Description Usage Arguments Details Value See Also Examples
These functions provide the density, distribution function, quantile function, and random generation for the half-Cauchy distribution.
1 2 3 4 | dhalfcauchy(x, scale=25, log=FALSE)
phalfcauchy(q, scale=25)
qhalfcauchy(p, scale=25)
rhalfcauchy(n, scale=25)
|
x,q |
These are each a vector of quantiles. |
p |
This is a vector of probabilities. |
n |
This is the number of observations, which must be a positive integer that has length 1. |
scale |
This is the scale parameter alpha, which must be positive. |
log |
Logical. If |
Application: Continuous Univariate
Density: p(theta) = 2alpha / pi(theta^2 + alpha^2), theta >= 0
Inventor: Derived from Cauchy
Notation 1: theta ~ HC(alpha)
Notation 2: p(theta) = HC(theta | alpha)
Parameter 1: scale parameter alpha > 0
Mean: E(theta) = does not exist
Variance: var(theta) = does not exist
Mode: mode(theta) = 0
The half-Cauchy distribution with scale alpha=25 is a recommended, default, weakly informative prior distribution for a scale parameter. Otherwise, the scale, alpha, is recommended to be set to be just a little larger than the expected standard deviation, as a weakly informative prior distribution on a standard deviation parameter.
The Cauchy distribution is known as a pathological distribution because its mean and variance are undefined, and it does not satisfy the central limit theorem.
dhalfcauchy
gives the density,
phalfcauchy
gives the distribution function,
qhalfcauchy
gives the quantile function, and
rhalfcauchy
generates random deviates.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 | library(LaplacesDemon)
x <- dhalfcauchy(1,25)
x <- phalfcauchy(1,25)
x <- qhalfcauchy(0.5,25)
x <- rhalfcauchy(1,25)
#Plot Probability Functions
x <- seq(from=0, to=20, by=0.1)
plot(x, dhalfcauchy(x,1), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dhalfcauchy(x,5), type="l", col="green")
lines(x, dhalfcauchy(x,10), type="l", col="blue")
legend(2, 0.9, expression(alpha==1, alpha==5, alpha==10),
lty=c(1,1,1), col=c("red","green","blue"))
|
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