Three kinds of discrete approximations of statistical multivariate distributions and their applications

Abstract

Discrete approximations of statistical continuous distributions have been widely requested in various fields. Using random samples generated by Monte Carlo (MC) method to infer the population has been dominant in statistics. The empirical distribution of a random sample can be regarded as a discrete approximation of the population distribution in a certain statistical sense. However, MC has a poor performance in many problems. This paper concerns some alternative methods, such as Quasi-Monte Carlo (QMC) F-numbers and Mean Square Error Representative Points (MSE-RPs), and constructs approximation distributions for elliptically contoured distributions and skew-normal distributions. Numerical comparisons are given for two geometric probability problems and for estimation accuracy by resampling from the discrete approximation distributions obtained by MC, QMC and MSE-RPs. Our simulation results indicate that QMC and MSE-RPs have better performance in most comparisons. These results show that QMC and MSE-RPs have high potential in statistical inference. In addition, we also discuss the relationship between principal component analysis and MSE-RPs for elliptically contoured distributions, as well as its potential applications.