Model checking for parametric single‐index quantile models

Abstract

In this work, we construct a lack‐of‐fit test for testing parametric single‐index quantile regression models. We apply the kernel smoothing technique for the multivariate nonparametric estimation involved in this task. To avoid the “curse of dimensionality” in multivariate nonparametric estimation and to fully utilize the information contained in the model, we employ a sufficient dimension reduction technique to identify the corresponding dimensionally reduced subspace and then construct our test statistic in this subspace. At different quantile levels, the test statistics given in this paper can quickly detect local alternative hypotheses, which are different from the null hypothesis for small and moderate sample sizes. A new wild bootstrap method is applied to approximate the critical values of the quantile regression model test. The effectiveness of the method is verified by simulation experiments and a real data application.

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

“…Figure 1 shows the changes in the empirical size and power curves of different test methods with the exact ratio . The test statistic from Yuan et al (2020) is denoted as . From Figure 1, we have the following findings:…”

“…For example, under the assumption of random missingness, we can adopt the approach of Section 2.2 in Han et al (2019) and take advantage of the fact that they perform conditional quantile estimation for missing data. Naturally, we can also consider using the method in Yuan et al (2020) after multiple imputations of the missing responses or use the local linear conditional quantile estimation method proposed in Guerre and Sabbah (2012) for nonparametric estimation of the conditional quantiles. However, these approaches are not the subject of this paper, and different methods and their related properties must be further explored.…”

“… also has similar conclusions. Yuan et al (2020) conducted a formal study on this property in fully observed samples.…”

“…We aim to incorporate the advantages of the test statistic in Yuan et al (2020) into the proposed test statistics. Unlike the method used in Yuan et al (2020), to make our test statistics adaptable to different quantile regression models, we address problems caused by missing response variables. First, we construct the estimated of the basis matrix of the central quantile subspace under the given missing mechanism and then construct the test statistics based on the and missing mechanism.…”

“…Under this assumption, our observed sample is , where is an observational indicator function of , satisfying (2). We aim to incorporate the advantages of the test statistic in Yuan et al (2020) into the proposed test statistics. Unlike the method used in Yuan et al (2020), to make our test statistics adaptable to different quantile regression models, we address problems caused by missing response variables.…”