Grouped Semi-Supervised Estimation of Two-Sample Functionals via Single-Index Conditional Imputation
Abstract
Estimating two-sample comparison functionals within heterogeneous groups is challenging when outcomes are observed only for a small labelled subset while covariates are widely available. We develop semi-supervised estimation of group-specific two-sample comparison functionals and their prespecified aggregates. The proposed estimator combines single-index conditional imputation from labelled cross-arm pairs with averaging over all available covariate pairs. To handle index-score densities that may vanish at attainable endpoints, we use centred stabilization, with a smooth stabilizer in the profile criterion and a shrinking lower truncation of the kernel denominator (the hard floor) in the point estimator. The outer average retains all covariate pairs. Under a correctly specified single-index conditional-mean model and the stated regularity conditions, we establish consistency and asymptotic normality for bounded comparison functionals. A four-role original-observation influence representation accounts for sample overlap and identifies the variance component that unlabelled covariates can reduce relative to supervised estimation. We further establish conditional validity of observation-level multinomial perturbation resampling with complete refitting and shared labelled counts, distinguishing ideal variance consistency from finite-replicate variance estimation. Simulations show small bias and lower mean squared error than supervised estimation across the point-estimation settings, together with near-nominal coverage in the perturbation experiment. A descriptive NHANES 2011-2018 illustration comparing serum creatinine by hypertension history across age groups yields point estimates similar to their supervised counterparts and smaller semi-supervised standard errors for all reported comparisons.
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BibTeX
@misc{xu2026grouped,
title = {{Grouped Semi-Supervised Estimation of Two-Sample Functionals via Single-Index Conditional Imputation}},
author = {Hengyao Xu and Tao Tan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25789},
eprint = {2609.25789},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25789},
}
RIS
TY - UNPB
TI - Grouped Semi-Supervised Estimation of Two-Sample Functionals via Single-Index Conditional Imputation
AU - Hengyao Xu
AU - Tao Tan
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25789
UR - https://arxiv.org/abs/2609.25789
AB - Estimating two-sample comparison functionals within heterogeneous groups is challenging when outcomes are observed only for a small labelled subset while covariates are widely available. We develop semi-supervised estimation of group-specific two-sample comparison functionals and their prespecified aggregates. The proposed estimator combines single-index conditional imputation from labelled cross-arm pairs with averaging over all available covariate pairs. To handle index-score densities that may vanish at attainable endpoints, we use centred stabilization, with a smooth stabilizer in the profile criterion and a shrinking lower truncation of the kernel denominator (the hard floor) in the point estimator. The outer average retains all covariate pairs. Under a correctly specified single-index conditional-mean model and the stated regularity conditions, we establish consistency and asymptotic normality for bounded comparison functionals. A four-role original-observation influence representation accounts for sample overlap and identifies the variance component that unlabelled covariates can reduce relative to supervised estimation. We further establish conditional validity of observation-level multinomial perturbation resampling with complete refitting and shared labelled counts, distinguishing ideal variance consistency from finite-replicate variance estimation. Simulations show small bias and lower mean squared error than supervised estimation across the point-estimation settings, together with near-nominal coverage in the perturbation experiment. A descriptive NHANES 2011-2018 illustration comparing serum creatinine by hypertension history across age groups yields point estimates similar to their supervised counterparts and smaller semi-supervised standard errors for all reported comparisons.
ER -