RelDists (version 1.0.1)

dWALD: 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)

Value

dWALD gives the density, pWALD gives the distribution function, qWALD gives the quantile function, rWALD

generates random deviates.

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.

Author

Sofia Cuartas,

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 \(x < 0\).

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)