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().
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
)A list of class LambertW_fit:
data y,
scalar; log-likelihood evaluated at the optimum \(\widehat{\theta}_{MLE}\),
list; starting values for numerical optimization,
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),
list; MLE for \(\theta\),
see Arguments,
Hessian matrix; used to calculate standard errors (only if hessian = TRUE,
otherwise NULL),
function call,
see Arguments,
message from the optimization method. What kind of convergence?,
estimation method; here "MLE".
a numeric vector of real values.
character; name of input distribution; see
get_distnames.
type of Lambert W \(\times\) F distribution: skewed "s";
heavy-tail "h"; or skewed heavy-tail "hh".
a list of fixed parameters in the optimization; default
only alpha = 1.
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.
a list containing the starting values of \((\alpha,
\boldsymbol \beta, \gamma, \delta)\) for the numerical optimization;
default: see get_initial_theta.
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).
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.
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.
logical; if TRUE, it restricts delta or
gamma to the non-negative reals. See theta2unbounded
for details.