# The Vasicek distribution: quantile parameterization 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. ```r 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") ``` ## 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. ## Returns `VASIQ()` returns a `gamlss.family` object that can be used to fit a Vasicek distribution using the `gamlss` function. ## 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. ## Note For `VASIQ()`, $\mu$ corresponds to the $\tau$-th quantile and $\sigma$ is a shape parameter. Parameter estimation is performed using the `gamlss` function. ## Examples ```r 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) ``` ## 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 `VASIM` ## Author(s) Josmar Mazucheli jmazucheli@gmail.com Bruna Alves pg402900@uem.br

VASIQ function