Quantile Jensen’s inequalities

Abstract

Abstract Quantiles 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 ≥ 0.5 $q\geq 0.5$ ) 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.

Keywords

• C-function
• Confidence interval
• Jensen’s inequality
• Quantile