Abstract
Missing responses is a common type of data where the interested outcomes are not always observed. In this paper, we develop two new kernel machines to handle such a case, which can be used for both regression and classification. The first proposed kernel machine uses $\text{only}$ the complete cases where both response and covariates are observed. It is, however, subject to some assumption limitations. Our second proposed doubly-robust kernel machine overcomes such limitations regardless of the misspecification of either the missing mechanism or the conditional distribution of the response. Theoretical properties, including the oracle inequalities for the excess risk, universal consistency, and learning rates are established. We demonstrate the superiority of the proposed methods to some existing methods by simulation and illustrate their application to a real data set concerning a survey about homeless people.
Highlights
We consider the problem of statistical learning in the presence of missing responses
Missing responses occurred in those tracts while having their covariates still available. (More details can be found in Kriegler and Berk (2010).) Another example incurring missing responses concerns a biomedical study where genetic information is collected on all participants, but the level of a biomarker is collected only on a subset of them based on the corresponding genetic information
The proposed inverse-probability complete-case estimator can be applied under any convex loss function
Summary
We consider the problem of statistical learning in the presence of missing responses. For the missing responses problem, various methods have been developed, including the augmented inverse probability weighting (AIPW) methods, semiparametric methods, and kernel machine methods, among others. Wang et al (2004) extended a semiparametric regression analysis method to include missing responses and built a doubly-robust estimator for the population mean. While for the kernel machines approach, the augmented inverse probability weighted loss should ensure the doubly-robust property and guarantee the convexity of the augmented loss function. We first propose a family of kernel machines that use the estimated inverse probabilities of the observed cases to weight the loss function of the complete cases. We call it ‘inverse-weighted-probability complete-case estimator’ (Tsiatis, 2006). An R package called KM4ICD that integrates to the package mlr (Machine Learning in R) for the kernel machine estimators is given in the Supplementary Materials
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