delta_Taylor {LambertW} | R Documentation |
Estimate of delta by Taylor approximation
Description
Computes an initial estimate of \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 |
distname |
string; name of the distribution. Currently only supports |
Details
The second order Taylor approximation of the theoretical kurtosis of a
heavy tail Lambert W x Gaussian RV around \delta = 0
equals
\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 \delta
yields (positive root)
\widehat{\delta}_{Taylor} = \frac{1}{66} \cdot \left( \sqrt{66
\widehat{\gamma}_2(\mathbf{y}) - 162}-6 \right),
where \widehat{\gamma}_2(\mathbf{y})
is the empirical kurtosis of \mathbf{y}
.
Since the kurtosis is finite only for \delta < 1/4
,
delta_Taylor
upper-bounds the returned estimate by 0.25
.
Value
scalar; estimated \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