On Test of Independence Between a Nonparametric Covariate and the Error under MAR: IPW,AIPW and Spacing-Aware Extensions

Abstract

This article extends prior work on testing independence between a nonparametric covariate and the regression error in a semiparametric partially linear model with responses missing at random (MAR). We incorporate two additional imputation/estimation strategies: inverse probability weighting (IPW) and augmented inverse probability weighting (AIPW). For each method we propose a spacing-aware variant that leverages local spacings of the nonparametric covariate $X$ when forming weights or imputations. We describe the resulting estimators for the parametric component $\beta$ and the nonparametric function $m(\cdot )$ , adapting the third-order difference proxy for errors. We show how the U-statistic based tests (Kendall’s $\tau$ , Bergsma–Dassios’ ${\tau }^{*}$ , and Blum–Kiefer–Rosenblatt’s (BKR) $D$ ) may be implemented with these imputations. A comprehensive simulation design is provided for comparative power analysis by the estimators via Nadaraya–Watson (NW) estimator, integrated local linear smoothing (ILLS), IPW, AIPW and their spacing-aware counterparts. Implementation details and bootstrap calibration for degenerate U-statistics are also furnished.