VASIQ: The Vasicek distribution: quantile parameterization

Description

The function VASIQ() defines the Vasicek distribution as a gamlss.family object to be used in GAMLSS fitting. In this parameterization, $\mu$ corresponds to the fixed $\tau$-th quantile, and $\sigma$ is a shape parameter. The functions dVASIQ, pVASIQ, qVASIQ, and rVASIQ define the density, distribution function, quantile function, and random generation for the Vasicek distribution, respectively.

Usage

dVASIQ(x, mu, sigma, tau = 0.5, log = FALSE)
pVASIQ(q, mu, sigma, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
qVASIQ(p, mu, sigma, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
rVASIQ(n, mu, sigma, tau = 0.5)
VASIQ(mu.link = "logit", sigma.link = "logit")

Value

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

function.

Arguments

x, q: Vector of quantiles in the interval $(0,1)$.
mu: Vector of $\tau$-th quantile parameter values.
sigma: Vector of shape parameter values.
tau: Fixed quantile level $\tau$ used in the $d$, $p$, $q$, and $r$ functions for VASIQ.
log, log.p: Logical; if TRUE, probabilities are returned on the log scale.
lower.tail: Logical; if TRUE (default), $P(X \le x)$ is returned; otherwise, $P(X > x)$.
p: Vector of probabilities.
n: Number of observations. If length(n) > 1, the length is taken to be the number required.
mu.link: Link function for the $\mu$ parameter.
sigma.link: Link function for the $\sigma$ parameter.

Author

Josmar Mazucheli jmazucheli@gmail.com

Bruna Alves pg402900@uem.br

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.

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.

Examples

Run this code

set.seed(123)
x <- rVASIQ(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, dVASIQ(x = S, mu = 0.50, sigma = 0.69, tau = 0.50), col = 2)

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

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

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

tau <- 0.5
fit <- gamlss(y ~ 1, data = data,
              family = VASIQ(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 <- rVASIQ(n, mu, sigma, tau = 0.5)
data <- data.frame(y, x, tau = 0.5)

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

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

sapply(fittaus, summary)

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