New Estimation and Model Selection Procedures for Semiparametric Modeling in Longitudinal Data Analysis

TLDR

Semiparametric regression models are useful for longitudinal data analysis, but their complexity and the data structure pose new challenges to parametric inference and model selection. The article proposes two new approaches: one for estimating regression coefficients and another innovative class of variable selection procedures for identifying significant variables in semiparametric models. The methods involve estimating regression coefficients and variable selection via penalized least squares, establishing convergence rates, deriving a robust sandwich‑based standard error, and using local polynomial regression to estimate the baseline function. The estimators are asymptotically normal, simultaneously select significant variables and estimate unknown parameters, and with appropriate regularization perform as well as an oracle estimator. Keywords include local polynomial regression, partial linear model, penalized least squares, profile least squares, and smoothly clipped absolute deviation.

Abstract

AbstractSemiparametric regression models are very useful for longitudinal data analysis. The complexity of semiparametric models and the structure of longitudinal data pose new challenges to parametric inferences and model selection that frequently arise from longitudinal data analysis. In this article, two new approaches are proposed for estimating the regression coefficients in a semiparametric model. The asymptotic normality of the resulting estimators is established. An innovative class of variable selection procedures is proposed to select significant variables in the semiparametric models. The proposed procedures are distinguished from others in that they simultaneously select significant variables and estimate unknown parameters. Rates of convergence of the resulting estimators are established. With a proper choice of regularization parameters and penalty functions, the proposed variable selection procedures are shown to perform as well as an oracle estimator. A robust standard error formula is derived using a sandwich formula and is empirically tested. Local polynomial regression techniques are used to estimate the baseline function in the semiparametric model.KEY WORDS : Local polynomial regressionPartial linear modelPenalized least squaresProfile least squaresSmoothly clipped absolute deviation