IGMM  R Documentation 
An iterative method of moments estimator to find this τ = (μ_x,
σ_x, γ) for type = 's'
(τ = (μ_x, σ_x,
δ) for type = 'h'
or τ = (μ_x, σ_x, δ_l,
δ_r) for type = "hh"
) which minimizes the distance between
the sample and theoretical skewness (or kurtosis) of \boldsymbol x
and X.
This algorithm is only welldefined for data with finite mean and variance
input X. See analyze_convergence
and references therein
for details.
IGMM( y, type = c("h", "hh", "s"), skewness.x = 0, kurtosis.x = 3, tau.init = get_initial_tau(y, type), robust = FALSE, tol = .Machine$double.eps^0.25, location.family = TRUE, not.negative = NULL, max.iter = 100, delta.lower = 1, delta.upper = 3 )
y 
a numeric vector of real values. 
type 
type of Lambert W \times F distribution: skewed 
skewness.x 
theoretical skewness of input X; default 
kurtosis.x 
theoretical kurtosis of input X; default 
tau.init 
starting values for IGMM algorithm; default:

robust 
logical; only used for 
tol 
a positive scalar specifiying the tolerance level for terminating
the iterative algorithm. Default: 
location.family 
logical; tell the algorithm whether the underlying
input should have a location family distribution (for example, Gaussian
input); default: 
not.negative 
logical; if 
max.iter 
maximum number of iterations; default: 
delta.lower, delta.upper 
lower and upper bound for

For algorithm details see the References.
A list of class LambertW_fit
:
tol 
see Arguments 
data 
data 
n 
number of observations 
type 
see Arguments 
tau.init 
starting values for τ 
tau 
IGMM estimate for τ 
tau.trace 
entire iteration trace of τ^{(k)}, k = 0, ..., K, where

sub.iterations 
number of iterations only performed in GMM algorithm to find optimal γ (or δ) 
iterations 
number of iterations to update μ_x and σ_x. See References for detals. 
hessian 
Hessian matrix (obtained from simulations; see References) 
call 
function call 
skewness.x, kurtosis.x 
see Arguments 
distname 
a character string describing distribution characteristics given
the target theoretical skewness/kurtosis for the input. Same information as 
location.family 
see Arguments 
message 
message from the optimization method. What kind of convergence? 
method 
estimation method; here: 
Georg M. Goerg
delta_GMM
, gamma_GMM
, analyze_convergence
# estimate tau for the skewed version of a Normal y < rLambertW(n = 1000, theta = list(beta = c(2, 1), gamma = 0.2), distname = "normal") fity < IGMM(y, type = "s") fity summary(fity) plot(fity) # estimate tau for the skewed version of an exponential y < rLambertW(n = 1000, theta = list(beta = 1, gamma = 0.5), distname = "exp") fity < IGMM(y, type = "s", skewness.x = 2, location.family = FALSE) fity summary(fity) plot(fity) # estimate theta for the heavytailed version of a Normal = Tukey's h y < rLambertW(n = 500, theta = list(beta = c(2, 1), delta = 0.2), distname = "normal") system.time( fity < IGMM(y, type = "h") ) fity summary(fity) plot(fity)
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