# The Vasicek distribution - mean parameterization The function `VASIM()` define the Vasicek distribution for a `gamlss.family` object to be used in GAMLSS fitting. `VASIM()` has mean equal to the parameter mu and sigma as shape parameter. The functions `dVASIM`, `pVASIM`, `qVASIM` and `rVASIM` define the density, distribution function, quantile function and random generation for Vasicek distribution. ```r 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") ``` ## Arguments - `x, q`: vector of quantiles on the (0,1) interval. - `mu`: vector of the mean parameter values. - `sigma`: vector of shape parameter values. - `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 `VASIM()` return a gamlss.family object which can be used to fit a Vasicek distribution by gamlss() function. ## Details Probability density function $$ f(x\mid \mu ,\sigma )=\sqrt{\frac{1-\sigma }{\sigma }}\exp \left\{ \frac{1}{2}\left[ \Phi ^{-1}\left( x\right) ^{2}-\left( \frac{\Phi ^{-1}\left( x\right) \sqrt{1-\sigma }-\Phi ^{-1}\left( \mu \right) }{\sqrt{\sigma }}\right) ^{2}\right] \right\} $$ Cumulative distribution function $$ F(x\mid \mu ,\sigma )=\Phi \left( \frac{\Phi ^{-1}\left( x\right) \sqrt{1-\sigma }-\Phi ^{-1}\left( \mu \right) }{\sqrt{\sigma }}\right) $$ Quantile function $$ Q(\tau \mid \mu ,\sigma )=F^{-1}(\tau \mid \mu ,\sigma )=\Phi \left(\frac{\Phi ^{-1}\left(\mu\right) +\Phi ^{-1}\left( \tau \right) \sqrt{\sigma }}{\sqrt{1-\sigma }}\right) $$ Expected value $$ E(X) = \mu $$ Variance $$ Var(X) = \Phi_2\left ( \Phi^{-1}(\mu),\Phi^{-1}(\mu),\sigma \right )-\mu^2 $$ where $0<(x, \mu, \tau, \sigma)<1$ and $\Phi_2(\cdot)$ is the probability distribution function for the standard bivariate normal distribution with correlation $\sigma$. ## 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 <- rVASIM(n = 1000, mu = 0.50, sigma = 0.69) R <- range(x) S <- seq(from = R[1], to = R[2], length.out = 1000) hist(x, prob = TRUE, main = 'Vasicek') lines(S, dVASIM(x = S, mu = 0.50, sigma = 0.69), col = 2) plot(ecdf(x)) lines(S, pVASIM(q = S, mu = 0.50, sigma = 0.69), col = 2) plot(quantile(x, probs = S), type = "l") lines(qVASIM(p = S, mu = 0.50, sigma = 0.69), col = 2) library(gamlss) set.seed(123) data <- data.frame(y = rVASIM(n = 100, mu = 0.5, sigma = 0.69)) fit <- gamlss(y ~ 1, data = data, mu.link = 'logit', sigma.link = 'logit', family = VASIM) 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 <- 0.1; y <- rVASIM(n, mu, sigma) data <- data.frame(y, x) fit <- gamlss(y ~ x, data = data, family = VASIM, mu.link = 'logit', sigma.link = 'logit'); ``` ## 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 `VASIQ`, `pmvnorm`. ## Author(s) Josmar Mazucheli jmazucheli@gmail.com Bruna Alves pg402900@uem.br

VASIM function