We use a Stein identity to define a new class of parametric distributions which we call ``independent additive weighted bias distributions.'' We investigate related $L^2$-type discrepancy measures, empirical versions of which not only encompass traditional ODE-based procedures but also offer novel methods for conducting goodness-of-fit tests in composite hypothesis testing problems. We determine critical values for these new procedures using a parametric bootstrap approach and evaluate their power through Monte Carlo simulations. As an illustration, we apply these procedures to examine the compatibility of two real data sets with a compound Poisson Gamma distribution.

翻译：我们利用Stein恒等式定义了一类新的参数分布，称之为“独立加性加权偏差分布”。我们研究了相关的$L^2$型差异度量，其经验形式不仅涵盖了传统的基于常微分方程的方法，还为复合假设检验问题中的拟合优度检验提供了新方法。我们通过参数自举法确定了这些新方法的临界值，并利用蒙特卡洛模拟评估了其检验功效。作为示例，我们将这些方法应用于检验两个真实数据集与复合泊松伽马分布的相容性。