Threshold detection under a semiparametric regression model

Linear regression models have been extensively considered in the literature. However, in some practical applications they may not be appropriate all over the range of the covariate. In this paper, a more flexible model is introduced by considering a regression model $Y=r(X)+\epsilon$ where the regression function $r(\cdot )$ is assumed to be linear for large values in the domain of the predictor variable $X$. More precisely, we assume that $r(x)={\alpha }_0+{\beta }_{0}x$ for $x>u_0$, where the value $u_0$ is identified as the smallest value satisfying such a property. A penalized procedure is introduced to estimate the threshold $u_0$. The considered proposal focusses on a semiparametric approach since no parametric model is assumed for the regression function for values smaller than $u_0$. Consistency properties of both the threshold estimator and the estimators of $({\alpha }_0,{\beta }_0)$ are derived, under mild assumptions. Through a numerical study, the small sample properties of the proposed procedure and the importance of introducing a penalization are investigated. The analysis of a real data set allows us to demonstrate the usefulness of the penalized estimators.