Consider a pair of cumulative distribution functions $F$ and $G$, where $F$ is unknown and $G$ is a known reference distribution. Given a sample from $F$, we propose tests to detect the convexity or the concavity of $G^{-1}\circ F$ versus equality in distribution (up to location and scale transformations). This framework encompasses well-known cases, including increasing hazard rate distributions, as well as some other relevant families that have garnered attention more recently, for which no tests are currently available. We introduce test statistics based on the estimated probability that the random variable of interest does not exceed a given expected order statistic, which, in turn, is estimated via L-estimation. The tests are unbiased, consistent, and exhibit monotone power with respect to the convex transform order. To ensure consistency, we show that our L-estimators satisfy a strong law of large numbers, even when the mean is not finite, thereby making the tests suitable for heavy-tailed distributions. Unlike other approaches, these tests are broadly applicable, regardless of the choice of $G$ and without support restrictions. The performance of the method under various conditions is demonstrated via simulations, and its applicability is illustrated through a concrete example.

翻译：考虑一对累积分布函数 $F$ 和 $G$，其中 $F$ 未知，$G$ 为已知参考分布。给定来自 $F$ 的样本，我们提出检验方法以检测 $G^{-1}\circ F$ 的凸性或凹性，其对立假设为分布相等（允许位置与尺度变换）。该框架涵盖了若干经典情形，包括递增失效率分布，以及近期受到关注但尚无可用检验方法的其他相关分布族。我们引入了基于估计概率的检验统计量，该概率表示随机变量不超过给定期望顺序统计量，而期望顺序统计量本身通过 L 估计进行估计。所提检验具有无偏性、相合性，且关于凸变换序具有单调功效。为保证相合性，我们证明了所采用的 L 估计量满足强大数定律，即使均值不存在时亦然，从而使该检验适用于重尾分布。与其他方法不同，这些检验具有广泛适用性，不依赖于 $G$ 的选择且无需支撑集限制。通过模拟验证了该方法在不同条件下的性能，并通过具体实例说明了其适用性。