dist.HuangWand {LaplacesDemon}  R Documentation 
HuangWand Distribution
Description
These are the density and random generation functions for the HuangWand prior distribution for a covariance matrix.
Usage
dhuangwand(x, nu=2, a, A, log=FALSE)
dhuangwandc(x, nu=2, a, A, log=FALSE)
rhuangwand(nu=2, a, A)
rhuangwandc(nu=2, a, A)
Arguments
x 
This is a 
nu 
This is a scalar degrees of freedom parameter

a 
This is a positiveonly vector of scale parameters

A 
This is a positiveonly vector of of scale hyperparameters

log 
Logical. If 
Details
Application: Continuous Multivariate
Density:
p(\theta) = \mathcal{W}^{1}_{\nu+k1}(2 \nu diag(1/a)) \mathcal{G}^{1}(1/2, 1/A^2)
Inventor: Huang and Wand (2013)
Notation 1:
\theta \sim \mathcal{HW}_\nu(\textbf{a}, \textbf{A})
Notation 2:
p(\theta) \sim \mathcal{HW}_\nu(\theta  \textbf{a}, \textbf{A})
Parameter 1: degrees of freedom
\nu
Parameter 2: scale
a > 0
Parameter 3: scale
A > 0
Mean:
Variance:
Mode:
Huang and Wand (2013) proposed a prior distribution for a covariance matrix that uses a hierarchical inverse Wishart. This is a more flexible alternative to the inverse Wishart distribution, and the HuangWand prior retains conjugacy. The Cholesky parameterization is also provided here.
The HuangWand prior distribution alleviates two main limitations of an inverse Wishart distribution. First, the uncertainty in the diagonal variances of a covariance matrix that is inverse Wishart distributed is represented with only one degrees of freedom parameter, which may be too restrictive. The HuangWand prior overcomes this limitation. Second, the inverse Wishart distribution imposes a dependency between variance and correlation. The HuangWand prior lessens, but does not fully remove, this dependency.
The standard deviations of a HuangWand distributed covariance matrix
are halft distributed, as \mathcal{HT}(\nu, \textbf{A})
. This is in accord with modern assumptions about distributions of
scale parameters, and is also useful for sparse covariance matrices.
The rhuangwand
function allows either a
or A
to be
missing. When a
is missing, the covariance matrix is generated
from the hyperparameters. When A
is missing, the covariance
matrix is generated from the parameters.
Value
dhuangwand
and dhuangwandc
give the density, and
rhuangwand
and rhuangwandc
generate random deviates.
References
Huang, A., Wand, M., et al. (2013), "Simple Marginally Noninformative Prior Distributions for Covariance Matrices". Bayesian Analysis, 8, p. 439–452.
See Also
dhalft
and
dinvwishart
Examples
library(LaplacesDemon)
dhuangwand(diag(3), nu=2, a=runif(3), A=rep(1e6,3), log=TRUE)
rhuangwand(nu=2, A=rep(1e6, 3)) #Missing a
rhuangwand(nu=2, a=runif(3)) #Missing A