Semiparametric partially linear varying coefficient modal regression

Abstract

We in this paper propose a semiparametric partially linear varying coefficient (SPLVC) modal regression, in which the conditional mode function of the response variable given covariates admits a partially linear varying coefficient structure. In comparison to existing regressions, the newly developed SPLVC modal regression captures the “most likely” effect and provides superior prediction performance when the data distribution is skewed. The consistency and asymptotic properties of the resultant estimators for both parametric and nonparametric parts are rigorously established. We employ a kernel-based objective function to simplify the computation and a modified modal-expectation–maximization (MEM) algorithm to estimate the model numerically. Furthermore, taking the residual sums of modes as the loss function, we construct a goodness-of-fit testing statistic for hypotheses on the coefficient functions, whose limiting null distribution is shown to follow an asymptotically χ2-distribution with a scale dependent on density functions. To achieve sparsity in the high-dimensional SPLVC modal regression, we develop a regularized estimation procedure by imposing a penalty on the coefficients in the parametric part to eliminate the irrelevant variables. Monte Carlo simulations and two real-data applications are conducted to examine the performance of the suggested estimation methods and hypothesis test. We also briefly explore the extension of the SPLVC modal regression to the case where some varying coefficient functions admit higher-order smoothness.