Description Usage Arguments Details Value See Also Examples
These functions provide the density, distribution function, quantile function, and random generation for the half-t distribution.
1 2 3 4 |
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. |
nu |
This is the scalar degrees of freedom parameter, which is usually represented as nu. |
log |
Logical. If |
Application: Continuous Univariate
Density: p(theta) = (1 + (1/nu)*(theta/alpha)^2)^(-(nu+1)/2), theta >= 0
Inventor: Derived from the Student t
Notation 1: theta ~ HT(alpha,nu)
Notation 2: p(theta) = HT(theta | alpha,nu)
Parameter 1: scale parameter alpha > 0
Parameter 2: degrees of freedom parameter nu
Mean: E(theta) = unknown
Variance: var(theta) = unknown
Mode: 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
.
dhalft
gives the density,
phalft
gives the distribution function,
qhalft
gives the quantile function, and
rhalft
generates random deviates.
dhalfcauchy
,
dst
,
dt
,
dunif
, and
is.proper
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | library(LaplacesDemonCpp)
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"))
|
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