UGOM {ugomquantreg}

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

x, q

vector of quantiles on the (0,1) interval.

mu

vector of quantile parameter values.

sigma

vector of shape parameter values.

tau

the \tau-th fixed quantile in [d-p-q-r]-UGOM function.

log, log.p

logical; If TRUE, probabilities p are given as log(p).

lower.tail

logical; If TRUE, (default), P(X \leq{x}) are returned, otherwise P(X > x).

p

vector of probabilities.

n

the number of observations. If length(n) > 1, the length is taken to be the number required.

mu.link

the mu link function with default logit.

sigma.link

the sigma link function with default logit.

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 [e-mail]

Bruna Alves [e-mail]

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"))