Smoothed empirical likelihood for the difference of two quantiles with the paired sample

In this paper, we propose a novel smoothed empirical likelihood method for the difference of quantiles with paired samples. While the empirical likelihood for the difference of two quantiles with independent samples has been studied, it is crucial to develop a statistical procedure that accounts for the dependence between paired samples from $${\varvec{X}}=(X_1, X_2)$$ X = ( X 1 , X 2 ) . To this end, we propose two estimating equations for the difference of two quantiles and introduce a nuisance parameter in our smoothed empirical likelihood framework. We demonstrate that our approach yields a limiting distribution that follows the standard $$\chi ^2$$ χ 2 distribution. Extensive simulation studies confirm that our smoothed empirical likelihood method outperforms the normal approximation and method M (Wilcox and Erceg-Hurn in J Appl Stat 39(12):2655–2664, 2012) in most cases. Finally, we illustrate the usefulness of our proposed method by applying it to a real-world data set, estimating the interval of the quantile difference of GDP between different years.