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May 2022 Inference in latent factor regression with clusterable features
Xin Bing, Florentina Bunea, Marten Wegkamp
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Bernoulli 28(2): 997-1020 (May 2022). DOI: 10.3150/21-BEJ1374

Abstract

Regression models, in which the observed features XRp and the response YR depend, jointly, on a lower dimensional, unobserved, latent vector ZRK, with Kp, 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 βRK 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 2-norm minimax lower bound, and show that our proposed estimator βˆ is minimax-rate adaptive. Our main contribution is the provision of a unified analysis of the component-wise Gaussian asymptotic distribution of βˆ 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.

Funding Statement

Bunea and Wegkamp are supported in part by NSF grants DMS-1712709 and DMS-2015195.
Bing is supported in part by NSF grant DMS-1407600.

Acknowledgements

We thank the two anonymous referees for their many insightful and helpful suggestions. We are grateful to Jishnu Das for help with the interpretation of our data analysis results.

Citation

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Xin Bing. Florentina Bunea. Marten Wegkamp. "Inference in latent factor regression with clusterable features." Bernoulli 28 (2) 997 - 1020, May 2022. https://doi.org/10.3150/21-BEJ1374

Information

Received: 1 April 2020; Revised: 1 March 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

Digital Object Identifier: 10.3150/21-BEJ1374

Keywords: adaptive estimation , high dimensional regression , Identification , latent factor model , minimax estimation , post clustering inference/regression , pure variables , uniform inference

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 2 • May 2022
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