Single-Index Quantile Regression with Left Truncated Data

The purpose of this paper is two fold. First, the authors investigate quantile regression (QR) estimation for single-index QR models when the response is subject to random left truncation. The random weights are introduced to deal with left truncated data and the associated iteration estimation method is proposed. The asymptotic properties for the proposed QR estimates of the index parameter and unknown link function are both obtained. Further, by combining the QR loss function and the adaptive LASSO penalization, a variable selection procedure for the index parameter is introduced and its oracle property is established. Second, a weighted empirical log-likelihood ratio of the index parameter based on the QR method is introduced and is proved to be asymptotic standard chi-square distribution. Furthermore, confidence regions of the index parameter can be constructed. The finite sample performance of the proposed methods are demonstrated. A real data analysis is also conducted to show the usefulness of the proposed approaches.