RelDists (version 1.0.1)

dExWALD: 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)

Value

dExWALD gives the density, pExWALD gives the distribution function, qExWALD gives the quantile function, rExWALD

generates random deviates.

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.

Author

Freddy Hernandez,

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).

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)