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 ( $q\geq 0.5$ ) 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. $q= 0.5$ ) was given. We also refine this inequality in the case where $q<0.5$ . 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.

中文翻译：

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分位数詹森的不等式

随机变量的分位数是至关重要的量，与均值和中位数等相比，它们提供有关分布的更精细的信息。我们建立随机变量的q分位数（$ q \ geq 0.5 $）的Jensen不等式，其中包括Merkle（Stat。Probab。Lett。71（3）：277–281，2005）作为特例，其中Jensen的不等式约为给出了中位数（即$ q = 0.5 $）。我们还优化了$ q <0.5 $的情况下的不等式。借助于对具有所需概率的分位数和区间之间关系的严格描述，还考虑了对枢轴数量参数置信区间的应用。