Description Usage Arguments Details Value Note Author(s) References See Also Examples
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.
1 2 3 4 5 6 7 8 9 |
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 ≤q{x}) are returned, otherwise P(X > x). |
p |
vector of probabilities. |
n |
number of observations. If |
mu.link |
the mu link function with default logit. |
sigma.link |
the sigma link function with default logit. |
Probability density function
f(x\mid μ ,σ )=√{\frac{1-σ }{σ }}\exp ≤ft\{ \frac{1}{2}≤ft[ Φ ^{-1}≤ft( x\right) ^{2}-≤ft( \frac{Φ ^{-1}≤ft( x\right) √{1-σ }-Φ ^{-1}≤ft( μ \right) }{√{σ }}\right) ^{2}\right] \right\}
Cumulative distribution function
F(x\mid μ ,σ )=Φ ≤ft( \frac{Φ ^{-1}≤ft( x\right) √{1-σ }-Φ ^{-1}≤ft( μ \right) }{√{σ }}\right)
Quantile function
Q(τ \mid μ ,σ )=F^{-1}(τ \mid μ ,σ )=Φ ≤ft(\frac{Φ ^{-1}≤ft(μ\right) +Φ ^{-1}≤ft( τ \right) √{σ }}{√{1-σ }}\right)
Expected value
E(X) = μ
Variance
Var(X) = Φ_2≤ft ( Φ^{-1}(μ),Φ^{-1}(μ),σ \right )-μ^2
where 0<(x, μ, τ, σ)<1 and Φ_2(\cdot) is the probability distribution function for the standard bivariate normal distribution with correlation σ.
VASIM()
return a gamlss.family object which can be used to fit a Vasicek distribution by gamlss() function.
Note that for VASIQ()
, mu is the τ-th quantile and sigma a shape parameter. The gamlss
function is used for parameters estimation.
Josmar Mazucheli jmazucheli@gmail.com
Bruna Alves pg402900@uem.br
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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | 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');
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