The function ExWALD() defines the Ex-wALD distribution, three-parameter
continuous distribution for a gamlss.family object to be used in
GAMLSS fitting using the function gamlss().
Value
Returns a gamlss.family object which can be used to fit a
Ex-WALD distribution in the gamlss() function.
Details
The Ex-Wald distribution with parameters \(\mu\), \(\sigma\) and \(\nu\) has density given by
\(f(x |\mu, \sigma, \nu) = \frac{1}{\nu} \exp(\frac{-x}{\nu} + \sigma(\mu-k)) F_W(x|k, \sigma) \, \text{for} \, k \geq 0\)
\(f(x |\mu, \sigma, \nu) = \frac{1}{\nu} \exp\left( \frac{-(\sigma-\mu)^2}{2x} \right) Re \left( w(k^\prime \sqrt{x/2} + \frac{\sigma i}{\sqrt{2x}}) \right) \, \text{for} \, k < 0\)
where \(k=\sqrt{\mu^2-\frac{2}{\nu}}\), \(k^\prime=\sqrt{\frac{2}{\nu}-\mu^2}\) and \(F_W\) corresponds to the cumulative function of the Wald distribution.
More details about those expressions can be found on page 680 from Heathcote (2004).
References
Schwarz, W. (2001). The ex-Wald distribution as a descriptive model of response times. Behavior Research Methods, Instruments, & Computers, 33, 457-469.
Heathcote, A. (2004). Fitting Wald and ex-Wald distributions to response time data: An example using functions for the S-PLUS package. Behavior Research Methods, Instruments, & Computers, 36, 678-694.
Author
Freddy Hernandez, fhernanb@unal.edu.co
Examples
# Example 1
# Generating random values with
# known mu, sigma and nu
mu <- 0.20
sigma <- 70
nu <- 115
set.seed(123)
y <- rExWALD(n=100, mu, sigma, nu)
library(gamlss)
mod1 <- gamlss(y~1, family=ExWALD,
control=gamlss.control(n.cyc=1000, trace=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 1258.666
exp(coef(mod1, what="mu"))
#> (Intercept)
#> 0.2392296
exp(coef(mod1, what="sigma"))
#> (Intercept)
#> 74.6907
exp(coef(mod1, what="nu"))
#> (Intercept)
#> 129.982
# Example 2
# Generating random values under some model
# \donttest{
# A function to simulate a data set with Y ~ ExWALD
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(-1 + 2.8 * x1) # 1.5 approximately
sigma <- exp( 1 - 1.2 * x2) # 1.5 approximately
nu <- 2
y <- rExWALD(n=n, mu=mu, sigma=sigma, nu=nu)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(1234)
dat <- gendat(n=200)
# Fitting the model
mod2 <- gamlss(y ~ x1,
sigma.fo = ~ x2,
nu.fo = ~ 1,
family = ExWALD,
data = dat,
control = gamlss.control(n.cyc=1000,
trace=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 822.0462
#> GAMLSS-RS iteration 2: Global Deviance = 819.875
#> GAMLSS-RS iteration 3: Global Deviance = 818.6914
#> GAMLSS-RS iteration 4: Global Deviance = 817.9948
#> GAMLSS-RS iteration 5: Global Deviance = 817.6103
#> GAMLSS-RS iteration 6: Global Deviance = 817.4151
#> GAMLSS-RS iteration 7: Global Deviance = 817.3221
#> GAMLSS-RS iteration 8: Global Deviance = 817.28
#> GAMLSS-RS iteration 9: Global Deviance = 817.2608
#> GAMLSS-RS iteration 10: Global Deviance = 817.2515
#> GAMLSS-RS iteration 11: Global Deviance = 817.2476
#> GAMLSS-RS iteration 12: Global Deviance = 817.246
#> GAMLSS-RS iteration 13: Global Deviance = 817.2453
summary(mod2)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family: c("ExWALD", "Ex-Wald")
#>
#> Call: gamlss(formula = y ~ x1, sigma.formula = ~x2, nu.formula = ~1,
#> family = ExWALD, data = dat, control = gamlss.control(n.cyc = 1000,
#> trace = TRUE))
#>
#> Fitting method: RS()
#>
#> ------------------------------------------------------------------
#> Mu link function: log
#> Mu Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -0.9079 0.1767 -5.139 6.59e-07 ***
#> x1 2.2407 0.3472 6.453 8.25e-10 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ------------------------------------------------------------------
#> Sigma link function: log
#> Sigma Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.8353 0.1504 5.553 8.95e-08 ***
#> x2 -0.6647 0.2863 -2.322 0.0212 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ------------------------------------------------------------------
#> Nu link function: log
#> Nu Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.63266 0.09008 7.023 3.36e-11 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ------------------------------------------------------------------
#> No. of observations in the fit: 200
#> Degrees of Freedom for the fit: 5
#> Residual Deg. of Freedom: 195
#> at cycle: 13
#>
#> Global Deviance: 817.2453
#> AIC: 827.2453
#> SBC: 843.7368
#> ******************************************************************
# }