Quantile Jensen’s inequalities

Abstract

AbstractQuantiles of random variable are crucial quantities that give more delicate information about distribution than mean and median and so on. We establish Jensen’s inequality for q-quantile ($q\geq 0.5$ q ≥ 0.5 ) of a random variable, which includes as a special case Merkle (Stat. Probab. Lett. 71(3):277–281, 2005) where Jensen’s inequality about median (i.e. $q= 0.5$ q = 0.5 ) was given. We also refine this inequality in the case where $q<0.5$ q < 0.5 . An application to the confidence interval of parameters in pivotal quantity is also considered by virtue of the rigorous description on the relationship between quantiles and intervals that have required probability.