Integral, mean and covariance of the simplex-truncated multivariate normal distribution

Area of Science:

• Statistics
• Data Science
• Computational Statistics

Background:

• Compositional data, representing fractions or probabilities, are prevalent across scientific disciplines.
• The simplex-truncated multivariate normal distribution models the spread of such data when normally distributed.
• Accurate estimation of integral, mean, and covariance is vital for analyzing these truncated distributions.

Purpose of the Study:

• To describe and compare three distinct computational approaches for estimating key characteristics of the simplex-truncated multivariate normal distribution.
• To evaluate the computational efficiency of each method across varying dimensions.

Main Methods:

• Naive rejection sampling.
• A hybrid method combining subset simulation, the Holmes-Diaconis-Ross algorithm, and analytical elliptical slice sampling (Gessner et al.).
• A semi-analytical method utilizing integrals of hyperrectangularly-truncated multivariate normal distributions.

Main Results:

• All three methods demonstrated strong agreement in their estimations.
• Computational efficiency varied significantly with the dimensionality of the distribution.
• The semi-analytical method is efficient for low-dimensional problems.
• The Gessner et al. method emerges as the most practically efficient for high-dimensional distributions.

Conclusions:

• The choice of computational method for simplex-truncated multivariate normal distributions depends on the specific application's dimensionality and implementation.
• For low dimensions, the semi-analytical approach offers speed, while for high dimensions, the Gessner et al. method is recommended for practical efficiency.