UC San Diego

The detection of sparse heterogeneous mixtures becomes important in settings where a small proportion of a population may be affected by a given treatment, for example. The situation is typically formalized as a contamination model. We consider such models in asymptotic regimes where the contamination proportion tends to zero at various rates. We study the following three settings: the contamination manifests itself as a change in variance, the contamination manifests itself as a positive dependence between the variables in the bivariate setting, and the effect is a shift in mean without knowing the null distribution. In each setting, we study how large the effect needs to be in order to reliably distinguish the null hypothesis and the alternative hypothesis. We show that the corresponding higher criticism test is first-order comparable to the likelihood ratio test, while other classical tests are suboptimal. In particular, we make connections between the first two settings. We consider the dependence problem from both parametric and nonparametric perspectives. In the last chapter, we consider a different problem, that is to examine the extent to which the causal inference resulting estimate is sensitive to the unmeasured confounders for survival and competing risks data.