dStudent: Density of a Multivariate Student t Distribution

In student: Multivariate Student t Distribution

Description

Evaluate the density of a multivariate Student t distribution.

Usage

dStudent(x, df, loc = rep(0, d), scale,
         factor = tryCatch(factorize(scale), error = function(e) e),
         log = FALSE)

Arguments

x	(n, d)-matrix of evaluation points.
df	degrees of freedom (positive real number or Inf in which case the density of a multivariate normal distribution with mean vector loc and covariance matrix scale is evaluated).
loc	location vector of dimension d (the number of columns of x).
scale	covariance matrix of dimension (d, d).
factor	factorization matrix of the covariance matrix scale. A matrix R with d rows such that R^T R equals scale. Although not tested (for efficiency reasons), R (thus factor) has to be upper triangular (otherwise the determinant of scale involved in the density is not computed correctly). Also note that factor can be NULL or inherit from "error" in which case the degenerate density is computed as Inf at loc and 0 everywhere else.
log	logical indicating whether the logarithm of the density is to be computed.

Details

Internally used is factor, so scale is not required to be provided if factor is given.

The default factorization used is the Cholesky decomposition. To this end, scale needs to have full rank.

Value

dStudent() returns an n-vector with the density values (default) or log-density values (if log) of the multivariate Student t distribution with df degrees of freedom, location vector loc and scale matrix scale (a covariance matrix).

Author(s)

Marius Hofert

References

McNeil, A. J., Frey, R., and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.

See Also

pStudent(), rStudent()

Examples

## Generate a random correlation matrix in three dimensions
d <- 3
set.seed(271)
A <- matrix(runif(d * d), ncol = d)
P <- cov2cor(A %*% t(A))
## Evaluate t_{3.5} density
df <- 3.5
x <- matrix(1:12/12, ncol = d) # evaluation points
dt <- dStudent(x, df = df, scale = P)
stopifnot(all.equal(dt, c(0.013266542, 0.011967156, 0.010760575, 0.009648682),
                    tol = 1e-7))
## Evaluate normal density
dn <- dStudent(x, df = Inf, scale = P)
stopifnot(all.equal(dn, c(0.013083858, 0.011141923, 0.009389987, 0.007831596),
                    tol = 1e-7))

## Missing data
x[3,2] <- NA
x[4,3] <- NA
dt <- dStudent(x, df = df, scale = P)
stopifnot(is.na(dt) == rep(c(FALSE, TRUE), each = 2))

## Univariate case
x <- matrix(1:10/10, ncol = 1)
dt <- dStudent(x, df = df, factor = 1)
dt. <- dt(as.vector(x), df = df)
stopifnot(all.equal(dt, dt.))

student documentation built on May 2, 2019, 6:48 p.m.