A fractional order statistic towards defining a smooth quantile function for discrete data

This work is motivated in part by a recent publication by Ma et al. (2011) who resolved the asymptotic non-normality problem of the classical sample quantiles for discrete data through defining a new mid-distribution based quantile function. This work is the motivation for defining a new and improved smooth population quantile function given discrete data. Our definition is based on the theory of fractional order statistics. The main advantage of our definition as compared to its competitors is the capability to distinguish the u  th quantile across different discrete distributions over the whole interval, u∈(0,1)u∈(0,1). In addition, we define the corresponding estimator of the smooth population quantiles and demonstrate the convergence and asymptotic normal distribution of the corresponding sample quantiles. We verify our theoretical results through a Monte Carlo simulation, and illustrate the utilization of our quantile function in a Q–Q plot for discrete data.