Description Usage Arguments Details Value Author(s) References See Also Examples
This function provides the joint density function for the truncated multivariate Student t
distribution with mean vector equal to mean
, covariance matrix
sigma
, degrees of freedom parameter df
and
lower and upper truncation points lower
and upper
.
1 2 3 |
x |
Vector or matrix of quantiles. If |
mean |
Mean vector, default is |
sigma |
Covariance matrix, default is |
df |
degrees of freedom parameter |
lower |
Vector of lower truncation points,
default is |
upper |
Vector of upper truncation points,
default is |
log |
Logical; if |
The Truncated Multivariate Student t Distribution is a conditional Multivariate Student t distribution subject to (linear) constraints a ≤ \bold{x} ≤ b.
The density of the p-variate Multivariate Student t distribution with nu degrees of freedom is
f(\bold{x}) = \frac{Γ((ν + p)/2)}{(πν)^{p/2} Γ(ν/2) \|Σ\|^{1/2}} [ 1 + \frac{1}{ν} (x - μ)^T Σ^{-1} (x - μ) ]^{- (ν + p) / 2}
The density of the truncated distribution f_{a,b}(x) with constraints a <= x <= b is accordingly
f_{a,b}(x) = \frac{f(\bold{x})} {P(a ≤ x ≤ b)}
a numeric vector with density values
Stefan Wilhelm [email protected]
Geweke, J. F. (1991) Efficient simulation from the multivariate normal and Student-t distributions subject to linear constraints and the evaluation of constraint probabilities. http://www.biz.uiowa.edu/faculty/jgeweke/papers/paper47/paper47.pdf
Samuel Kotz, Saralees Nadarajah (2004). Multivariate t Distributions and Their Applications. Cambridge University Press
ptmvt
and rtmvt
for probabilities and random number generation in the truncated case,
see dmvt
, rmvt
and pmvt
for the untruncated multi-t distribution.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | # Example
x1 <- seq(-2, 3, by=0.1)
x2 <- seq(-2, 3, by=0.1)
mean <- c(0,0)
sigma <- matrix(c(1, -0.5, -0.5, 1), 2, 2)
lower <- c(-1,-1)
density <- function(x)
{
z=dtmvt(x, mean=mean, sigma=sigma, lower=lower)
z
}
fgrid <- function(x, y, f)
{
z <- matrix(nrow=length(x), ncol=length(y))
for(m in 1:length(x)){
for(n in 1:length(y)){
z[m,n] <- f(c(x[m], y[n]))
}
}
z
}
# compute multivariate-t density d for grid
d <- fgrid(x1, x2, function(x) dtmvt(x, mean=mean, sigma=sigma, lower=lower))
# compute multivariate normal density d for grid
d2 <- fgrid(x1, x2, function(x) dtmvnorm(x, mean=mean, sigma=sigma, lower=lower))
# plot density as contourplot
contour(x1, x2, d, nlevels=5, main="Truncated Multivariate t Density",
xlab=expression(x[1]), ylab=expression(x[2]))
contour(x1, x2, d2, nlevels=5, add=TRUE, col="red")
abline(v=-1, lty=3, lwd=2)
abline(h=-1, lty=3, lwd=2)
|
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.