beta-utils function

Utilities for parameter vector beta of the input distribution

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

beta2tau converts $\boldsymbol \beta$ 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 $\boldsymbol \beta$

(and may not even exist for use.mean.variance = TRUE; in this case beta2tau will throw an error).

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

estimate_beta estimates $\boldsymbol \beta$ 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 $\boldsymbol \beta$.

Depending on the distribution $\boldsymbol \beta$ 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").

beta2tau(beta, distname, use.mean.variance = TRUE)

check_beta(beta, distname)

estimate_beta(x, distname)

get_beta_names(distname)

Arguments

• beta: numeric; vector $\boldsymbol \beta$ 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 $\boldsymbol \beta$ 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).

Returns

beta2tau returns a numeric vector, which is c("$\\tau =\n$", "$\\tau(\\boldsymbol \\beta)$") implied by beta and distname.

check_beta throws an error if $\boldsymbol \beta$ 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 $\boldsymbol \beta$ given x.

get_beta_names returns a vector of characters.

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 $\widehat{\boldsymbol x}_{\widehat{\theta}}$ by IGMM, and then getting the MLE of $\boldsymbol \beta$

for this input data c("$\\widehat{\\boldsymbol x}_{\\widehat{\\theta}} \\sim\n$", "$F_X(x \\mid \\boldsymbol \\beta)$") (usually using fitdistr).

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")

See Also

tau-utils, theta-utils