Bias-Corrected Empirical-Likelihood Inference for Single-Index Expectile Regression Under Missing-at-Random Responses

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.