

See Also

IGMM to estimate all parameters jointly.

Examples


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


[Package LambertW version 0.6.9-1 Index]