A New Regression Model: Modal Linear Regression

Abstract

The mode of a distribution provides an important summary of data and is often estimated based on some non-parametric kernel density estimator. This article develops a new data analysis tool called modal linear regression in order to explore highdimensional data. Modal linear regression models the conditional mode of a response Y given a set of predictors x as a linear function of x. Modal linear regression differs from standard linear regression in that standard linear regression models the conditional mean (as opposed to mode) of Y as a linear function of x. We propose an Expectation-Maximization algorithm in order to estimate the regression coefficients of modal linear regression. We also provide asymptotic properties for the proposed estimator without the symmetric assumption of the error density. Our empirical studies with simulated data and real data demonstrate that the proposed modal regression gives shorter predictive intervals than mean linear regression, median linear regression, and MM-estimators.

A robust and efficient estimation and variable selection method for partially linear single-index models

“…[24,25,31,15] systematically studied the modal regression for the linear model, univariate nonparametric regression model, SPLVCM and single-index model, respectively. In this paper, we devote to extending modal regression to PLSIM and studying the variable selection for the parametric components to achieve robust and efficient sparse estimators.…”

Robust estimation for partially linear single-index model

“…For the linear regression model y i = x T i β + ε i , we can obtain estimator of the regression parameter β by maximizing (Yao and Li, 2014)…”

Robust variable selection for nonlinear models with diverging number of parameters

“…Throughout this paper, we will assume that φ(t) is the standard normal density (for the simplicity of computation). Thus, based on the idea in Yao and Li (2013), the robust modal estimator β n of model (1.2) is to maximise…”

“…For the linear model y i = x T i β + ε i , Yao and Li (2013) proposed to estimate the modal regression parameter β by maximising…”

“…More recently, Yao and Li (2013) proposed a new regression model called modal linear regression (MODLR) that assumes that the mode of f (y|x) is a linear function of the predictor x. A distinguishing characteristic of this method is that it introduces an additional tuning parameter which is automatically selected using the observed data to achieve both robustness and efficiency of the resulting estimate.…”