Utilities for the parameter vector of Lambert W\times F distributions

Description

These functions work with \boldsymbol θ = (\boldsymbol β, γ, δ, α), which fully parametrizes Lambert W\times F distributions.

See Details for more background information on some functions.

check_theta checks if θ = (α, \boldsymbol β, γ, δ) describes a well-defined Lambert W distribution.

complete_theta completes missing values in a parameters list so users don't have to specify everything in detail. If not supplied, then alpha = 1, gamma = 0, and delta = 0 will be set by default.

flatten_theta and unflatten_theta convert between the list theta and its vector-style flattened type. The flattened version is required for several optimization routines, since they optimize over multivariate vectors – not lists.

get_initial_theta provides initial estimates for α, \boldsymbol β, γ, and δ, which are then used in maximum likelihood (ML) estimation (MLE_LambertW).

get_theta_bounds returns lower and upper bounds for θ (necessary for optimization such as MLE_LambertW).

theta2tau converts θ to the transformation vector τ = (μ_x, σ_x, γ, δ, α).

theta2unbounded transforms θ from the bounded space to an unrestricted space (by \log-transformation on σ_x, δ, and α; note that this restricts γ ≥q 0, δ ≥q 0, and α ≥q 0.).

Usage

check_theta(theta, distname)

complete_theta(theta = list(), LambertW.input = NULL)

flatten_theta(theta)

get_initial_theta(
  y,
  distname,
  type = c("h", "hh", "s"),
  theta.fixed = list(alpha = 1),
  method = c("Taylor", "IGMM"),
  use.mean.variance = TRUE
)

get_theta_bounds(
  distname,
  beta,
  type = c("s", "h", "hh"),
  not.negative = FALSE
)

theta2tau(theta = list(beta = c(0, 1)), distname, use.mean.variance = TRUE)

theta2unbounded(theta, distname, type = c("h", "hh", "s"), inverse = FALSE)

unflatten_theta(theta.flattened, distname, type)

Arguments

Details

get_initial_theta obtains a quick initial estimate of θ by first finding the (approximate) input \widehat{\boldsymbol x}_{\widehat{θ}} by IGMM, and then estimating \boldsymbol β for this input data \widehat{\boldsymbol x}_{\widehat{θ}} \sim F_X(x \mid \boldsymbol β) (see estimate_beta).

Converting theta to an unbouded space is especially useful for optimization routines (like nlm), which can be performed over an unconstrained space. The obtained optimum can be converted back to the original space using the inverse transformation (set inverse = TRUE transforms it via \exp) – this guarantees that the estimate satisfies non-negativity constraints (if required). The main advantage is that this avoids using optimization routines with boundary constraints – since they are much slower compared to uncostrained optimization.

Value

check_theta throws an error if list theta does not define a proper Lambert W \times F distribution; does nothing otherwise.

complete_theta returns a list containing:

get_initial_theta returns a list containing:

get_theta_bounds returns a list containing two vectors:

See Also

check_beta

estimate_beta, get_initial_tau

beta2tau

Examples

## Not run: 
check_theta(theta = list(beta =  c(1, 1, -1)), distname = "t")

## End(Not run)

check_theta(theta = list(beta =  c(1, 1)), distname = "normal") # ok

params <- list(beta = c(2, 1), delta = 0.3) # alpha and gamma are missing
complete_theta(params) # added default values

params <- list(beta = c(2, 1), delta = 0.3, alpha = c(1, 2))
params <- complete_theta(params)
check_theta(params, distname = 'normal')

###
x <- rnorm(1000)
get_initial_theta(x, distname = "normal", type = "h")
get_initial_theta(x, distname = "normal", type = "s")

# starting values for the skewed version of an exponential
y <- rLambertW(n = 1000, distname = "exp", beta = 2, gamma = 0.1)
get_initial_theta(y, distname = "exp", type = "s")

# starting values for the heavy-tailed version of a Normal = Tukey's h
y <- rLambertW(n = 1000, beta = c(2, 1), distname = "normal", delta = 0.2)
get_initial_theta(y, distname = "normal", type = "h")#' 

###
get_theta_bounds(type = "hh", distname = "normal", beta = c(0, 1))

### 
theta.restr <- theta2unbounded(list(beta = c(-1, 0.1), 
                                    delta = c(0.2, 0.2)), 
                                    distname = "normal")
theta.restr
# returns again the beta and delta from above
theta2unbounded(theta.restr, inverse = TRUE, distname = "normal")