Bias-Corrected Empirical-Likelihood Inference for Single-Index Expectile Regression Under Missing-at-Random Responses
- 1School of Science, Xi’an Polytechnic University, Xi’an 710048, China
Abstract
We study bias-corrected empirical-likelihood inference for single-index expectile regression when responses are missing at random. Local-linear expectile estimation is combined with conditional centering to construct three procedures based on complete cases, inverse-probability weighting, and cross-fitted imputation. The expectile score is continuous and globally Lipschitz, which permits the bias-correction argument without further smoothing of the score. Under the stated conditions, the index estimators are asymptotically normal and each empirical-likelihood ratio has a standard chi-squared limit. Joint confidence regions can therefore be formed without estimating the limiting covariance matrix. For the imputation construction considered here, I-BCEL and CC-BCEL have the same first-order limiting covariance. In the simulations, the estimation error decreases with sample size; differences among the incomplete-data methods are more apparent when the missing-response rate is 40%, where small-sample undercoverage also remains. An analysis of the New York air-quality data illustrates the three procedures with naturally missing responses.
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BibTeX
@article{liu2026biascorrected,
title = {{Bias-Corrected Empirical-Likelihood Inference for Single-Index Expectile Regression Under Missing-at-Random Responses}},
author = {Pingping Liu and Shuanghua Luo and Junxiang Lu},
journal = {Axioms},
year = {2026},
month = nov,
doi = {10.3390/axioms15110816},
url = {https://www.mdpi.com/2075-1680/15/11/816},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Bias-Corrected Empirical-Likelihood Inference for Single-Index Expectile Regression Under Missing-at-Random Responses
AU - Pingping Liu
AU - Shuanghua Luo
AU - Junxiang Lu
PY - 2026
DA - 2026/11/02
JO - Axioms
DO - 10.3390/axioms15110816
UR - https://www.mdpi.com/2075-1680/15/11/816
AB - We study bias-corrected empirical-likelihood inference for single-index expectile regression when responses are missing at random. Local-linear expectile estimation is combined with conditional centering to construct three procedures based on complete cases, inverse-probability weighting, and cross-fitted imputation. The expectile score is continuous and globally Lipschitz, which permits the bias-correction argument without further smoothing of the score. Under the stated conditions, the index estimators are asymptotically normal and each empirical-likelihood ratio has a standard chi-squared limit. Joint confidence regions can therefore be formed without estimating the limiting covariance matrix. For the imputation construction considered here, I-BCEL and CC-BCEL have the same first-order limiting covariance. In the simulations, the estimation error decreases with sample size; differences among the incomplete-data methods are more apparent when the missing-response rate is 40%, where small-sample undercoverage also remains. An analysis of the New York air-quality data illustrates the three procedures with naturally missing responses.
SN - 2075-1680
ER -