We consider testing marginal independence versus conditional independence in a trivariate Gaussian setting. The two models are non-nested and their intersection is a union of two marginal independences. We consider two sequences of such models, one from each type of independence, that are closest to each other in the Kullback-Leibler sense as they approach the intersection. They become indistinguishable if the signal strength, as measured by the product of two correlation parameters, decreases faster than the standard parametric rate. Under local alternatives at such rate, we show that the asymptotic distribution of the likelihood ratio depends on where and how the local alternatives approach the intersection. To deal with this non-uniformity, we study a class of "envelope" distributions by taking pointwise suprema over asymptotic cumulative distribution functions. We show that these envelope distributions are well-behaved and lead to model selection procedures with rate-free uniform error guarantees and near-optimal power. To control the error even when the two models are indistinguishable, rather than insist on a dichotomous choice, the proposed procedure will choose either or both models.

中文翻译：

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关于测试边际独立性与条件独立性

我们考虑在三变量高斯设置中测试边际独立性与条件独立性。这两个模型是非嵌套的，它们的交集是两个边际独立性的并集。我们考虑这样的模型的两个序列，一个来自每种类型的独立性，当它们接近交叉点时，在 Kullback-Leibler 意义上彼此最接近。如果由两个相关参数的乘积测量的信号强度比标准参数速率下降得更快，则它们变得无法区分。在这种速率的局部替代方案下，我们表明似然比的渐近分布取决于局部替代方案接近交叉点的位置和方式。为了处理这种不均匀性，我们研究了一类“信封” 通过在渐近累积分布函数上取逐点上位来计算分布。我们表明这些包络分布表现良好，并导致模型选择程序具有无速率均匀误差保证和接近最优的功率。为了在两个模型无法区分时控制误差，而不是坚持二分法选择，建议的程序将选择一个或两个模型。