R: Huang-Wand Distribution

dist.HuangWand {LaplacesDemon} | R Documentation

## Huang-Wand Distribution

### Description

These are the density and random generation functions for the Huang-Wand 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 k x k positive-definite covariance matrix Sigma for `dhuangwand`, or the Cholesky factor U of the covariance matrix for `dhuangwandc`.
`nu` | This is a scalar degrees of freedom parameter nu. The default is `nu=2`, which is an uninformative prior, resulting in marginal uniform distributions on the correlation matrix.
`a` | This is a positive-only vector of scale parameters a of length k.
`A` | This is a positive-only vector of of scale hyperparameters A of length k. Larger values result in a more uninformative prior. A default, uninformative prior is `A=rep(1e6,k)`.
`log` | Logical. If `log=TRUE`, then the logarithm of the density is returned.

### Details

- Application: Continuous Multivariate

- Density: p(theta) = W^(-1)[nu+k-1](2*nu*diag(1/a)) G^(-1)(1/2, 1/A^2)

- Inventor: Huang and Wand (2013)

- Notation 1: theta ~ HW[nu](a, A)

- Notation 2: p(theta) ~ HW[nu](theta | a, 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 Huang-Wand prior retains conjugacy. The Cholesky parameterization is also provided here.

The Huang-Wand 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 Huang-Wand prior overcomes this limitation. Second, the inverse Wishart distribution imposes a dependency between variance and correlation. The Huang-Wand prior lessens, but does not fully remove, this dependency.

The standard deviations of a Huang-Wand distributed covariance matrix are half-t distributed, as HT(nu, 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

[Package LaplacesDemon version 16.1.1 Index]