Abstract: We propose to estimate a parametric regression with truncated data built on the mode value, where the dependent variable is subject to left truncation by another random variable. We construct a kernel mode-based objective function with a constant bandwidth for estimation and suggest a modified mode expectation-maximization algorithm to numerically estimate the model. The asymptotic normal distribution of the proposed estimator is derived under mild conditions. To efficiently construct confidence intervals for the resulting estimator, we develop a mode-based empirical likelihood method, where the asymptotic distribution of the empirical log-likelihood ratio is shown to follow a chi-square distribution. Furthermore, by combining the kernel mode-based objective function with the SCAD penalty, a variable selection procedure for the parameters is introduced and its oracle property is established. Monte Carlo simulations and real data analysis related to housing market are presented to show the finite sample performance of the developed estimation and variable selection procedures.

Key words and phrases: Empirical likelihood, mode-based regression, random truncation, robust estimation, variable selection.