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

MLE_LambertW {LambertW}

R Documentation

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