Inference in latent factor regression with clusterable features

Regression models, in which the observed features X ∈ R p and the response Y ∈ R depend, jointly, on a lower dimensional, unobserved, latent vector Z ∈ R K , with K (cid:3) p , are popular in a large array of applications, and mainly used for predicting a response from correlated features. In contrast, methodology and theory for inference on the regression coefficient β ∈ R K relating Y to Z are scarce, since typically the un-observable factor Z is hard to interpret. Furthermore, the determination of the asymptotic variance of an estimator of β is a long-standing problem, with solutions known only in a few particular cases. To address some of these outstanding questions, we develop inferential tools for β in a class of factor regression models in which the observed features are signed mixtures of the latent factors. The model specifications are both practically desirable, in a large array of applications, render interpretability to the components of Z , and are sufficient for parameter identifiability. Without assuming that the number of latent factors K or the structure of the mixture is known in advance, we construct computationally efficient estimators of β , along with estimators of other important model parameters. We benchmark the rate of convergence of β by first establishing its (cid:3) 2 -norm minimax lower bound, and show that our proposed estimator (cid:2) β is minimax-rate adaptive. Our main contribution is the provision of a unified analysis of the component-wise Gaussian asymptotic distribution of (cid:2) β and, especially, the derivation of a closed form expression of its asymptotic variance, together with consistent variance estimators. The resulting inferential tools can be used when both K and p are independent of the sample size n , and also when both, or either, p and K vary with n , while allowing for p > n . This complements the only asymptotic normality results obtained for a particular case of the model under consideration, in the regime K = O( 1 ) and p → ∞ , but without a variance estimate. As an application, we provide, within our model specifications, a statistical platform for inference in regression on latent cluster centers, thereby increasing the scope of our theoretical results. We benchmark the newly developed methodology on a recently collected data set for the study of the effectiveness of a new SIV vaccine. Our analysis enables the determination of the top latent antibody-centric mechanisms associated with the vaccine response.