The function WALD() defines the wALD distribution, two-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 WALD distribution in the gamlss() function.
Details
The Wald distribution with parameters \(\mu\) and \(sigma\) has density given by
\(\operatorname{f}(x |\mu, \sigma)=\frac{\sigma}{\sqrt{2 \pi x^3}} \exp \left[-\frac{(\sigma-\mu x)^2}{2x}\right ], x>0 \)
References
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
Sofia Cuartas García, scuartasg@unal.edu.co
Examples
# Example 1
# Generating random values with
# known mu and sigma
require(gamlss)
mu <- 1.5
sigma <- 4.0
y <- rWALD(10000, mu, sigma)
mod1 <- gamlss(y~1, sigma.fo=~1, family="WALD",
control=gamlss.control(n.cyc=5000, trace=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 27669.01
#> GAMLSS-RS iteration 2: Global Deviance = 27668.59
#> GAMLSS-RS iteration 3: Global Deviance = 27668.34
#> GAMLSS-RS iteration 4: Global Deviance = 27668.21
#> GAMLSS-RS iteration 5: Global Deviance = 27668.13
#> GAMLSS-RS iteration 6: Global Deviance = 27668.08
#> GAMLSS-RS iteration 7: Global Deviance = 27668.06
#> GAMLSS-RS iteration 8: Global Deviance = 27668.04
#> GAMLSS-RS iteration 9: Global Deviance = 27668.03
#> GAMLSS-RS iteration 10: Global Deviance = 27668.03
#> GAMLSS-RS iteration 11: Global Deviance = 27668.03
#> GAMLSS-RS iteration 12: Global Deviance = 27668.03
#> GAMLSS-RS iteration 13: Global Deviance = 27668.02
exp(coef(mod1, what="mu"))
#> (Intercept)
#> 1.513483
exp(coef(mod1, what="sigma"))
#> (Intercept)
#> 4.046919
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ WALD
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(0.75 - 0.69 * x1) # Approx 1.5
sigma <- exp(0.5 - 0.64 * x2) # Approx 1.20
y <- rWALD(n, mu, sigma)
data.frame(y=y, x1=x1, x2=x2)
}
dat <- gendat(n=200)
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=WALD, data=dat,
control=gamlss.control(n.cyc=5000, trace=TRUE))
#> GAMLSS-RS iteration 1: Global Deviance = 220.34
#> GAMLSS-RS iteration 2: Global Deviance = 218.9899
#> GAMLSS-RS iteration 3: Global Deviance = 218.7139
#> GAMLSS-RS iteration 4: Global Deviance = 218.6434
#> GAMLSS-RS iteration 5: Global Deviance = 218.628
#> GAMLSS-RS iteration 6: Global Deviance = 218.6233
#> GAMLSS-RS iteration 7: Global Deviance = 218.6217
#> GAMLSS-RS iteration 8: Global Deviance = 218.6214
summary(mod2)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family: c("WALD", "Wald")
#>
#> Call: gamlss(formula = y ~ x1, sigma.formula = ~x2, family = WALD,
#> data = dat, control = gamlss.control(n.cyc = 5000, trace = TRUE))
#>
#> Fitting method: RS()
#>
#> ------------------------------------------------------------------
#> Mu link function: log
#> Mu Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.8445 0.1017 8.306 1.54e-14 ***
#> x1 -0.8676 0.2003 -4.332 2.34e-05 ***
#> ---
#> 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) 0.62330 0.07169 8.694 1.33e-15 ***
#> x2 -0.85636 0.12141 -7.054 2.85e-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: 4
#> Residual Deg. of Freedom: 196
#> at cycle: 8
#>
#> Global Deviance: 218.6214
#> AIC: 226.6214
#> SBC: 239.8147
#> ******************************************************************