In several applications in surveys, it would be helpful to be able to generate a set of $R$ $n$-dimensional variates with the following properties:
- Has mean vector $1$
- Has a specified variance-covariance matrix ($\Sigma$, a positive semi-definite matrix, with rank $\leq R$)
- Is nonnegative
Some First Thoughts about Solutions
The first two requirements are easy to satisfy. We can just draw from a multivariate Normal distribution, as discussed here: Generating data with a given sample covariance matrix
But if we add the additional requirement that all of the variates are nonnegative, then a multivariate Normal distribution won't work. In the applications I have in mind (described below), typically the diagonal of $\Sigma$ has a few entries which are $1$ or even as large as $1.5$, so a multivariate Normal will easily generate negative values.
In this application, it doesn't matter at all what the skew or kurtosis are, and it doesn't matter whether the random variates come from a particular distribution. All that matters is that they're nonnegative, have mean vector $1$, and have the specified covariance matrix (ideally, it would have the exact specified sample covariance matrix, but it would be OK if it just had the specified covariance matrix in expectation).
The multivariate lognormal and Gamma distributions are nonnegative and seem like fairly natural options, except that there are constraints on their precise shape (due to their density functions) which mean that they will often not be able to attain the desired variance-covariance. So these parametric distributions, at least, seem unnecessarily limiting.
In low dimensions, one can generate random variates and then "fix them up" to satisfy constraints, at least approximately. This StackExchange gets at this kind of approach, but it doesn't really cover the constraint of nonnegativity and is really focused on a univariate case.
In survey statistics, resampling methods such as the jackknife or bootstrap are typically implemented by generating replicate weights, which are random variates with a specific mean vector and covariance matrix. To be concrete, if one has a dataset of size $n$, then one generates $R$ sets of replicate weights, which can be represented as a matrix of dimension $R \times n$. In general, we want these replicate weights to be nonnegative, since the weights will frequently be used in statistical procedures that require nonnegative weights and usually even for many procedures that allow negative weights, the software implementations won't allow negative weights because they're just unexpected.
This R package vignette describes the generation of replicate weights for the "generalized survey bootstrap", and describes how multivariate normal distributions are used to generate replicate weights that have mean vector $1$ and a specified variance-covariance matrix, but which can sometimes be negative. https://cran.r-project.org/web/packages/svrep/vignettes/bootstrap-replicates.html#forming-adjustment-factors It also describes a rescaling adjustments that makes the random variates nonnegative and with mean vector $1$, but unfortunately that rescaling adjustment increases the variance-covariance by a constant (which is a problem).