Description Usage Arguments Details Value Author(s) See Also Examples
These functions provide the density and random number generation for the multivariate t distribution, otherwise called the multivariate Student distribution, given the Cholesky parameterization.
1 2 |
x |
This is either a vector of length k or a matrix with a number of columns, k, equal to the number of columns in scale matrix S. |
n |
This is the number of random draws. |
mu |
This is a numeric vector or matrix representing the location
parameter,mu (the mean vector), of the multivariate
distribution (equal to the expected value when |
U |
This is the k x k upper-triangular matrix
that is Cholesky factor U of scale matrix
S, such that |
df |
This is the degrees of freedom, and is often represented with nu. |
log |
Logical. If |
Application: Continuous Multivariate
Density:
p(theta) = Gamma[(nu+k)/2] / {Gamma(nu/2)nu^(k/2)pi^(k/2)|Sigma|^(1/2)[1 + (1/nu)(theta-mu)^T*Sigma^(-1)(theta-mu)]^[(nu+k)/2]}
Inventor: Unknown (to me, anyway)
Notation 1: theta ~ t[k](mu, Sigma, nu)
Notation 2: p(theta) = t[k](theta | mu, Sigma, nu)
Parameter 1: location vector mu
Parameter 2: positive-definite k x k scale matrix Sigma
Parameter 3: degrees of freedom nu > 0 (df in the functions)
Mean: E(theta) = mu, for nu > 1, otherwise undefined
Variance: var(theta) = (nu / (nu - 2))*Sigma, for nu > 2
Mode: mode(theta) = mu
The multivariate t distribution, also called the multivariate Student or
multivariate Student t distribution, is a multidimensional extension of the
one-dimensional or univariate Student t distribution. A random vector is
considered to be multivariate t-distributed if every linear
combination of its components has a univariate Student t-distribution.
This distribution has a mean parameter vector mu of length
k, and an upper-triangular k x k matrix that is
Cholesky factor U, as per the chol
function for Cholesky decomposition. When degrees of freedom
nu=1, this is the multivariate Cauchy distribution.
In practice, U is fully unconstrained for proposals
when its diagonal is log-transformed. The diagonal is exponentiated
after a proposal and before other calculations. Overall, the Cholesky
parameterization is faster than the traditional parameterization.
Compared with dmvt
, dmvtc
must additionally
matrix-multiply the Cholesky back to the scale matrix, but it
does not have to check for or correct the scale matrix to
positive-definiteness, which overall is slower. The same is true when
comparing rmvt
and rmvtc
.
dmvtc
gives the density and
rmvtc
generates random deviates.
Statisticat, LLC. software@bayesian-inference.com
chol
,
dinvwishartc
,
dmvc
,
dmvcp
,
dmvtp
,
dst
,
dstp
, and
dt
.
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