Overview of the ugomquantreg package

Description

The ugomquantreg package implements the probability density function, quantile function, cumulative distribution function and random number generation function for unit-Gompertz distribution parameterized as a function of its \tau-th quantile, 0 <\tau<1. Some function are written in C++ using Rcpp.

Details

ammonia: Ammonia oxidized to acid nitric data set.

bodyfat: Body fat data set.

UGOM: For quantile modeling (con/in)ditional on covariate(s).

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

Bruna Alves pg402900@uem.br

The unit-Gompertz distribution - quantile parameterization

Description

The function UGOM() define the unit-Gompertz distribution for a gamlss.family object to be used in GAMLSS fitting. UGOM() has the \tau-th quantile equal to the parameter mu and sigma as the shape parameter. The functions dUGOM, pUGOM, qUGOM and rUGOM define the density, distribution function, quantile function and random generation for unit-Gompertz distribution.

Usage

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

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

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

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

UGOM(mu.link = "logit", sigma.link = "log")

Arguments

Details

Probability density function

f\left( {x\mid \mu ,\sigma ,\tau } \right)=\left ( \frac{\log \left( \tau \right) }{1-\mu ^{-\sigma }} \right ) \sigma x^{-\left( 1+\sigma \right) }\exp \left[ \left ( \frac{\log \left( \tau \right) }{1-\mu ^{-\sigma }} \right ) \left( 1-x^{-\sigma }\right) \right]

Cumulative distribution function

F\left({x\mid \mu ,\sigma ,\tau } \right) = \exp \left[ \left ( \frac{\log \left( \tau \right) }{1-\mu ^{-\sigma }} \right ) \left( 1-x^{-\sigma }\right) \right]

Mean

E(X)=\left( \frac{\log \left( \tau \right) }{1-\mu ^{-\sigma }}\right) ^{\frac{1}{\theta }}\exp \left(\frac{\log \left( \tau \right) }{1-\mu ^{-\sigma }}\right)\Gamma \left( \frac{\sigma -1}{\sigma },\frac{\log \left( \tau \right) }{ 1-\mu ^{-\sigma }}\right)

where 0 < (x, \mu)<1, \mu is, for a fixed and known value of \tau, the \tau-th quantile, \sigma is the shape parameter and \Gamma(a, b) is the upper incomplete gamma function.

Value

UGOM() return a gamlss.family object which can be used to fit a unit-Gompertz distribution by gamlss() function.

Note

Note that for UGOM(), mu is the \tau-th quantile and sigma a shape parameter. The gamlss function is used for parameters estimation.

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., Alve, B. (2021). The Unit-Gompertz quantile regression model for bounded responses. preprint, 0(0), 1-20.

Mazucheli, J., Menezes, A. F. and Dey S. (2019). Unit-Gompertz distribution with applications. Statistica, 79(1), 25–43.

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.

Examples

set.seed(123)
x <- rUGOM(n = 1000, mu = 0.50, sigma = 1.69, tau = 0.50)
R <- range(x)
S <- seq(from = R[1], to = R[2], length.out = 1000)

hist(x, prob = TRUE, main = 'unit-Gompertz')
lines(S, dUGOM(x = S, mu = 0.50, sigma = 1.69, tau = 0.50), col = 2)

plot(ecdf(x))
lines(S, pUGOM(q = S, mu = 0.50, sigma = 1.69, tau = 0.50), col = 2)

plot(quantile(x, probs = S), type = "l")
lines(qUGOM(p = S, mu = 0.50, sigma = 1.69, tau = 0.50), col = 2)

library(gamlss)
set.seed(123)
data <- data.frame(y =  rUGOM(n = 100, mu = 0.5, sigma = 2.0, tau = 0.5))

tau <- 0.50
fit <- gamlss(y ~ 1, data = data, family = UGOM)

set.seed(123)
n <- 100
x <- rbinom(n, size = 1, prob = 0.5)
eta <- 0.5 + 1 * x;
mu <- 1 / (1 + exp(-eta));
sigma <- 1.5;
y <- rUGOM(n, mu, sigma, tau = 0.5)
data <- data.frame(y, x)

tau <- 0.50
fit <- gamlss(y ~ x, data = data, family = UGOM(mu.link = "logit", sigma.link = "log"))

Ammonia oxidized to acid nitric data set

Description

The data come from experiments with a plant where ammonia is oxidized to acid nitric.

Usage

data(ammonia, package = "ugomquantreg")

Format

A data-frame with 21 observations and 4 columns:

• stackloss: the percentage of ammonia lost.

• airflow: the air flow to the plant.

• watertemp: the cooling water inlet temperature.

• acidconc: the acid concentration.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

Bruna Alves pg402900@uem.br

Source

https://support.sas.com/rnd/app/stat/examples/BayesQuantile/quantile.htm

References

Brownlee, K. A., (1965). Statistical Theory and Methodology in Science and Engineering. New York: John Wiley & Sons.

Yu, K., and Moyeed, R. A., (2001). Bayesian quantile regression. Statistics and Probability Letters, 54(4) 437–447.

Examples

data(ammonia, package = "ugomquantreg")

library(gamlss)

tau <- 0.50
fit.logit <- gamlss(stackloss ~ airflow + watertemp + acidconc, data = ammonia, 
family = UGOM(sigma.link="identity"))

tau <- 0.50
fit.probit <- gamlss(stackloss ~ airflow + watertemp + acidconc, 
data = ammonia, family = UGOM(mu.link = "probit", sigma.link = "log"))

fittaus <- lapply(c(0.10, 0.25, 0.50, 0.75, 0.90), function(Tau){
 tau <<- Tau;
 gamlss(stackloss ~ airflow + watertemp + acidconc, data = ammonia, 
 family = UGOM(mu.link = "logit", sigma.link = "log"))
})

sapply(fittaus, coef)

Percentage of body fat data set

Description

The body fat percentage of individuals assisted in a public hospital in Curitiba, Paraná, Brazil.

Usage

data(bodyfat, package = "ugomquantreg")

Format

A data-frame with 298 observations and 9 columns:

• ARMS: arms fat percentage.

• LEGS: legs fat percentage.

• BODY: body fat percentage.

• ANDROID: android fat percentage.

• GYNECOID: ginecoid fat percentage.

• AGE: age of individuals.

• BMI: body mass index.

• SEX: 1 for female, 2 for male.

• IPAQ: 0 for IPAQ = sedentary, 1 for IPAQ = insufficiently active and 2 for IPAQ = active.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

Bruna Alves pg402900@uem.br

Source

http://www.leg.ufpr.br/doku.php/publications:papercompanions:multquasibeta

References

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, 1–15, (preprint).

Examples

data(bodyfat, package = "ugomquantreg")

library(gamlss)

tau <- 0.50
fit.logit <- gamlss(ARMS ~ AGE + I(BMI / 100) + as.factor(SEX) + as.factor(IPAQ), 
data = bodyfat, family = UGOM(mu.link = "logit", sigma.link = "log"))

tau <- 0.50;
fit.probit <- gamlss(ARMS ~ AGE + I(BMI / 100) + as.factor(SEX) + as.factor(IPAQ), 
data = bodyfat, family = UGOM(mu.link = "probit", sigma.link = "log"))