A Review of Representative Points of Statistical Distributions and Their Applications

Statistical modeling relies on a diverse range of statistical distributions, encompassing both univariate and multivariate distributions and/or discrete and continuous distributions. In the literature, numerous statistical methods have been proposed to approximate continuous distributions. The most commonly used approach is the use of the empirical distribution which is obtained from a random sample drawn from the distribution. However, it is very likely that the empirical distribution suffers from an accuracy problem when used to approximate the underlying distribution, especially if the sample size is not sufficient. In order to improve statistical inferences, various alternative forms of discrete approximation to the distribution were proposed in the literature. The choice of support points for the discrete approximation, known as Representative Points (RPs), becomes extremely important in terms of distribution approximations. In this paper we give a review of the three main methods for constructing RPs, namely based on the Monte Carlo method, the number-theoretic method (or quasi-Monte Carlo method), and the mean square error method, aiming to introduce such important methods to the statistical or mathematical community. Additional approaches for forming RPs are also briefly discussed. The review focuses on certain critical aspects such as theoretical properties and computational algorithms for constructing RPs. We also address the issue of the application of RPs through studying practical problems and provide evidence of RPs' advantages over random samples in approximating the distribution.