U-utils function

Zero-mean, unit-variance version of standard distributions

Density, distribution function, quantile function and random number generation for the shifted and scaled U of the (location-)scale family input $X \sim F_X(x \mid \boldsymbol \beta)$

• see References.

Since the normalized random variable U is one of the main building blocks of Lambert W $\times$ F distributions, these functions are wrappers used by other functions such as dLambertW or rLambertW.

dU(u, beta, distname, use.mean.variance = TRUE)

pU(u, beta, distname, use.mean.variance = TRUE)

qU(p, beta, distname, use.mean.variance = TRUE)

rU(n, beta, distname, use.mean.variance = TRUE)

Arguments

• u: vector of quantiles.
• beta: numeric vector (deprecated); parameter $\boldsymbol \beta$ of the input distribution. See check_beta on how to specify beta for each distribution.
• 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.
• p: vector of probability levels
• n: number of samples

Returns

dU evaluates the pdf at y, pU evaluates the cdf, qU is the quantile function, and rU generates random samples from U.

Examples

# a zero-mean, unit variance version of the t_3 distribution.
curve(dU(x, beta = c(1, 1, 3), distname = "t"), -4, 4,
      ylab = "pdf", xlab = "u",
      main = "student-t \n zero-mean, unit variance")
# cdf of unit-variance version of an exp(3) -> just an exp(1)
curve(pU(x, beta = 3, distname = "exp"), 0, 4, ylab = "cdf", xlab = "u",
      main = "Exponential \n unit variance", col = 2, lwd = 2) 
curve(pexp(x, rate = 1), 0, 4, add = TRUE, lty = 2)
# all have (empirical) variance 1
var(rU(n = 1000, distname = "chisq", beta = 2))
var(rU(n = 1000, distname = "normal", beta = c(3, 3)))
var(rU(n = 1000, distname = "exp", beta = 1))
var(rU(n = 1000, distname = "unif", beta = c(0, 10)))