Package {vasicekreg}

Overview of the vasicekreg package

Description

The vasicekreg package provides distribution functions and GAMLSS families for Vasicek-type distributions on the unit interval. Three base families are available:

• NVASIM: normal kernel with mean parameterization, where \mu=E(Y).

• NVASIQ: normal kernel with quantile parameterization, where \mu=Q_Y(\tau) for a fixed \tau\in(0,1).

• LVASIQ: logistic kernel with quantile parameterization, where \mu=Q_Y(\tau) for a fixed \tau\in(0,1).

For responses observed at the boundaries, the normal-kernel mean model is also available as:

• ZANVASIM: point mass at zero and a continuous component on (0,1).

• OANVASIM: point mass at one and a continuous component on (0,1).

• ZOANVASIM: point masses at zero and one and a continuous component on (0,1).

The parameter \sigma\in(0,1) controls dispersion in the continuous Vasicek component. The corresponding d, p, q, and r functions provide density or probability mass values, cumulative probabilities, quantiles, and random observations, respectively.

Details

bodyfat: Body fat dataset.

NVASIM: Normal-kernel mean parameterization and GAMLSS family. In regression models, covariates describe the conditional mean through \mu.

NVASIQ: Normal-kernel quantile parameterization and GAMLSS family. For a fixed quantile level \tau, covariates describe the conditional \tau-th quantile through \mu.

LVASIQ: Logistic-kernel quantile parameterization and GAMLSS family. For a fixed quantile level \tau, covariates describe the conditional \tau-th quantile through \mu. A logistic-kernel mean-regression family is not provided because the mean has no closed-form expression and does not equal \mu under this parameterization.

ZANVASIM: Zero-adjusted normal-kernel mean family. Here \nu=P(Y=0), \mu=E(Y\mid Y>0), and the marginal mean is E(Y)=(1-\nu)\mu.

OANVASIM: One-adjusted normal-kernel mean family. Here \nu=P(Y=1), \mu=E(Y\mid Y<1), and the marginal mean is E(Y)=\nu+(1-\nu)\mu. The parameters \mu and \nu therefore have the same interpretations as their counterparts in the one-inflated beta family BEOI. The shape parameter \sigma is distribution-specific and should not be compared directly between these families.

ZOANVASIM: Zero-and-one-adjusted normal-kernel mean family. Here \nu=P(Y=0), \tau=P(Y=1\mid Y>0), and \mu=E(Y\mid 0<Y<1). Consequently, P(Y=1)=(1-\nu)\tau and E(Y)=(1-\nu)[\tau+(1-\tau)\mu].

The distribution functions dNVASIM, pNVASIM, qNVASIM, dNVASIQ, pNVASIQ, qNVASIQ, dLVASIQ, pLVASIQ, and qLVASIQ call compiled C++ routines through Rcpp. The boundary-adjusted distribution functions are implemented in R and reuse the compiled NVASIM functions for their continuous component. Parameter validation, the GAMLSS family definitions, and all log-likelihood derivatives are implemented in R. The mean and variance components of the LVASIQ() family object are obtained by numerical quadrature because these moments have no closed-form expressions.

For the distribution functions associated with NVASIQ and LVASIQ, tau is supplied as an argument. For GAMLSS fitting, NVASIQ() and LVASIQ() require tau to be defined as a scalar variable in the global environment. The same value must be retained when residuals or other post-fit quantities are computed. This fixed quantile level is distinct from the parameter tau in ZOANVASIM(), which represents the conditional probability at one among nonzero observations.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

Bruna Alves pg402900@uem.br

Examples

if (requireNamespace("betareg", quietly = TRUE)) {
    data("ReadingSkills", package = "betareg")
    ReadingSkills$dyslexia <- stats::relevel(
        factor(ReadingSkills$dyslexia), ref = "no"
    )

    control <- gamlss::gamlss.control(n.cyc = 200, trace = FALSE)

    ## In both models, mu = E(Y | Y < 1) and nu = P(Y = 1).
    fit_oanvasim <- gamlss::gamlss(
        accuracy1 ~ dyslexia * iq,
        sigma.formula = ~ dyslexia + iq,
        nu.formula = ~ 1,
        family = OANVASIM(),
        data = ReadingSkills,
        control = control
    )

    fit_beoi <- gamlss::gamlss(
        accuracy1 ~ dyslexia * iq,
        sigma.formula = ~ dyslexia + iq,
        nu.formula = ~ 1,
        family = gamlss.dist::BEOI(),
        data = ReadingSkills,
        control = control
    )

    n <- nrow(ReadingSkills)
    comparison <- data.frame(
        family = c("OANVASIM", "BEOI"),
        logLik = -c(
            fit_oanvasim$G.deviance,
            fit_beoi$G.deviance
        ) / 2,
        AIC = c(
            gamlss::GAIC(fit_oanvasim, k = 2),
            gamlss::GAIC(fit_beoi, k = 2)
        ),
        BIC = c(
            gamlss::GAIC(fit_oanvasim, k = log(n)),
            gamlss::GAIC(fit_beoi, k = log(n))
        )
    )
    comparison
}

L-Vasicek distribution (logistic kernel) with quantile parameterization

Description

The function LVASIQ() defines the logistic-kernel Vasicek distribution as a gamlss.family object for conditional quantile regression. The functions dLVASIQ, pLVASIQ, qLVASIQ, and rLVASIQ give the density, distribution function, quantile function, and random generation. The parameter \mu is the conditional \tau-th quantile (0<\mu<1), \sigma is a dispersion parameter (0<\sigma<1), and \tau\in(0,1) is fixed by the user. For GAMLSS fitting, tau must be defined as a scalar variable in the global environment before LVASIQ() is evaluated.

Usage

dLVASIQ(x, mu, sigma, tau = 0.5, log = FALSE)

pLVASIQ(q, mu, sigma, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qLVASIQ(p, mu, sigma, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rLVASIQ(n, mu, sigma, tau = 0.5)

LVASIQ(mu.link = "logit", sigma.link = "logit")

Arguments

Details

Let \mathrm{logit}(u)= \log\left(\frac{u}{1-u}\right) and \Lambda(z)=\frac{1}{1+e^{-z}}, with \lambda(z)= \Lambda(z)\left[1-\Lambda(z)\right]. Define

z = \sqrt{\frac{1-\sigma}{\sigma}} \left[\mathrm{logit}(x)-\mathrm{logit}(\mu)\right] +\mathrm{logit}(\tau).

Cumulative distribution function

F(x\mid\mu,\sigma,\tau)=\Lambda(z).

Probability density function

f(x\mid\mu,\sigma,\tau)= \sqrt{\frac{1-\sigma}{\sigma}} \frac{\lambda(z)}{x(1-x)}.

Quantile function

Q(p\mid\mu,\sigma,\tau)= \Lambda\!\left\{ \mathrm{logit}(\mu) +\sqrt{\frac{\sigma}{1-\sigma}} \left[\mathrm{logit}(p)-\mathrm{logit}(\tau)\right] \right\}.

By construction Q(\tau)=\mu, i.e. \mu is the \tau-th quantile. Note that, unlike the normal-kernel Vasicek distribution, the logistic kernel does not yield a closed-form mean; in particular E(X)\neq\mu in general.

The GAMLSS family uses analytical derivatives. For one observation, let

a=\sqrt{\frac{1-\sigma}{\sigma}},\qquad d=\mathrm{logit}(y)-\mathrm{logit}(\mu),\qquad P=\Lambda\left\{ad+\mathrm{logit}(\tau)\right\},

and define

V=P(1-P),\qquad b=\frac{1}{2\sigma(1-\sigma)},\qquad g=\frac{1}{\mu(1-\mu)}.

If \ell denotes the individual log-likelihood contribution, the first derivatives are

\frac{\partial\ell}{\partial\mu} =-ag(1-2P)

and

\frac{\partial\ell}{\partial\sigma} =-b\left\{1+ad(1-2P)\right\}.

The second and cross derivatives are

\frac{\partial^2\ell}{\partial\mu^2} =g^2\left\{ a(1-2\mu)(1-2P)-2a^2V \right\},

\frac{\partial^2\ell}{\partial\mu\,\partial\sigma} =abg\left\{ (1-2P)-2adV \right\},

and

\frac{\partial^2\ell}{\partial\sigma^2} =b^2\left\{ 2(1-2\sigma) +ad(3-4\sigma)(1-2P) -2a^2d^2V \right\}.

These expressions are evaluated directly by LVASIQ(); numerical differentiation is not used. The mean and variance components of the family object use numerical quadrature because the corresponding moments do not have elementary closed forms.

Value

dLVASIQ gives the density, pLVASIQ the distribution function, qLVASIQ the quantile function, and rLVASIQ generates random deviates. LVASIQ() returns a gamlss.family object.

Note

The global variable tau must remain equal to the quantile level associated with a fitted model when residuals or other post-fit quantities are computed.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

References

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

Vasicek, O. A. (2002). The distribution of loan portfolio value. Risk, 15(12), 1–10.

Examples

set.seed(123)
x <- rLVASIQ(n = 1000, mu = 0.50, sigma = 0.25, tau = 0.5)
S <- seq(min(x), max(x), length.out = 1000)

hist(x, prob = TRUE, main = "L-Vasicek (logistic kernel)")
lines(S, dLVASIQ(x = S, mu = 0.50, sigma = 0.25, tau = 0.5), col = 2)

plot(ecdf(x))
lines(S, pLVASIQ(q = S, mu = 0.50, sigma = 0.25, tau = 0.5), col = 2)

data <- data.frame(
    y = rLVASIQ(n = 100, mu = 0.50, sigma = 0.25, tau = 0.50)
)
tau <- 0.50
fit <- gamlss::gamlss(
    y ~ 1,
    data = data,
    family = LVASIQ(mu.link = "logit", sigma.link = "logit")
)
fitted(fit, what = "mu")[1:5]
rm(tau)

N-Vasicek distribution (normal kernel) with mean parameterization

Description

Defines the normal-kernel Vasicek distribution under a mean parameterization for use as a gamlss.family. The parameter \mu represents the mean of the distribution, with 0 < \mu < 1, and \sigma is a shape parameter.

The density, distribution function, quantile function and random number generation are provided by dNVASIM(), pNVASIM(), qNVASIM() and rNVASIM(), respectively.

Usage

dNVASIM(x, mu, sigma, log = FALSE)

pNVASIM(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)

qNVASIM(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)

rNVASIM(n, mu, sigma)

NVASIM(mu.link = "logit", sigma.link = "logit")

Arguments

Details

Probability density function

f(x\mid \mu ,\sigma )=\sqrt{\frac{1-\sigma }{\sigma }}\exp \left\{ \frac{1}{2}\left[ \Phi ^{-1}\left( x\right) ^{2}-\left( \frac{\Phi ^{-1}\left( x\right) \sqrt{1-\sigma }-\Phi ^{-1}\left( \mu \right) }{\sqrt{\sigma }}\right) ^{2}\right] \right\}

Cumulative distribution function

F(x\mid \mu ,\sigma )=\Phi \left( \frac{\Phi ^{-1}\left( x\right) \sqrt{1-\sigma }-\Phi ^{-1}\left( \mu \right) }{\sqrt{\sigma }}\right)

Quantile function

Q(\tau \mid \mu ,\sigma )=F^{-1}(\tau \mid \mu ,\sigma )=\Phi \left(\frac{\Phi ^{-1}\left(\mu\right) +\Phi ^{-1}\left( \tau \right) \sqrt{\sigma }}{\sqrt{1-\sigma }}\right)

Expected value

E(X) = \mu

Variance

Var(X) = \Phi_2\left ( \Phi^{-1}(\mu),\Phi^{-1}(\mu),\sigma \right )-\mu^2

where (x, \mu, \sigma, \tau) \in (0,1) and \Phi_2(\cdot) is the probability distribution function for the standard bivariate normal distribution with correlation \sigma.

Value

NVASIM() returns a gamlss.family object.

Note

In the NVASIM() parameterization, \mu corresponds to the mean of the distribution and \sigma is a shape parameter.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com Bruna Alves pg402900@uem.br

References

Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models. Chapman and Hall, London.

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

Rigby, R. A. and Stasinopoulos, D. M. (2005). Generalized additive models for location, scale and shape (with discussion). Applied Statistics, 54(3), 507–554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019). Distributions for Modeling Location, Scale, and Shape: Using GAMLSS in R. Chapman and Hall/CRC.

Stasinopoulos, D. M. and Rigby, R. A. (2007). Generalized additive models for location, scale and shape (GAMLSS) in R. Journal of Statistical Software, 23(7), 1–45.

Stasinopoulos, D. M., Rigby, R. A., Heller, G., Voudouris, V., and De Bastiani, F. (2017). Flexible Regression and Smoothing: Using GAMLSS in R. Chapman and Hall/CRC.

Vasicek, O. A. (1987). Probability of loss on loan portfolio. KMV Corporation.

Vasicek, O. A. (2002). The distribution of loan portfolio value. Risk, 15(12), 1–10.

See Also

NVASIQ, pmvnorm

Examples

set.seed(123)
x <- rNVASIM(n = 1000, mu = 0.50, sigma = 0.69)
R <- range(x)
S <- seq(from = R[1], to = R[2], length.out = 1000)

hist(x, prob = TRUE, main = 'Vasicek')
lines(S, dNVASIM(x = S, mu = 0.50, sigma = 0.69), col = 2)

plot(ecdf(x))
lines(S, pNVASIM(q = S, mu = 0.50, sigma = 0.69), col = 2)

plot(quantile(x, probs = S), type = "l")
lines(qNVASIM(p = S, mu = 0.50, sigma = 0.69), col = 2)

library(gamlss)
set.seed(123)
data <- data.frame(y =  rNVASIM(n = 100, mu = 0.5, sigma = 0.69))

fit <- gamlss(y ~ 1, data = data, mu.link = 'logit', sigma.link = 'logit', family = NVASIM)
1 /(1 + exp(-fit$mu.coefficients))
1 /(1 + exp(-fit$sigma.coefficients))

## Not run: 
library(gamlss)
set.seed(123)

n <- 1000
x <- rbinom(n, size = 1, prob = 0.5)
eta <- 0.5 + 1 * x;
mu <- 1 / (1 + exp(-eta));
sigma <- 0.5;
y <- rNVASIM(n, mu, sigma)
data <- data.frame(y, x)

fit <- gamlss(y ~ x, data = data, family = NVASIM, mu.link = 'logit', sigma.link = 'logit', 
control = gamlss.control(n.cyc = 200))
summary(fit)

## End(Not run)

N-Vasicek distribution (normal kernel) with quantile parameterization

Description

The function NVASIQ() defines the normal-kernel Vasicek distribution as a gamlss.family object. In this parameterization, \mu corresponds to the fixed \tau-th quantile and \sigma is a shape parameter. For GAMLSS fitting, tau must be defined as a scalar variable in the global environment before NVASIQ() is evaluated. The functions dNVASIQ, pNVASIQ, qNVASIQ, and rNVASIQ define the density, distribution function, quantile function, and random generation for the Vasicek distribution, respectively.

Usage

dNVASIQ(x, mu, sigma, tau = 0.5, log = FALSE)

pNVASIQ(q, mu, sigma, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qNVASIQ(p, mu, sigma, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rNVASIQ(n, mu, sigma, tau = 0.5)

NVASIQ(mu.link = "logit", sigma.link = "logit")

Arguments

Details

Probability density function:

f\left(x \mid \mu, \sigma, \tau\right) = \sqrt{\frac{1-\sigma}{\sigma}} \exp\left\{\frac{1}{2}\left[\Phi^{-1}(x)^2 - \left(\frac{\sqrt{1-\sigma}\left(\Phi^{-1}(x)-\Phi^{-1}(\mu)\right) - \sqrt{\sigma}\,\Phi^{-1}(\tau)}{\sqrt{\sigma}}\right)^2\right]\right\}.

Cumulative distribution function:

F\left(x \mid \mu, \sigma, \tau\right) = \Phi\left(\frac{\sqrt{1-\sigma}\left(\Phi^{-1}(x)-\Phi^{-1}(\mu)\right) - \sqrt{\sigma}\,\Phi^{-1}(\tau)}{\sqrt{\sigma}}\right).

where 0 < (x, \mu, \tau, \sigma) < 1, \mu is the \tau-th quantile, and \sigma is the shape parameter.

Value

NVASIQ() returns a gamlss.family object that can be used to fit a Vasicek distribution using the gamlss function.

Note

For NVASIQ(), \mu corresponds to the \tau-th quantile and \sigma is a shape parameter. The global variable tau must remain set to the quantile level associated with a fitted model when residuals or other post-fit quantities are computed.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

Bruna Alves pg402900@uem.br

References

Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models. Chapman and Hall, London.

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.

Rigby, R. A. and Stasinopoulos, D. M. (2005). Generalized additive models for location, scale and shape (with discussion). Applied Statistics, 54(3), 507–554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019). Distributions for Modeling Location, Scale, and Shape: Using GAMLSS in R. Chapman and Hall/CRC.

Stasinopoulos, D. M. and Rigby, R. A. (2007). Generalized additive models for location, scale and shape (GAMLSS) in R. Journal of Statistical Software, 23(7), 1–45.

Stasinopoulos, D. M., Rigby, R. A., Heller, G., Voudouris, V., and De Bastiani, F. (2017). Flexible Regression and Smoothing: Using GAMLSS in R. Chapman and Hall/CRC.

Vasicek, O. A. (1987). Probability of loss on loan portfolio. KMV Corporation.

Vasicek, O. A. (2002). The distribution of loan portfolio value. Risk, 15(12), 1–10.

See Also

NVASIM

Examples

set.seed(123)
x <- rNVASIQ(n = 1000, mu = 0.50, sigma = 0.69, tau = 0.50)
R <- range(x)
S <- seq(from = R[1], to = R[2], length.out = 1000)

hist(x, prob = TRUE, main = "Vasicek")
lines(S, dNVASIQ(x = S, mu = 0.50, sigma = 0.69, tau = 0.50), col = 2)

plot(ecdf(x))
lines(S, pNVASIQ(q = S, mu = 0.50, sigma = 0.69, tau = 0.50), col = 2)

plot(quantile(x, probs = S), type = "l")
lines(qNVASIQ(p = S, mu = 0.50, sigma = 0.69, tau = 0.50), col = 2)

library(gamlss)
set.seed(123)
data <- data.frame(y = rNVASIQ(n = 100, mu = 0.50, sigma = 0.69, tau = 0.50))

tau <- 0.5
fit <- gamlss(y ~ 1, data = data,
              family = NVASIQ(mu.link = "logit",
                             sigma.link = "logit"))
1 / (1 + exp(-fit$mu.coefficients))
1 / (1 + exp(-fit$sigma.coefficients))

set.seed(123)
n <- 100
x <- rbinom(n, size = 1, prob = 0.5)
eta <- 0.5 + 1 * x
mu <- 1 / (1 + exp(-eta))
sigma <- 0.5
y <- rNVASIQ(n, mu, sigma, tau = 0.5)
data <- data.frame(y, x)

tau <- 0.5
fit <- gamlss(y ~ x, data = data, family = NVASIQ)

fittaus <- lapply(c(0.10, 0.25, 0.50, 0.75, 0.90), function(Tau) {
  tau <<- Tau
  gamlss(y ~ x, data = data, family = NVASIQ)
})

sapply(fittaus, summary)

One-adjusted N-Vasicek distribution with mean parameterization

Description

Defines a one-adjusted normal-kernel Vasicek distribution for responses in (0,1]. The parameter \nu is the probability at one. Conditional on an observation in (0,1), the distribution is NVASIM with mean \mu and shape parameter \sigma.

Usage

d1NVASIM(x, mu = 0.5, sigma = 0.5, nu = 0.1, log = FALSE)

p1NVASIM(q, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

q1NVASIM(p, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

r1NVASIM(n, mu = 0.5, sigma = 0.5, nu = 0.1)

dOANVASIM(x, mu = 0.5, sigma = 0.5, nu = 0.1, log = FALSE)

pOANVASIM(q, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

qOANVASIM(p, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

rOANVASIM(n, mu = 0.5, sigma = 0.5, nu = 0.1)

OANVASIM(mu.link = "logit", sigma.link = "logit", nu.link = "logit")

Arguments

Details

Let Y_c\sim\mathrm{NVASIM}(\mu,\sigma) and let 0<\nu<1. The BEOI-type one-adjusted distribution is defined by

P(Y=1)=\nu

and

f_Y(y)=(1-\nu)f_{Y_c}(y\mid\mu,\sigma),\quad 0<y<1.

Consequently,

E(Y)=\nu+(1-\nu)\mu

and

\mathrm{Var}(Y)=(1-\nu)\mathrm{Var}(Y_c)+ \nu(1-\nu)(1-\mu)^2.

Thus, \mu=E(Y\mid 0<Y<1) is the mean of the continuous component, whereas \nu+(1-\nu)\mu is the marginal mean.

Value

OANVASIM() returns a gamlss.family object. The functions d1NVASIM(), p1NVASIM(), q1NVASIM(), and r1NVASIM() return probability mass or density values, cumulative probabilities, quantiles, and random observations, respectively. dOANVASIM(), pOANVASIM(), qOANVASIM(), and rOANVASIM() are equivalent names following the GAMLSS family-name convention.

References

Ospina, R. and Ferrari, S. L. P. (2010). Inflated beta distributions. Statistical Papers, 51, 111–126.

Rigby, R. A. and Stasinopoulos, D. M. (2005). Generalized additive models for location, scale and shape. Applied Statistics, 54(3), 507–554.

See Also

NVASIM, ZANVASIM, BEOI

Examples

set.seed(123)
y <- r1NVASIM(1000, mu = 0.60, sigma = 0.30, nu = 0.20)
mean(y == 1)
mean(y)
0.20 + (1 - 0.20) * 0.60

## Not run: 
library(gamlss)
fit <- gamlss(
  y ~ 1,
  sigma.formula = ~ 1,
  nu.formula = ~ 1,
  family = OANVASIM(),
  control = gamlss.control(trace = FALSE)
)

## End(Not run)

Zero-adjusted N-Vasicek distribution with mean parameterization

Description

Defines a zero-adjusted normal-kernel Vasicek distribution for responses in [0,1). The parameter \nu is the probability of a structural zero. Conditional on a positive response, the distribution is NVASIM with mean \mu and shape parameter \sigma.

Usage

d0NVASIM(x, mu = 0.5, sigma = 0.5, nu = 0.1, log = FALSE)

p0NVASIM(q, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

q0NVASIM(p, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

r0NVASIM(n, mu = 0.5, sigma = 0.5, nu = 0.1)

dZANVASIM(x, mu = 0.5, sigma = 0.5, nu = 0.1, log = FALSE)

pZANVASIM(q, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

qZANVASIM(p, mu = 0.5, sigma = 0.5, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

rZANVASIM(n, mu = 0.5, sigma = 0.5, nu = 0.1)

ZANVASIM(mu.link = "logit", sigma.link = "logit", nu.link = "logit")

Arguments

Details

Let Y_+\sim\mathrm{NVASIM}(\mu,\sigma) and let 0<\nu<1. The zero-adjusted distribution is defined by

P(Y=0)=\nu

and

f_Y(y)=(1-\nu)f_{Y_+}(y\mid\mu,\sigma),\quad 0<y<1.

Its cumulative distribution function is

F_Y(y)=\nu+(1-\nu)F_{Y_+}(y\mid\mu,\sigma),\quad 0<y<1.

Consequently,

E(Y)=(1-\nu)\mu

and

\mathrm{Var}(Y)=(1-\nu)\mathrm{Var}(Y_+)+ \nu(1-\nu)\mu^2.

Thus, \mu is the mean conditional on Y>0; it is not the marginal mean when \nu>0. The marginal mean is (1-\nu)\mu.

Value

ZANVASIM() returns a gamlss.family object. The functions d0NVASIM(), p0NVASIM(), q0NVASIM(), and r0NVASIM() return density or probability mass values, cumulative probabilities, quantiles, and random observations, respectively. dZANVASIM(), pZANVASIM(), qZANVASIM(), and rZANVASIM() are equivalent names following the GAMLSS family-name convention.

References

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

Ospina, R. and Ferrari, S. L. P. (2010). Inflated beta distributions. Statistical Papers, 51, 111–126.

Rigby, R. A. and Stasinopoulos, D. M. (2005). Generalized additive models for location, scale and shape. Applied Statistics, 54(3), 507–554.

See Also

NVASIM, BEZI

Examples

set.seed(123)
y <- r0NVASIM(1000, mu = 0.60, sigma = 0.30, nu = 0.20)
mean(y == 0)
mean(y)
(1 - 0.20) * 0.60

library(gamlss)
fit <- gamlss(
  y ~ 1,
  sigma.formula = ~ 1,
  nu.formula = ~ 1,
  family = ZANVASIM(),
  control = gamlss.control(trace = FALSE)
)
fitted(fit, what = "mu")[1]
fitted(fit, what = "sigma")[1]
fitted(fit, what = "nu")[1]

Zero-and-one-adjusted N-Vasicek distribution

Description

Defines a normal-kernel Vasicek distribution augmented by point masses at zero and one. Conditional on an observation in (0,1), the continuous component is NVASIM with mean \mu and shape parameter \sigma.

Usage

d01NVASIM(x, mu = 0.5, sigma = 0.5, nu = 0.1, tau = 0.1, log = FALSE)

p01NVASIM(
  q,
  mu = 0.5,
  sigma = 0.5,
  nu = 0.1,
  tau = 0.1,
  lower.tail = TRUE,
  log.p = FALSE
)

q01NVASIM(
  p,
  mu = 0.5,
  sigma = 0.5,
  nu = 0.1,
  tau = 0.1,
  lower.tail = TRUE,
  log.p = FALSE
)

r01NVASIM(n, mu = 0.5, sigma = 0.5, nu = 0.1, tau = 0.1)

dZOANVASIM(x, mu = 0.5, sigma = 0.5, nu = 0.1, tau = 0.1, log = FALSE)

pZOANVASIM(
  q,
  mu = 0.5,
  sigma = 0.5,
  nu = 0.1,
  tau = 0.1,
  lower.tail = TRUE,
  log.p = FALSE
)

qZOANVASIM(
  p,
  mu = 0.5,
  sigma = 0.5,
  nu = 0.1,
  tau = 0.1,
  lower.tail = TRUE,
  log.p = FALSE
)

rZOANVASIM(n, mu = 0.5, sigma = 0.5, nu = 0.1, tau = 0.1)

ZOANVASIM(
  mu.link = "logit",
  sigma.link = "logit",
  nu.link = "logit",
  tau.link = "logit"
)

Arguments

Details

Let Y_c\sim\mathrm{NVASIM}(\mu,\sigma). Write p_0, p_1, and p_c for the probabilities of zero, one, and the continuous component. The sequential BEOI-type parameterization is

\nu=P(Y=0),\qquad \tau=P(Y=1\mid Y>0).

Hence,

p_0=\nu,\qquad p_1=(1-\nu)\tau,\qquad p_c=(1-\nu)(1-\tau).

The distribution is

P(Y=0)=p_0,\qquad P(Y=1)=p_1

and

f_Y(y)=p_c f_{Y_c}(y\mid\mu,\sigma),\quad 0<y<1.

Its marginal mean and variance are

E(Y)=(1-\nu)\left[\tau+(1-\tau)\mu\right]

and

\mathrm{Var}(Y)= (1-\nu)\left[(1-\tau)\left\{\mathrm{Var}(Y_c)+\mu^2\right\} +\tau\right] -\left\{(1-\nu)\left[\tau+(1-\tau)\mu\right]\right\}^2.

Logit links for \nu and \tau guarantee valid probabilities. If \nu=0, the model reduces to the BEOI-type one-adjusted model; if \tau=0, it reduces to the zero-adjusted model.

Value

ZOANVASIM() returns a four-parameter gamlss.family object. The functions d01NVASIM(), p01NVASIM(), q01NVASIM(), and r01NVASIM() return probability mass or density values, cumulative probabilities, quantiles, and random observations, respectively. dZOANVASIM(), pZOANVASIM(), qZOANVASIM(), and rZOANVASIM() are equivalent names following the GAMLSS family-name convention.

References

Ospina, R. and Ferrari, S. L. P. (2010). Inflated beta distributions. Statistical Papers, 51, 111–126.

Rigby, R. A. and Stasinopoulos, D. M. (2005). Generalized additive models for location, scale and shape. Applied Statistics, 54(3), 507–554.

See Also

NVASIM, ZANVASIM, OANVASIM, BEOI

Examples

set.seed(123)
y <- r01NVASIM(
  1000, mu = 0.60, sigma = 0.30, nu = 0.20, tau = 0.25
)
c(zero = mean(y == 0), one = mean(y == 1))
mean(y)
(1 - 0.20) * (0.25 + (1 - 0.25) * 0.60)

## Not run: 
library(gamlss)
fit <- gamlss(
  y ~ 1,
  sigma.formula = ~ 1,
  nu.formula = ~ 1,
  tau.formula = ~ 1,
  family = ZOANVASIM(),
  control = gamlss.control(trace = FALSE)
)

## End(Not run)

Percentage of Body Fat Dataset

Description

Percentage of body fat measurements from individuals assisted in a public hospital in Curitiba, Paraná, Brazil.

Usage

bodyfat

Format

A data frame with 298 observations and 9 variables:

• ARMS: arms fat percentage.

• LEGS: legs fat percentage.

• BODY: body fat percentage.

• ANDROID: android fat percentage.

• GYNECOID: gynoid fat percentage.

• AGE: age of individuals.

• BMI: body mass index.

• SEX: 1 for female and 2 for male.

• IPAQ: physical activity level according to IPAQ (0 = sedentary, 1 = insufficiently active, 2 = active).

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

Bruna Alves pg402900@uem.br

References

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.

Mazucheli, J., Leiva, V., Alves, B., and Menezes, A. F. B. (2021). A new quantile regression for modeling bounded data under a unit Birnbaum-Saunders distribution with applications in medicine and politics. Symmetry, 13(4), 1–21.

Petterle, R. R., Bonat, W. H., Scarpin, C. T., Jonasson, T., and Borba, V. Z. C. (2020). Multivariate quasi-beta regression models for continuous bounded data. The International Journal of Biostatistics, 17(1), 39–53.

Examples

data(bodyfat, package = "vasicekreg")

bodyfat$AGE <- bodyfat$AGE - 46.00
bodyfat$BMI <- bodyfat$BMI - 24.72
bodyfat$SEX <- as.factor(bodyfat$SEX)
bodyfat$IPAQ<- as.factor(bodyfat$IPAQ)

library(gamlss)

## Mean regression model
fitmean <- gamlss(
  ARMS ~ AGE + BMI + SEX + IPAQ,
  data = bodyfat,
  family = NVASIM(mu.link = "logit", sigma.link = "logit")
)

## Not run: 
tau_levels <- c(0.10, 0.25, 0.50, 0.75, 0.90)

## Quantile regression models with the normal kernel
fit_normal <- lapply(tau_levels, function(Tau) {
  tau <<- Tau
  gamlss(
    ARMS ~ AGE + BMI + SEX + IPAQ,
    data = bodyfat,
    family = NVASIQ(
      mu.link = "logit",
      sigma.link = "logit"
    )
  )
})

## Quantile regression models with the logistic kernel
fit_logistic <- lapply(tau_levels, function(Tau) {
  tau <<- Tau
  gamlss(
    ARMS ~ AGE + BMI + SEX + IPAQ,
    data = bodyfat,
    family = LVASIQ(
      mu.link = "logit",
      sigma.link = "logit"
    )
  )
})

lapply(fit_normal, summary)
lapply(fit_logistic, summary)

## End(Not run)