On Testing Marginal versus Conditional Independence
Abstract
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 KullbackLeibler 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 nonuniformity, we study a class of "envelope" distributions by taking pointwise suprema over asymptotic cumulative distribution functions. We show that these envelope distributions are wellbehaved and lead to model selection procedures with ratefree uniform error guarantees and nearoptimal 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.01850
 Bibcode:
 2019arXiv190601850G
 Keywords:

 Mathematics  Statistics Theory;
 Statistics  Methodology
 EPrint:
 Revisions and updated references