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.

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