Many flexible families of positive random variables exhibit non-closed forms of the density and distribution functions and this feature is considered unappealing for modelling purposes. However, such families are often characterized by a simple expression of the corresponding Laplace transform. Relying on the Laplace transform, we propose to carry out parameter estimation and goodness-of-fit testing for a general class of non-standard laws. We suggest a novel data-driven inferential technique, providing parameter estimators and goodness-of-fit tests, whose large-sample properties are derived. The implementation of the method is specifically considered for the positive stable and Tweedie distributions. A Monte Carlo study shows good finite-sample performance of the proposed technique for such laws.

翻译：许多灵活的正随机变量族表现出密度函数与分布函数的非闭合形式，这一特性在建模中被视为不利因素。然而，此类分布族常以相应拉普拉斯变换的简洁表达式为特征。基于拉普拉斯变换，我们提出针对一类广义非标准分布进行参数估计与拟合优度检验的方法。我们提出一种新型数据驱动推断技术，该技术可提供参数估计量及拟合优度检验，并推导了其大样本性质。该方法在正稳定分布与Tweedie分布中的实施得到专门探讨。蒙特卡洛研究表明，所提技术对此类分布具有良好的有限样本性能。