BCSreg (version 1.1.1)

ZABCS: The Zero-Adjusted Box-Cox Symmetric Distributions

Description

Density function, distribution function, quantile function, and random generation for the class of the zero-adjusted Box-Cox symmetric (ZABCS) distributions.

Usage

dZABCS(x, alpha, mu, sigma, lambda, zeta, family = "NO", log = FALSE)
pZABCS(
  q,
  alpha,
  mu,
  sigma,
  lambda,
  zeta,
  family = "NO",
  lower.tail = TRUE,
  log.p = FALSE
)
qZABCS(
  p,
  alpha,
  mu,
  sigma,
  lambda,
  zeta,
  family = "NO",
  lower.tail = TRUE,
  log.p = FALSE
)
rZABCS(n, alpha, mu, sigma, lambda, zeta, family = "NO")

Value

dZABCS returns the density function, pZABCS gives the cumulative distribution function, qZABCS provides the quantile function, and rZABCS generates random variables.

Arguments

x, q

vector of positive quantiles.

alpha

vector of zero-adjusted parameters, with values on (0, 1).

mu

vector of strictly positive scale parameters.

sigma

vector of strictly positive relative dispersion parameters.

lambda

vector of real-valued skewness parameters. If lambda = 0, the BCS distribution reduces to the corresponding log-symmetric distribution with parameters mu and sigma (and a possible extra parameter zeta).

zeta

strictly positive extra parameter. It must be specified with only one value in cases where the BCS distribution has an extra parameter. See “Details” below.

family

a character that specifies the symmetric generating family of the BCS distribution. Available options are: "NO" (default), "ST", "LOI", "LOII", "PE", "SN", "HP", and "SL", corresponding to the normal, Student-t, type I logistic, type II logistic, power exponential, sinh-normal, hyperbolic, and slash distributions, respectively.

log, log.p

logical; if TRUE, probabilities p are given as log(p). Default is FALSE.

lower.tail

logical; if TRUE (default), probabilities are \(P(X \le x)\) otherwise, \(P(X > x)\).

p

vector of probabilities.

n

number of observations. If `n` is a vector, its length is used as the number of required observations.

Author

Francisco F. de Queiroz <>

Rodrigo M. R. de Medeiros <>

Details

The class of the ZABCS distributions was introduced by Medeiros and Queiroz (2025) as an extension of the Box-Cox symmetric (BCS) distributions (Ferrari and Fumes, 2017). The models consists of a broad class of probability distributions for positive continuous data which may include zeros.

Let \(Y\) be a positive continuous random variable with a ZABCS distribution with parameters \(\alpha \in (0, 1)\), \(\mu > 0\), \(\sigma > 0\), and \(\lambda \in \mathbb{R}\) and density generating function \(r\). The probability density function of \(Y\) is given by

\( f^{(0)}(y; \alpha, \mu, \sigma, \lambda) = \left\{ \begin{array}{rcl} \alpha, & y=0,\\ (1 - \alpha)f(y; \alpha, \mu, \sigma, \lambda), & y > 0,\\ \end{array} \right. \)

where

\( f(y; \mu, \sigma, \lambda) = \left\{\begin{array}{ll} \dfrac{y^{\lambda-1}}{\mu^\lambda \sigma} \dfrac{r(z^2)}{R\left(\frac{1}{\sigma |\lambda|}\right)}, & \mbox{ if } \lambda \neq 0,\\ \dfrac{1}{y\sigma} r(z^2), & \mbox{ if } \lambda = 0, \end{array}\right., \quad y > 0, \)

with

\( z = \left\{ \begin{array}{ll} \dfrac{1}{\sigma \lambda} \left\{\left(\frac{y}{\mu}\right)^\lambda - 1 \right\}, & \mbox{ if } \lambda \neq 0, \\ \dfrac{1}{\sigma} \log\left(\frac{y}{\mu}\right), & \mbox{ if } \lambda = 0, \end{array} \right. \)

\(r:[0,\infty) \longrightarrow [0, \infty)\) satisfies \(\int_0^\infty u^{-1/2}r(u)\textrm{d} u = 1\), and \(R(x) = \int_{-\infty}^x r(u^2)\textrm{d} u, x \in \mathbb{R}\).

The function \(r\) is called density generating function, and it specifies the generating symmetric family of \(Y\) within the class of the ZABCS probability models. This function can also depend on extra parameters, such as the zero-adjusted Box-Cox t and zero-adjusted Box-Cox power exponential distributions. We call these extra parameters zeta. The currently available ZABCS distributions in the BCSreg package are listed below:

Distribution	Family abbreviation	N. of extra parameters
Zero-adjusted Box-Cox Hyperbolic	"HP"	1
Zero-adjusted Box-Cox Type I Logistic	"LOI"	0
Zero-adjusted Box-Cox Type II Logistic	"LOII"	0
Zero-adjusted Box-Cox Normal	"NO"	0
Zero-adjusted Box-Cox Power Exponential	"PE"	1
Zero-adjusted Box-Cox Sinh-Normal	"SN"	1
Zero-adjusted Box-Cox Slash	"SL"	1
Zero-adjusted Box-Cox t	"ST"	1

References

Ferrari, S. L. P., and Fumes, G. (2017). Box-Cox symmetric distributions and applications to nutritional data. AStA Advances in Statistical Analysis, 101, 321-344.

Medeiros, R. M. R., and Queiroz, F. F. (2025). Flexible modeling of nonnegative continuous data: Box-Cox symmetric regression and its zero-adjusted extension.

See Also

BCS to access the density function, distribution function, quantile function, and a random number generator for the BCS distributions. BCSreg for estimating the parameters of a ZABCS regression model.

Examples

# Probability density function

## Right-skewed distributions
curve(dZABCS(x, 0.4, 3, 0.3, -1.5, family = "NO"), from = 0.001, to = 7,
      xlim = c(0, 7), ylim = c(0, 0.5), ylab = "Density")
curve(dZABCS(x, 0.4, 3, 0.3, -1.5, 4, family = "ST"), add = TRUE, col = 2, from = 0.001)
curve(dZABCS(x, 0.4, 3, 0.3, -1.5, 5, family = "PE"), add = TRUE, col = 4, from = 0.001)
points(0, 0.4, type = "h", lty = 2)
points(0, 0.4, pch = 16, lty = 2)
legend("topright", legend = c("BCNO", "BCT", "BCPE"), lty = 1, col = c(1, 2, 4))

## Truncated symmetric distributions (with support on (0, Inf))
curve(dZABCS(x, 0.4, 3, 0.3, 1, family = "NO"), from = 0.001, to = 7,
      xlim = c(0, 7), ylim = c(0, 0.5), ylab = "Density")
curve(dZABCS(x, 0.4, 3, 0.3, 1, 4, family = "ST"), add = TRUE, col = 2, from = 0.001)
curve(dZABCS(x, 0.4, 3, 0.3, 1, 5, family = "PE"), add = TRUE, col = 4, from = 0.001)
points(0, 0.4, type = "h", lty = 2)
points(0, 0.4, pch = 16, lty = 2)
legend("topright", legend = c("BCNO", "BCT", "BCPE"), lty = 1, col = c(1, 2, 4))

## Left-skewed distributions
curve(dZABCS(x, 0.4, 3, 0.3, 3, family = "NO"), from = 0.001, to = 7,
      xlim = c(0, 7), ylim = c(0, 0.5), ylab = "Density")
curve(dZABCS(x, 0.4, 3, 0.3, 3, 4, family = "ST"), add = TRUE, col = 2, from = 0.001)
curve(dZABCS(x, 0.4, 3, 0.3, 3, 5, family = "PE"), add = TRUE, col = 4, from = 0.001)
points(0, 0.4, type = "h", lty = 2)
points(0, 0.4, pch = 16, lty = 2)
legend("topright", legend = c("BCNO", "BCT", "BCPE"), lty = 1, col = c(1, 2, 4))


# Random generation

## Parameter setting
alpha <- 0.2   # zero-adjustment parameter
mu <- 5        # scale parameter
sigma <- 0.2   # relative dispersion parameter
lambda <- -0.2 # skewness parameter

## Generating family
family <- "NO"

## Visualization
x <- rZABCS(10000, alpha, mu, sigma, lambda, family = family)

hist(x, prob = TRUE, col = "white", main = "")

points(0, mean(x == 0), type = "h", lty = 2)
points(0, mean(x == 0), pch = 16, lty = 2)
curve(dZABCS(x, alpha, mu, sigma, lambda, zeta, family = family), col = "blue", add = TRUE)

plot(ecdf(x), main = "")
curve(pZABCS(x, alpha, mu, sigma, lambda, zeta, family = family), col = "blue", add = TRUE)