vasicekreg (version 1.0.2)

VASIM: Vasicek distribution with mean parameterization

Description

The function VASIM() defines the Vasicek distribution under a mean-based parameterization for use as a gamlss.family object in GAMLSS models. In this formulation, \(\mu\) represents the mean of the distribution and \(\sigma\) is a shape parameter. The functions dVASIM, pVASIM, qVASIM, and rVASIM provide the density, distribution, quantile, and random generation functions, respectively.

Usage

dVASIM(x, mu, sigma, log = FALSE)
pVASIM(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qVASIM(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rVASIM(n, mu, sigma)
VASIM(mu.link = "logit", sigma.link = "logit")

Value

VASIM() returns a gamlss.family object.

Arguments

x, q

Vector of quantiles in the interval \((0,1)\).

mu

Vector of mean values.

sigma

Vector of shape parameter values.

log, log.p

Logical; if TRUE, probabilities are given on the log scale.

lower.tail

Logical; if TRUE, probabilities \(P(X \le x)\) are returned.

p

Vector of probabilities.

n

Number of observations.

mu.link

Link function for the \(\mu\) parameter.

sigma.link

Link function for the \(\sigma\) parameter.

Author

Josmar Mazucheli

Bruna Alves

Details

The probability density function is given by $$f(x \mid \mu, \sigma) = \sqrt{\frac{1-\sigma}{\sigma}} \exp\left\{\frac{1}{2}\left[\Phi^{-1}(x)^2 - \left(\frac{\Phi^{-1}(x)\sqrt{1-\sigma}-\Phi^{-1}(\mu)} {\sqrt{\sigma}}\right)^2\right]\right\}.$$

The cumulative distribution function is $$F(x \mid \mu, \sigma) = \Phi\left(\frac{\Phi^{-1}(x)\sqrt{1-\sigma}-\Phi^{-1}(\mu)} {\sqrt{\sigma}}\right).$$

The quantile function is $$Q(\tau \mid \mu, \sigma) = \Phi\left(\frac{\Phi^{-1}(\mu)+\Phi^{-1}(\tau)\sqrt{\sigma}} {\sqrt{1-\sigma}}\right).$$

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. tools:::Rd_expr_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

VASIQ, pmvnorm

Examples

set.seed(123)
x <- rVASIM(n = 1000, mu = 0.5, sigma = 0.69)

hist(x, probability = TRUE, main = "Vasicek distribution")

if (FALSE) {
library(gamlss)
data <- data.frame(y = x[1:100])
fit <- gamlss(y ~ 1, data = data,
              family = VASIM(mu.link = "logit",
                             sigma.link = "logit"))
summary(fit)
}