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:在应用科学的不同领域,例如在生物学或分析生存时间时,数据按照Gompertz的分布是一种广泛应用的假设。由于给定的模型可能导致错误的结论,因此评估数据是否适合一个基本模型是至关重要的。我们建议建立一个适合Gompertz分布式家庭的基于特征的加权加权2美元类型测试的新体系,从而在参数未知时测试综合假设。特征的驱动因素是与Stein的分布近似法相联系的分布变换。我们在希尔伯特空间设置中提供了测试统计数据的无效分布限制,并且由于限制分布取决于未知参数,我们提议了一个参数布局程序。测试程序的一致性得到了证明。广泛的模拟研究以及对真实数据实例的应用显示了程序的实际好处:我们分析的第一组数据由果蝇的生命周期组成,第二套数据是1948年在德国出生的妇女生命表合成的。