# beta-utils: Utilities for parameter vector beta of the input distribution In LambertW: Probabilistic Models to Analyze and Gaussianize Heavy-Tailed, Skewed Data

 beta-utils R Documentation

## Utilities for parameter vector beta of the input distribution

### Description

The parameter \boldsymbol β specifies the input distribution X \sim F_X(x \mid \boldsymbol β).

beta2tau converts \boldsymbol β to the transformation vector τ = (μ_x, σ_x, γ = 0, α = 1, δ = 0), which defines the Lambert W\times F random variable mapping from X to Y (see tau-utils). Parameters μ_x and σ_x of X in general depend on \boldsymbol β (and may not even exist for use.mean.variance = TRUE; in this case beta2tau will throw an error).

check_beta checks if \boldsymbol β defines a valid distribution, e.g., for normal distribution 'sigma' must be positive.

estimate_beta estimates \boldsymbol β 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 \boldsymbol β.

Depending on the distribution \boldsymbol β 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)

### Arguments

 beta numeric; vector \boldsymbol β 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 \boldsymbol β 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 θ for Lambert W\times F distributions see get_initial_theta and get_initial_tau.

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

### Value

beta2tau returns a numeric vector, which is τ = τ(\boldsymbol β) implied by beta and distname.

check_beta throws an error if \boldsymbol β 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 \boldsymbol β given x.

get_beta_names returns a vector of characters.

tau-utils, theta-utils

### Examples

# By default: delta = gamma = 0 and alpha = 1
beta2tau(c(1, 1), distname = "normal")
## Not run:
beta2tau(c(1, 4, 1), distname = "t")

## End(Not run)
beta2tau(c(1, 4, 1), distname = "t", use.mean.variance = FALSE)
beta2tau(c(1, 4, 3), distname = "t") # no problem

## Not run:
check_beta(beta = c(1, 1, -1), distname = "normal")

## End(Not run)

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

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