On testing marginal versus conditional independence

We consider testing marginal independence versus conditional independence in a trivariate Gaussian setting. The two models are nonnested, 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 a 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 nonuniformity, 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.