This paper develops a smooth test of goodness-of-fit for elliptical distributions. The test is adaptively omnibus, invariant to affine-linear transformations and has a convenient expression that can be broken into components. These components have diagnostic capabilities and can be used to identify specific departures. This helps in correcting the null model when the test rejects. As an example, the results are applied to the multivariate normal distribution for which the R package ECGofTestDx is available. It is shown that the proposed test strategy encompasses and generalizes a number of existing approaches. Some other cases are studied, such as the bivariate Laplace, logistic and Pearson type II distribution. A simulation experiment shows the usefulness of the diagnostic tools.

中文翻译：

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具有诊断功能的椭圆分布的拟合优度检验

本文开发了椭圆分布拟合优度的平滑测试。该测试是自适应综合性的，对仿射线性变换具有不变性，并且具有可以分解为组件的方便表达式。这些组件具有诊断功能，可用于识别特定的离场。这有助于在测试拒绝时纠正空模型。例如，将结果应用于 R 包 ECGofTestDx 可用的多元正态分布。结果表明，所提出的测试策略涵盖并概括了许多现有方法。研究了其他一些情况，例如二元拉普拉斯分布、logistic 和 Pearson II 型分布。模拟实验显示了诊断工具的有用性。