We introduce a comprehensive method for establishing stochastic orders among order statistics in the i.i.d. case. This approach relies on the assumption that the underlying distribution is linked to a reference distribution through a transform order. Notably, this method exhibits broad applicability, particularly since several well-known nonparametric distribution families can be defined using relevant transform orders, including the convex and the star transform orders. In the context of convex-ordered families, we demonstrate that applying Jensen's inequality enables the derivation of bounds for the probability that a random variable exceeds the expected value of its corresponding order statistic.

翻译：本文提出了一种在独立同分布情形下建立顺序统计量间随机序的通用方法。该方法基于以下假设：基础分布通过某种变换序与参考分布相关联。值得注意的是，此方法具有广泛的适用性，特别是因为若干著名的非参数分布族（包括凸变换序与星形变换序）均可通过相关变换序进行定义。在凸序分布族的情境下，我们证明了应用Jensen不等式能够推导出随机变量超过其对应顺序统计量期望值的概率上界。