Limiting behavior of the gap between the largest two representative points of statistical distributions

Abstract

This paper explores the properties of the gap of representative points (RPs) in the sense of minimum mean square error for various univariate statistical distributions. We illustrate the relationship between RPs and doubly truncated mean residual life (DMRL) as well as mean residual life (MRL), which are widely used in survival analysis. The limiting behavior of the gap between the largest two RPs is discussed. In addition, an upper bound of the optimal MSE is given when the univariate random variable X has a domain on finite interval. In simulation studies, the performance of RPs for various distributions is assessed in terms of moment estimation and resampling technique. A brief discussion about the relationship between the tail of the distribution and the gap of RPs is also given.