dist.HalfCauchy | R Documentation |
These functions provide the density, distribution function, quantile function, and random generation for the half-Cauchy distribution.
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 |
log |
Logical. If |
Application: Continuous Univariate
Density: p(\theta) = \frac{2 \alpha}{\pi(\theta^2 +
\alpha^2)}, \quad \theta > 0
Inventor: Derived from Cauchy
Notation 1: \theta \sim \mathcal{HC}(\alpha)
Notation 2: p(\theta) = \mathcal{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.
dcauchy
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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