dExWALD: The Ex-Wald distribution

In ousuga/RelDists: Estimation for some Reliability Distributions

The Ex-Wald distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Ex-Wald distribution with parameter \mu, \sigma and \nu.

Usage

dExWALD(x, mu = 1.5, sigma = 1.5, nu = 2, log = FALSE)

pExWALD(q, mu = 1.5, sigma = 1.5, nu = 2, lower.tail = TRUE, log.p = FALSE)

qExWALD(p, mu = 1.5, sigma = 1.5, nu = 2)

rExWALD(n, mu = 1.5, sigma = 1.5, nu = 2)

Arguments

x, q	vector of (non-negative integer) quantiles.
mu	vector of the mu parameter.
sigma	vector of the sigma parameter.
nu	vector of the nu parameter.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
p	vector of probabilities.
n	number of random values to return.

Details

The 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

dExWALD gives the density, pExWALD gives the distribution function, qExWALD gives the quantile function, rExWALD generates random deviates.

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

ExWALD

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dExWALD(x, mu=0.15, sigma=52.5, nu=50), ylim=c(0, 0.005),
      from=0, to=1200, col="cadetblue3", las=1, ylab="f(x)")

curve(dExWALD(x, mu=0.20, sigma=70, nu=50),
      add=TRUE, col= "purple")

curve(dExWALD(x, mu=0.25, sigma=87.5, nu=50),
      add=TRUE, col="goldenrod")

curve(dExWALD(x, mu=0.20, sigma=70, nu=115),
      add=TRUE, col="tomato")

curve(dExWALD(x, mu=0.20, sigma=70, nu=35),
      add=TRUE, col="blue")

legend("topright", col=c("cadetblue3", "purple", "goldenrod",
                         "tomato", "blue"), 
       lty=1, bty="n",
       legend=c("mu=0.15, sigma=52.5, nu=50",
                "mu=0.20, sigma=70.0, nu=50",
                "mu=0.25, sigma=87.5, nu=50",
                "mu=0.20, sigma=70.0, nu=115",
                "mu=0.20, sigma=70.0, nu=35"))

# Example 2
# Checking if the cumulative curves converge to 1
curve(pExWALD(x, mu=0.15, sigma=52.5, nu=50), ylim=c(0, 1),
      from=0, to=1200, col="cadetblue3", las=1, ylab="F(x)")

curve(pExWALD(x, mu=0.20, sigma=70, nu=50),
      add=TRUE, col= "purple")

curve(pExWALD(x, mu=0.25, sigma=87.5, nu=50),
      add=TRUE, col="goldenrod")

curve(pExWALD(x, mu=0.20, sigma=70, nu=115),
      add=TRUE, col="tomato")

curve(pExWALD(x, mu=0.20, sigma=70, nu=35),
      add=TRUE, col="blue")

legend("bottomright", col=c("cadetblue3", "purple", "goldenrod",
                         "tomato", "blue"), 
       lty=1, bty="n",
       legend=c("mu=0.15, sigma=52.5, nu=50",
                "mu=0.20, sigma=70.0, nu=50",
                "mu=0.25, sigma=87.5, nu=50",
                "mu=0.20, sigma=70.0, nu=115",
                "mu=0.20, sigma=70.0, nu=35"))

# Example 3
# Checking the quantile function
mu <- 5
sigma <- 3
nu <- 2
p <- seq(from=0.1, to=0.99, length.out=100)
plot(x=qExWALD(p, mu=mu, sigma=sigma, nu=nu), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pExWALD(x, mu=mu, sigma=sigma, nu=nu), from=0, add=TRUE, col="red")

# Example 4
# Comparing the random generator output with
# the theoretical probabilities
mu <- 0.2
sigma <- 70
nu <- 35
x <- rExWALD(n=10000, mu=mu, sigma=sigma, nu=nu)
hist(x, freq=FALSE)
curve(dExWALD(x, mu=mu, sigma=sigma, nu=nu), col="tomato", add=TRUE)

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