By extrapolating the explicit formula of the zero-bias distribution occurring in the context of Stein’s method, we construct characterization identities for a large class of absolutely continuous univariate distributions. Instead of trying to derive characterizing distributional transformations that inherit certain structures for the use in further theoretic endeavors, we focus on explicit representations given through a formula for the density- or distribution function. The results we establish with this ambition feature immediate applications in the area of goodness-of-fit testing. We draw up a blueprint for the construction of tests of fit that include procedures for many distributions for which little (if any) practicable tests are known. To illustrate this last point, we construct a test for the Burr Type XII distribution for which, to our knowledge, not a single test is known aside from the classical universal procedures.

中文翻译：

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连续单变量概率分布的不动点表征及其应用

通过外推出现在 Stein 方法上下文中的零偏差分布的显式公式，我们为一大类绝对连续的单变量分布构建了特征恒等式。我们不是试图推导出继承某些结构以用于进一步理论研究的特征分布变换，而是专注于通过密度或分布函数的公式给出的显式表示。我们以此为目标建立的结果具有在拟合优度测试领域的直接应用。我们制定了构建拟合测试的蓝图，其中包括许多分布的程序，而这些分布的可行测试很少（如果有的话）。为了说明最后一点，我们为 Burr Type XII 分布构建了一个测试，其中，