# The Vasicek distribution - quantile parameterization The function `VASIQ()` define the Vasicek distribution for a `gamlss.family` object to be used in GAMLSS fitting. `VASIQ()` has the $\tau$-th quantile equal to the parameter mu and sigma as shape parameter. The functions `dVASIQ`, `pVASIQ`, `qVASIQ` and `rVASIQ` define the density, distribution function, quantile function and random generation for Vasicek distribution. ```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 on the (0,1) interval. - `mu`: vector of the $\tau$-th quantile parameter values. - `sigma`: vector of shape parameter values. - `tau`: the $\tau$-th fixed quantile in [d-p-q-r]-VASIQ 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`: 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. ## Returns `VASIQ()` return a gamlss.family object which can be used to fit a Vasicek distribution by gamlss() function. ## Details Probability density function $$ f\left( {x\mid \mu ,\sigma ,\tau } \right) = \sqrt {{\textstyle{{1 - \sigma } \over \sigma }}} \exp \left\{ {\frac{1}{2}\left[ {\Phi ^{ - 1} \left( x \right)^2 - \left( {\frac{{\sqrt {1 - \sigma } \left[ {\Phi ^{ - 1} \left( x \right) - \Phi ^{ - 1} \left( \mu \right)} \right] - \sqrt \sigma \,\Phi ^{ - 1} \left( \tau \right)}}{{\sqrt \sigma }}} \right)^2 } \right]} \right\} $$ Cumulative distribution function $$ F\left({x\mid \mu ,\sigma ,\tau } \right) = \Phi \left[ {\frac{{\sqrt {1 - \sigma } \left[ {\Phi ^{ - 1} \left( x \right) - \Phi ^{ - 1} \left( \mu \right)} \right] - \sqrt \sigma \,\Phi ^{ - 1} \left( \tau \right)}}{{\sqrt \sigma }}} \right] $$ where $0<(x, \mu, \tau, \sigma)<1$, $\mu$ is the $\tau$-th quantile and $\sigma$ is the shape parameter. ## Note Note that for `VASIQ()`, mu is the $\tau$-th quantile and sigma a shape parameter. The `gamlss` function is used for parameters estimation. ## 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. and Korkmaz, M. C. (2021). The Vasicek quantile regression model. **(under review)**. 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