dWALD: The Wald distribution

In ousuga/RelDists: Estimation for some Reliability Distributions

The Wald distribution

Description

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

Usage

dWALD(x, mu, sigma, log = FALSE)

pWALD(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)

qWALD(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)

rWALD(n, mu, sigma)

Arguments

x, q	vector of (non-negative integer) quantiles.
mu	vector of the mu parameter.
sigma	vector of the sigma 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 and \sigma has density given by

f(x |\mu, \sigma)=\frac{\sigma}{\sqrt{2 \pi x^3}} \exp \left[-\frac{(\sigma-\mu x)^2}{2x}\right ],

for x < 0.

Value

dWALD gives the density, pWALD gives the distribution function, qWALD gives the quantile function, rWALD generates random deviates.

Author(s)

Sofia Cuartas, [e-mail]

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.

See Also

WALD

WALD.

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dWALD(x, mu=1, sigma=1),
      from=0, to=3, col="cadetblue3", las=1, ylab="f(x)")

curve(dWALD(x, mu=1, sigma=2),
      add=TRUE, col= "purple")

curve(dWALD(x, mu=2, sigma=4),
      add=TRUE, col="goldenrod")

legend("topright", col=c("cadetblue3", "purple", "goldenrod"), 
       lty=1, bty="n",
       legend=c("mu=1, sigma=1",
                "mu=1, sigma=2",
                "mu=2, sigma=4"))

# Example 2
# Checking if the cumulative curves converge to 1
curve(pWALD(x, mu=1, sigma=1), ylim=c(0, 1),
      from=0, to=5, col="cadetblue3", las=1, ylab="F(x)")

curve(pWALD(x, mu=1, sigma=2),
      add=TRUE, col= "purple")

curve(pWALD(x, mu=2, sigma=4),
      add=TRUE, col="goldenrod")

legend("bottomright", col=c("cadetblue3", "purple", "goldenrod"), 
       lty=1, bty="n",
       legend=c("mu=1, sigma=1",
                "mu=1, sigma=2",
                "mu=2, sigma=4"))

# Example 3
# Checking the quantile function
mu <- 1
sigma <- 2
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qWALD(p, mu=mu, sigma=sigma), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pWALD(x, mu=mu, sigma=sigma), from=0, add=TRUE, col="red")

# Example 4
# Comparing the random generator output with
# the theoretical probabilities
mu <- 1
sigma <- 20
x <- rWALD(n=10000, mu=mu, sigma=sigma)
hist(x, freq=FALSE)
curve(dWALD(x, mu=mu, sigma=sigma), col="tomato", add=TRUE)

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