MLE_LambertW: Maximum Likelihood Estimation for Lambert W \times F...

In LambertW: Probabilistic Models to Analyze and Gaussianize Heavy-Tailed, Skewed Data

Maximum Likelihood Estimation for Lambert W \times F distributions

Description

Maximum Likelihood Estimation (MLE) for Lambert W \times F distributions computes \widehat{\theta}_{MLE}.

For type = "s", the skewness parameter \gamma is estimated and \delta = 0 is held fixed; for type = "h" the one-dimensional \delta is estimated and \gamma = 0 is held fixed; and for type = "hh" the 2-dimensional \delta is estimated and \gamma = 0 is held fixed.

By default \alpha = 1 is fixed for any type. If you want to also estimate \alpha (for type = "h" or "hh") set theta.fixed = list().

Usage

MLE_LambertW(
  y,
  distname,
  type = c("h", "s", "hh"),
  theta.fixed = list(alpha = 1),
  use.mean.variance = TRUE,
  theta.init = get_initial_theta(y, distname = distname, type = type, theta.fixed =
    theta.fixed, use.mean.variance = use.mean.variance, method = "IGMM"),
  hessian = TRUE,
  return.estimate.only = FALSE,
  optim.fct = c("optim", "nlm", "solnp"),
  not.negative = FALSE
)

Arguments

y	a numeric vector of real values.
distname	character; name of input distribution; see get_distnames.
type	type of Lambert W \times F distribution: skewed "s"; heavy-tail "h"; or skewed heavy-tail "hh".
theta.fixed	a list of fixed parameters in the optimization; default only alpha = 1.
use.mean.variance	logical; if TRUE it uses mean and variance implied by \boldsymbol \beta to do the transformation (Goerg 2011). If FALSE, it uses the alternative definition from Goerg (2016) with location and scale parameter.
theta.init	a list containing the starting values of (\alpha, \boldsymbol \beta, \gamma, \delta) for the numerical optimization; default: see get_initial_theta.
hessian	indicator for returning the (numerically obtained) Hessian at the optimum; default: TRUE. If the numDeriv package is available it uses numDeriv::hessian(); otherwise stats::optim(..., hessian = TRUE).
return.estimate.only	logical; if TRUE, only a named flattened vector of \widehat{\theta}_{MLE} will be returned (only the estimated, non-fixed values). This is useful for simulations where it is usually not necessary to give a nicely organized output, but only the estimated parameter. Default: FALSE.
optim.fct	character; which R optimization function should be used. Either 'optim' (default), 'nlm', or 'solnp' from the Rsolnp package (if available). Note that if 'nlm' is used, then not.negative = TRUE will be set automatically.
not.negative	logical; if TRUE, it restricts delta or gamma to the non-negative reals. See theta2unbounded for details.

Value

A list of class LambertW_fit:

data	data y,
loglik	scalar; log-likelihood evaluated at the optimum \widehat{\theta}_{MLE},
theta.init	list; starting values for numerical optimization,
beta	estimated \boldsymbol \beta vector of the input distribution via Lambert W MLE (In general this is not exactly identical to \widehat{\boldsymbol \beta}_{MLE} for the input data),
theta	list; MLE for \theta,
type	see Arguments,
hessian	Hessian matrix; used to calculate standard errors (only if hessian = TRUE, otherwise NULL),
call	function call,
distname	see Arguments,
message	message from the optimization method. What kind of convergence?,
method	estimation method; here "MLE".

Examples

# See ?LambertW-package

LambertW documentation built on May 29, 2024, 4:30 a.m.