That data follow a Gompertz distribution is a widely used assumption in diverse fields of applied sciences, e.g., in biology or when analysing survival times. Since misspecified models may lead to false conclusions, assessing the fit of the data to an underlying model is of central importance. We propose a new family of characterisation-based weighted $L^2$-type tests of fit to the family of Gompertz distributions, hence tests for the composite hypothesis when the parameters are unknown. The characterisation is motivated by distributional transforms connected to Stein's method of distributional approximation. We provide the limit null distribution of the test statistics in a Hilbert space setting and, since the limit distribution depends on the unknown parameters, we propose a parametric bootstrap procedure. Consistency of the testing procedure is shown. An extensive simulation study as well as applications to real data examples show practical benefits of the procedures: the first data set we analyse consists of lifetimes of fruitflies, the second has been synthetically generated from life tables for women born in Germany in 1948.

翻译：假设数据服从Gompertz分布在应用科学的多个领域（例如生物学或生存时间分析）中是一个广泛使用的假设。由于模型设定错误可能导致错误结论，因此评估数据与基础模型的拟合程度至关重要。我们提出了一类基于刻画特征的加权$L^2$型拟合优度检验新方法，用于检验复合假设（即参数未知时Gompertz分布族的拟合优度）。该刻画受与Stein分布逼近方法相关的分布变换启发。我们在希尔伯特空间框架下给出了检验统计量的极限零分布，并针对极限分布依赖于未知参数的情况提出了参数自举程序。同时证明了检验程序的一致性。广泛的模拟研究和实际数据应用展示了该方法的实用价值：第一个数据集包含果蝇的寿命数据，第二个数据集根据1948年德国出生女性的生命表合成生成。