We define generalized innovations associated with generalized error models having arbitrary distributions, that is, distributions that can be mixtures of continuous and discrete distributions. These models include stochastic volatility models and regime-switching models. We also propose statistics for testing independence between the generalized errors of these models, extending previous results of Duchesne, Ghoudi and Remillard (2012) obtained for stochastic volatility models. We define families of empirical processes constructed from lagged generalized errors, and we show that their joint asymptotic distributions are Gaussian and independent of the estimated parameters of the individual time series. Moebius transformations of the empirical processes are used to obtain tractable covariances. Several tests statistics are then proposed, based on Cramer-von Mises statistics and dependence measures, as well as graphical methods to visualize the dependence. In addition, numerical experiments are performed to assess the power of the proposed tests. Finally, to show the usefulness of our methodologies, examples of applications for financial data and crime data are given to cover both discrete and continuous cases. ll developed methodologies are implemented in the CRAN package IndGenErrors.

翻译：本文定义了与广义误差模型相关的广义新息，该模型允许任意分布，即可以包含连续分布与离散分布的混合分布。此类模型涵盖随机波动率模型和区制转换模型。同时，我们提出了用于检验这些模型广义误差之间独立性的统计量，拓展了Duchesne、Ghoudi和Remillard（2012）针对随机波动率模型所获得的结果。我们构建了基于滞后广义误差的经验过程族，并证明其联合渐近分布服从高斯分布且与各时间序列的估计参数无关。通过引入经验过程的莫比乌斯变换，我们得到了易于处理的协方差结构。随后，基于Cramer-von Mises统计量与依赖性度量，提出了多种检验统计量，并提供了可视化依赖关系的图形方法。此外，通过数值实验评估了所提出检验方法的功效。最后，为展示本方法体系的实际应用价值，分别针对金融数据与犯罪数据给出了应用示例，涵盖了离散与连续两种情形。所有发展的方法均已在CRAN软件包IndGenErrors中实现。