Title: Post-selection inference for partially linear high-dimensional single-index models
Authors: Pieter Willems - KU Leuven (Belgium) [presenting]
Gerda Claeskens - KU Leuven (Belgium)
Abstract: A post-model selection estimator and method for inference are introduced for the linear part of a partially linear model $Y=X\alpha+g(Z\gamma)+\varepsilon$. Typically the linear part consists of a fixed and limited number of variables $X=(X_1,\ldots,X_r)$, while the control variables $Z=(Z_1,\ldots,Z_p)$ might consist of a large number of variables, $p$, that is allowed to grow with the sample size $n$ and potentially exceeds $n$. The function $g(\cdot)$ is not specified beforehand and is estimated via B-splines in combination with a $l_1$ regularization. This research is relevant because there is clearly an interest in developing new procedures that provide inference which is valid after model selection, which here takes place via the $l_1$ regularizer. This methodology allows for imperfect variable sections and provides a confidence region that is uniformly valid under certain assumptions. Inferential properties are established for this method by proving that asymptotic multivariate normality holds for the newly introduced estimator in the context of a partially linear single-index model. Simulation studies were conducted in multiple settings and a comparison was made with the methodology introduced by other authors in order to illustrate the empirical properties of the newly introduced estimator.