LambertW (version 0.6.9-1)

beta-utils: Utilities for parameter vector beta of the input distribution

Description

The parameter $\mathit{\beta }$ specifies the input distribution $X\sim F_X(x\mid \mathit{\beta })$.

beta2tau converts $\mathit{\beta }$ to the transformation vector $\tau =({\mu }_x,{\sigma }_x,\gamma =0,\alpha =1,\delta =0)$, which defines the Lambert W$\times$ F random variable mapping from $X$ to $Y$ (see tau-utils). Parameters ${\mu }_x$ and ${\sigma }_x$ of $X$ in general depend on $\mathit{\beta }$ (and may not even exist for use.mean.variance = TRUE; in this case beta2tau will throw an error).

check_beta checks if $\mathit{\beta }$ defines a valid distribution, e.g., for normal distribution 'sigma' must be positive.

estimate_beta estimates $\mathit{\beta }$ for a given $F_X$ using MLE or methods of moments. Closed form solutions are used if they exist; otherwise the MLE is obtained numerically using fitdistr.

get_beta_names returns (typical) names for each component of $\mathit{\beta }$.

Depending on the distribution $\mathit{\beta }$ has different length and names: e.g., for a "normal" distribution beta is of length $2$ ("mu", "sigma"); for an "exp"onential distribution beta is a scalar (rate "lambda").

Usage

beta2tau(beta, distname, use.mean.variance = TRUE)
check_beta(beta, distname)
estimate_beta(x, distname)
get_beta_names(distname)

Value

beta2tau returns a numeric vector, which is $\tau =\tau (\mathit{\beta })$ implied by beta and distname.

check_beta throws an error if $\mathit{\beta }$ is not appropriate for the given distribution; e.g., if it has too many values or if they are not within proper bounds (e.g., beta['sigma'] of a

"normal" distribution must be positive).

estimate_beta returns a named vector with estimates for $\mathit{\beta }$ given x.

get_beta_names returns a vector of characters.

Arguments

beta

numeric; vector $\mathit{\beta }$ of the input distribution; specifications as they are for the R implementation of this distribution. For example, if distname = "exp", then beta = 2 means that the rate of the exponential distribution equals $2$; if distname = "normal" then beta = c(1,2) means that the mean and standard deviation are 1 and 2, respectively.

distname

character; name of input distribution; see get_distnames.

use.mean.variance

logical; if TRUE it uses mean and variance implied by $\mathit{\beta }$ to do the transformation (Goerg 2011). If FALSE, it uses the alternative definition from Goerg (2016) with location and scale parameter.

x

a numeric vector of real values (the input data).

Details

estimate_beta does not do any data transformation as part of the Lambert W$\times$ F input/output framework. For an initial estimate of $\theta$ for Lambert W$\times$ F distributions see get_initial_theta and get_initial_tau.

A quick initial estimate of $\theta$ is obtained by first finding the (approximate) input ${\hat{\mathit{x}}}_{\hat{\theta }}$ by IGMM, and then getting the MLE of $\mathit{\beta }$ for this input data ${\hat{\mathit{x}}}_{\hat{\theta }}\sim F_X(x\mid \mathit{\beta })$ (usually using fitdistr).

See Also

tau-utils, theta-utils

Examples

# By default: delta = gamma = 0 and alpha = 1
beta2tau(c(1, 1), distname = "normal") 
if (FALSE) {
  beta2tau(c(1, 4, 1), distname = "t")
}
beta2tau(c(1, 4, 1), distname = "t", use.mean.variance = FALSE)
beta2tau(c(1, 4, 3), distname = "t") # no problem


if (FALSE) {
check_beta(beta = c(1, 1, -1), distname = "normal")
}


set.seed(124)
xx <- rnorm(100)^2
estimate_beta(xx, "exp")
estimate_beta(xx, "chisq")