Powerful nonparametric checks for parametric single‐index quantile models with missing responses

This paper considers the lack‐of‐fit test of parametric single‐index quantile models when the response variable is missing at random. The model's coefficients are estimated by an estimation method suitable for the quantile regression coefficients of the missing data. Simultaneously, an algorithm for solving the central subspace of the multidimensional quantile regression model with missing responses is proposed. Based on the central quantile regression subspace, we construct two‐dimensional reduction adaptive‐to‐model test statistics suitable for randomly missing response variables to avoid the curse of dimensionality. Under the null hypothesis and local alternative hypothesis, the asymptotic properties of the test statistics are obtained. The proposed testing methods are shown to be consistent and able to detect local alternative hypothetical models converging to the null model at the rate of order . A consistent bootstrap method is proposed to determine the critical values, and its asymptotic properties are established. The simulation results show that the proposed method is superior to existing methods in terms of both empirical size and power in the case of multidimensional and even high‐dimensional covariables. The ACTG Protocol 175 data set is analyzed to demonstrate the application of the testing procedures. Supplementary materials for this article are available online.