ExWALD: The Ex-Wald family

In ousuga/RelDists: Estimation for some Reliability Distributions

The Ex-Wald family

Description

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().

Usage

ExWALD(mu.link = "log", sigma.link = "log", nu.link = "log")

Arguments

mu.link	defines the mu.link, with "log" link as the default for the mu parameter.
sigma.link	defines the sigma.link, with "log" link as the default for the sigma parameter.
nu.link	defines the nu.link, with "log" link as the default for the nu parameter.

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).

Value

Returns a gamlss.family object which can be used to fit a Ex-WALD distribution in the gamlss() function.

Author(s)

Freddy Hernandez, [e-mail]

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.

See Also

dExWALD.

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))

exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
exp(coef(mod1, what="nu"))

# Example 2
# Generating random values under some model


# 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))
summary(mod2)

ousuga/RelDists documentation built on July 4, 2025, 10:55 a.m.