Sample Random Draws From the Truncated Multivariate Normal Using the Algorithm Proposed by Yifang Li and Sujit K. Ghosh (2015)

Samples random numbers from a truncated multivariate normal distribution with parameters mean vector, covariance matrix, and linear inequality constraints of the form \(l \leq Bx \leq u\), where \(B\) is a constraint matrix and \(l\) and \(u\) are lower and upper bounds. The function uses a Gibbs sampling algorithm to generate draws from the constrained distribution.

The truncated multivariate normal is important for research in Bayesian statistics, econometrics, and any field requiring parameter estimation subject to inequality constraints. Common applications include censored regression models, portfolio optimization with constraints, and prior distributions with bounded support.

Usage

rtmvnorm(mean, sigma, blc, lower, upper, init, burn = 10)

Arguments

mean: an \(n\times p\) matrix of means. n is the number of draws to be sampled. p is the dimension of the draws. C++: an arma::mat object.
sigma: a \(p\times p\) covariance matrix for the draws. C++: an arma::mat object.
blc: an \(m\times p\) matrix of coefficients for linear inequality constraints. C++: an arma::mat object.
lower: an \(n\times m\) matrix of lower truncation bounds. C++: an arma::mat object.
upper: an \(n\times m\) matrix of upper truncation bounds. C++: an arma::mat object.
init: an \(n\times p\) matrix of initial values for the algorithm. C++: an arma::mat object.
burn: number of iterations used as burn-in. Defaults is 10. C++: an arma::uword object.

Value

An \(n\times p\) matrix of draws from the specified truncated multivariate normal. C++: an arma::mat object.

Details

This function is based on C++ code from the R package tmvtnsim by Lu (2025) and is using objects and commands from the armadillo library by Sanderson & Curtin (2025) thanks to the RcppArmadillo package by Eddelbuettel, Francois, Bates, Ni, & Sanderson (2025).

References

Eddelbuettel D., Francois R., Bates D., Ni B., Sanderson C. (2025). RcppArmadillo: 'Rcpp' Integration for the 'Armadillo' Templated Linear Algebra Library. R package version 15.0.2-2. <doi:10.32614/CRAN.package.RcppArmadillo>

Sanderson C., Curtin R. (2025). Armadillo: An Efficient Framework for Numerical Linear Algebra. International Conference on Computer and Automation Engineering, 303-307, <doi:10.1109/ICCAE64891.2025.10980539>

Li, Y., Ghosh, S.K. Efficient sampling methods for truncated multivariate normal and student-t distributions subject to linear inequality constraints. J Stat Theory Pract 9, 712–732 (2015). <doi:10.1080/15598608.2014.996690>

Lu K. (2025). tmvtnsim: Truncated Multivariate Normal and t Distribution Simulation. R package version 0.1.4, <doi:10.32614/CRAN.package.tmvtnsim>

Author

Filip Reierson filip.reierson@gmail.com

Examples

rtmvnorm(mean = matrix(c(0, 0), nrow = 1), sigma = diag(2), 
         blc = diag(2), lower = matrix(c(-Inf, -Inf), nrow = 1), 
         upper = matrix(c(1, 1), nrow = 1), init = matrix(c(0, 0), 
         nrow = 1), burn = 10)
#>           [,1]       [,2]
#> [1,] -1.048713 0.03810289