README.md

In vasicekreg: Regression Modeling Using Vasicek Distribution

vasicekreg

vasicekreg provides distribution functions and GAMLSS regression families for Vasicek-type distributions on the unit interval. The package supports mean and quantile regression under the standard normal kernel and quantile regression under the logistic kernel. Normal-kernel families augmented at zero, at one, or at both boundaries are available for responses containing exact boundary values.

Available families

| Family | Kernel | Response range | Parameterization | Interpretation of mu | |---|---|---|---|---| | NVASIM | Standard normal | (0, 1) | Mean | mu = E(Y) | | NVASIQ | Standard normal | (0, 1) | Quantile | mu = Q_Y(tau) | | LVASIQ | Logistic | (0, 1) | Quantile | mu = Q_Y(tau) | | ZANVASIM | Standard normal | [0, 1) | Zero-adjusted mean | mu = E(Y | Y > 0) | | OANVASIM | Standard normal | (0, 1] | One-adjusted mean | mu = E(Y | Y < 1) | | ZOANVASIM | Standard normal | [0, 1] | Zero-and-one-adjusted mean | mu = E(Y | 0 < Y < 1) |

For all six families, sigma lies in (0, 1) and controls dispersion. For ZANVASIM, nu = P(Y = 0) and the marginal mean is E(Y) = (1 - nu) * mu. For OANVASIM, nu = P(Y = 1) and the marginal mean is E(Y) = nu + (1 - nu) * mu. In ZOANVASIM, nu = P(Y = 0) and tau = P(Y = 1 | Y > 0). Its marginal mean is E(Y) = (1 - nu) * (tau + (1 - tau) * mu). The logistic-kernel mean does not have a closed-form expression and generally differs from mu; consequently, the package does not provide a logistic-kernel mean-regression family.

Each parameterization includes density, cumulative distribution, quantile, and random generation functions:

• dNVASIM(), pNVASIM(), qNVASIM(), and rNVASIM();
• dNVASIQ(), pNVASIQ(), qNVASIQ(), and rNVASIQ();
• dLVASIQ(), pLVASIQ(), qLVASIQ(), and rLVASIQ();
• d0NVASIM(), p0NVASIM(), q0NVASIM(), and r0NVASIM();
• d1NVASIM(), p1NVASIM(), q1NVASIM(), and r1NVASIM();
• d01NVASIM(), p01NVASIM(), q01NVASIM(), and r01NVASIM().

Installation

Install the released version from CRAN:

install.packages("vasicekreg")

Install the development version from GitHub:

install.packages("remotes")
remotes::install_github("jmazucheli/vasicekreg")

Because the package contains compiled C++ code, installation from source requires an appropriate compiler toolchain.

Distribution functions

The following example uses the normal-kernel mean parameterization:

library(vasicekreg)

set.seed(123)
y <- rNVASIM(n = 1000, mu = 0.50, sigma = 0.25)

dNVASIM(x = 0.50, mu = 0.50, sigma = 0.25)
pNVASIM(q = 0.50, mu = 0.50, sigma = 0.25)
qNVASIM(p = 0.50, mu = 0.50, sigma = 0.25)

For the quantile parameterizations, tau is supplied directly to the distribution functions:

qNVASIQ(p = 0.25, mu = 0.60, sigma = 0.25, tau = 0.25)
qLVASIQ(p = 0.25, mu = 0.60, sigma = 0.25, tau = 0.25)

Both calls return mu = 0.60 because mu represents the tau-th quantile.

GAMLSS mean regression

library(gamlss)
library(vasicekreg)

set.seed(123)
dat_mean <- data.frame(
  y = rNVASIM(n = 300, mu = 0.60, sigma = 0.25)
)

fit_mean <- gamlss(
  y ~ 1,
  data = dat_mean,
  family = NVASIM(
    mu.link = "logit",
    sigma.link = "logit"
  ),
  control = gamlss.control(trace = FALSE)
)

fitted(fit_mean, what = "mu")[1]

GAMLSS zero-adjusted mean regression

set.seed(123)
dat_zero <- data.frame(
  y = r0NVASIM(n = 500, mu = 0.60, sigma = 0.25, nu = 0.20)
)

fit_zero <- gamlss(
  y ~ 1,
  sigma.formula = ~ 1,
  nu.formula = ~ 1,
  data = dat_zero,
  family = ZANVASIM(
    mu.link = "logit",
    sigma.link = "logit",
    nu.link = "logit"
  ),
  control = gamlss.control(trace = FALSE)
)

fitted(fit_zero, what = "mu")[1]
fitted(fit_zero, what = "sigma")[1]
fitted(fit_zero, what = "nu")[1]

Here mu is the conditional mean of the positive component. The fitted marginal mean is (1 - fitted(fit_zero, what = "nu")) * fitted(fit_zero, what = "mu").

GAMLSS one-adjusted and zero-and-one-adjusted regression

dat_one <- data.frame(
  y = r1NVASIM(500, mu = 0.60, sigma = 0.25, nu = 0.20)
)

fit_one <- gamlss(
  y ~ 1,
  sigma.formula = ~ 1,
  nu.formula = ~ 1,
  data = dat_one,
  family = OANVASIM(),
  control = gamlss.control(trace = FALSE)
)

dat_boundary <- data.frame(
  y = r01NVASIM(
    500, mu = 0.60, sigma = 0.25, nu = 0.20, tau = 0.25
  )
)

fit_boundary <- gamlss(
  y ~ 1,
  sigma.formula = ~ 1,
  nu.formula = ~ 1,
  tau.formula = ~ 1,
  data = dat_boundary,
  family = ZOANVASIM(),
  control = gamlss.control(trace = FALSE)
)

For ZOANVASIM, p0 = nu, p1 = (1 - nu) * tau, and pc = (1 - nu) * (1 - tau). This parameterization guarantees valid probabilities while retaining logit links for both boundary parameters. Here tau is a model parameter and is unrelated to the global quantile level used by NVASIQ() and LVASIQ().

GAMLSS quantile regression

For NVASIQ() and LVASIQ(), the quantile level must be defined as a scalar variable named tau in the global environment. It is not passed as an argument to the GAMLSS family constructor.

Normal kernel

library(gamlss)
library(vasicekreg)

set.seed(123)
tau <- 0.50

dat_normal <- data.frame(
  y = rNVASIQ(
    n = 300,
    mu = 0.60,
    sigma = 0.25,
    tau = tau
  )
)

fit_normal <- gamlss(
  y ~ 1,
  data = dat_normal,
  family = NVASIQ(
    mu.link = "logit",
    sigma.link = "logit"
  ),
  control = gamlss.control(trace = FALSE)
)

fitted(fit_normal, what = "mu")[1]

Logistic kernel

set.seed(123)
tau <- 0.25

dat_logistic <- data.frame(
  y = rLVASIQ(
    n = 300,
    mu = 0.60,
    sigma = 0.25,
    tau = tau
  )
)

fit_logistic <- gamlss(
  y ~ 1,
  data = dat_logistic,
  family = LVASIQ(
    mu.link = "logit",
    sigma.link = "logit"
  ),
  control = gamlss.control(trace = FALSE)
)

fitted(fit_logistic, what = "mu")[1]

The global value of tau must remain equal to the quantile level associated with the fitted model when residuals or other post-fit quantities are computed. Reset tau before working with a model fitted at another quantile level.

Implementation

The density, cumulative distribution, and quantile functions call compiled C++ routines through Rcpp. Random generation for NVASIM and NVASIQ uses inverse transformation with the corresponding compiled quantile functions, whereas rLVASIQ() calls a compiled random-generation routine directly. The boundary-adjusted functions reuse the compiled NVASIM functions and add the required point masses in R.

The GAMLSS family definitions and analytical log-likelihood derivatives are implemented in R. Numerical differentiation is not used. The mean and variance components of LVASIQ() are evaluated by numerical quadrature because these moments have no closed-form expressions.

Testing

From the package root directory:

Rscript -e 'testthat::test_local(path = ".", reporter = "summary")'

cd ..
R CMD build vasicekreg
R CMD check vasicekreg_1.1.0.tar.gz

Citation

To cite the package, use:

citation("vasicekreg")

The main methodological reference is:

Mazucheli, J., Alves, B., Korkmaz, M. Ç., and Leiva, V. (2022). Vasicek quantile and mean regression models for bounded data: New formulation, mathematical derivations, and numerical applications. Mathematics, 10, 1389. https://doi.org/10.3390/math10091389

License

vasicekreg is distributed under the MIT License.

vasicekreg documentation built on Aug. 20, 2026, 9:08 a.m.