# MLE_LambertW: Maximum Likelihood Estimation for Lambert W \times F... In LambertW: Probabilistic Models to Analyze and Gaussianize Heavy-Tailed, Skewed Data

 MLE_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



LambertW documentation built on Sept. 22, 2022, 5:07 p.m.