Sampling from the Cholesky factorization of a Wishart random variable, sampling from the inverse Wishart distribution, sampling from the Cholesky factorization of an inverse Wishart random variable, sampling from the pseudo Wishart distribution, sampling from the generalized inverse Wishart distribution, computing densities for the Wishart and inverse Wishart distributions, and computing the multivariate gamma and digamma functions.

The most common use for this package is likely fast sampling from the inverse Wishart or use of the multivariate gamma or digamma functions. This is a package for fast computation of various functions related to the Wishart distribution, such as sampling from the Cholesky factorization of the Wishart, sampling from the inverse Wishart, sampling from the Cholesky factorization of the inverse Wishart, sampling from the pseudo Wishart, computing densities for the Wishart and inverse Wishart, and computing a few auxiliary functions such as the multivariate gamma and digamma functions. Many of these functions are written in C to maximize efficiency.

The output of the sampling functions is in the same format as the output of `stats::rWishart()`

.

The main idea: sampling for multivariate or matrix variate statistics often makes use of distributions related to the Wishart. There are implementations in a few packages but they are often in R and much slower than the basic `stats::rWishart()`

or there is a lot of associated overhead in the package. Here, then, is a small package with some of those distributions and related functions. As the name suggests, the initial purpose was sampling from the Cholesky factorization of a Wishart distribution.

The package offers the following distributions:

- The inverse Wishart:

```
rInvWishart(n = 1,df = 5,Sigma = diag(4)))
```

- The Cholesky decomposition of the inverse Wishart:

```
rInvCholWishart(n = 1,df = 5,Sigma = diag(4)))
```

- The Cholesky decomposition of the Wishart

```
rCholWishart(n = 1,df = 5,Sigma = diag(4)))
```

- The pseudo Wishart and its pseudo-inverse, the generalized inverse Wishart (
`df`

less than the dimension of`Sigma`

minus 1):

```
rPseudoWishart(n = 1,df = 3,Sigma = diag(5)))
rGenInvWishart(n = 1,df = 3,Sigma = diag(5)))
```

- The package also offers density functions for the Wishart and inverse Wishart distributions:

```
dInvWishart(x = diag(4), df = 5, Sigma = diag(4), log = TRUE)
dWishart(x = diag(4), df = 5, Sigma = diag(4), log = TRUE)
```

In addition to this, the package offers the multivariate gamma function and multivariate digamma function, the multivariate counterparts of the gamma and digamma (derivative of the log of the gamma) functions.

```
mvgamma(x = 5, p = 3)
mvdigamma(x = 5, p = 3)
```

I suspect the most useful functions for most people will be the `mvgamma`

and `rInvWishart`

functions.

Now available on CRAN, install it at:

```
install.packages('CholWishart')
```

Install the latest development version at: devtools::install_github("gzt/CholWishart")

Please note that the 'CholWishart' project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.

- Tweak the documentation
- Port pseudo-Wishart to C, gen inv based on pseudo-Wishart

- Add new functions to the vignette
- Add generalized inverse Wishart (pseudo inverse of the pseudo Wishart)
- Add pseudo-Wishart (Wishart distribution based on fewer observations than the dimension of the covariance matrix).
- Add contributor guidelines and code of conduct.

- Minor update to internal functions

- Adding possibility of array input to density functions.

- Finalize edits to documentation including additional references.

- Add more documentation, add more references to documentation, clean LaTeX equations in documentation.

- Add additional tests for
`dWishart`

and`dInvWishart`

functions - Add references and equations to help files
- Add additional tests for complex entries (should fail) and other erroneous input

- Feature complete, fully documented, and the math should be correct.

- Breaking off from
`matrixdist`