# dist.HuangWand: Huang-Wand Distribution

### Description

These are the density and random generation functions for the Huang-Wand prior distribution for a covariance matrix.

### Usage

 ```1 2 3 4``` ```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.

`dhalft` and `dinvwishart`

### Examples

 ```1 2 3 4``` ```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 ```

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