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{θ}_{MLE}.

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

By default α = 1 is fixed for any `type`. If you want to also estimate α (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 β 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 (α, \boldsymbol β, γ, δ) 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{θ}_{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{θ}_{MLE},
`theta.init` | list; starting values for numerical optimization,
`beta` | estimated \boldsymbol β vector of the input distribution via Lambert W MLE (In general this is not exactly identical to \widehat{\boldsymbol β}_{MLE} for the input data),
`theta` | list; MLE for θ,
`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

[Package LambertW version 0.6.7 Index]