dist.Halft {LaplacesDemon} | R Documentation |
Half-t Distribution
Description
These functions provide the density, distribution function, quantile function, and random generation for the half-t distribution.
Usage
dhalft(x, scale=25, nu=1, log=FALSE)
phalft(q, scale=25, nu=1)
qhalft(p, scale=25, nu=1)
rhalft(n, scale=25, nu=1)
Arguments
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 |
nu |
This is the scalar degrees of freedom parameter, which is
usually represented as |
log |
Logical. If |
Details
Application: Continuous Univariate
Density:
p(\theta) = (1 + \frac{1}{\nu} (\theta / \alpha)^2)^{-(\nu+1)/2}, \quad \theta \ge 0
Inventor: Derived from the Student t
Notation 1:
\theta \sim \mathcal{HT}(\alpha, \nu)
Notation 2:
p(\theta) = \mathcal{HT}(\theta | \alpha, \nu)
Parameter 1: scale parameter
\alpha > 0
Parameter 2: degrees of freedom parameter
\nu
Mean:
E(\theta)
= unknownVariance:
var(\theta)
= unknownMode:
mode(\theta) = 0
The half-t distribution is derived from the Student t distribution, and
is useful as a weakly informative prior distribution for a scale
parameter. It is more adaptable than the default recommended
half-Cauchy, though it may also be more difficult to estimate due to its
additional degrees of freedom parameter, \nu
. When
\nu=1
, the density is proportional to a proper half-Cauchy
distribution. When \nu=-1
, the density becomes an improper,
uniform prior distribution. For more information on propriety, see
is.proper
.
Wand et al. (2011) demonstrated that the half-t distribution may be represented as a scale mixture of inverse-gamma distributions. This representation is useful for conjugacy.
Value
dhalft
gives the density,
phalft
gives the distribution function,
qhalft
gives the quantile function, and
rhalft
generates random deviates.
References
Wand, M.P., Ormerod, J.T., Padoan, S.A., and Fruhwirth, R. (2011). "Mean Field Variational Bayes for Elaborate Distributions". Bayesian Analysis, 6: p. 847–900.
See Also
dhalfcauchy
,
dst
,
dt
,
dunif
, and
is.proper
.
Examples
library(LaplacesDemon)
x <- dhalft(1,25,1)
x <- phalft(1,25,1)
x <- qhalft(0.5,25,1)
x <- rhalft(10,25,1)
#Plot Probability Functions
x <- seq(from=0.1, to=20, by=0.1)
plot(x, dhalft(x,1,-1), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dhalft(x,1,0.5), type="l", col="green")
lines(x, dhalft(x,1,500), type="l", col="blue")
legend(2, 0.9, expression(paste(alpha==1, ", ", nu==-1),
paste(alpha==1, ", ", nu==0.5), paste(alpha==1, ", ", nu==500)),
lty=c(1,1,1), col=c("red","green","blue"))