We analyze the (unconditional) distribution of a linear predictor that is constructed after a data-driven model selection step in a linear regression model. First, we derive the exact finite-sample cumulative distribution function (cdf) of the linear predictor, and a simple approximation to this (complicated) cdf. We then analyze the large-sample limit behavior of these cdfs, in the fixed-parameter case and under local alternatives.
In this paper, we propose methods for functional predictor selection and the estimation of smooth functional coefficients simultaneously in a scalar-on-function regression problem under high-dimensional multivariate functional data setting. In particular, we develop two methods for functional group-sparse regression under a generic Hilbert space of infinite dimension. We show the convergence of algorithms and the consistency of the estimation and the selection (oracle property) under infinite-dimensional Hilbert spaces. Simulation studies show the effectiveness of the methods in both the selec…
Institute of Computer Science, Polish Academy of Sciences（波兰科学院计算机科学研究所）；Warsaw University of Technology（华沙理工大学）
We consider selection of random predictors for high-dimensional regression problem with binary response for a general loss function. Important special case is when the binary model is semiparametric and the response function is misspecified under parametric model fit. Selection for such a scenario aims at recovering the support of the minimizer of the associated risk with large probability. We propose a two-step selection procedure which consists of screening and ordering predictors by Lasso method and then selecting a subset of predictors which minimizes Generalized Information Criterion on t…
We consider the joint sparse estimation of regression coefficients and the covariance matrix for covariates in a high-dimensional regression model, where the predictors are both relevant to a response variable of interest and functionally related to one another via a Gaussian directed acyclic graph (DAG) model. Gaussian DAG models introduce sparsity in the Cholesky factor of the inverse covariance matrix, and the sparsity pattern in turn corresponds to specific conditional independence assumptions on the underlying predictors. A variety of methods have been developed in recent years for Bayesi…
Variable selection methods have been developed in linear regression to provide sparse solutions. Recent studies have focused on further interpretations on the sparse solutions in terms of false positive control. In this paper, we consider false negative control for variable selection with the goal to efficiently select a high proportion of relevant predictors. Different from existing studies in power analysis and sure screening, we propose to directly estimate the false negative proportion (FNP) of a decision rule and select the smallest subset of predictors that has the estimated FNP less tha…
This paper develops a new approach to post-selection inference for screening high-dimensional predictors of survival outcomes. Post-selection inference for right-censored outcome data has been investigated in the literature, but much remains to be done to make the methods both reliable and computationally-scalable in high-dimensions. Machine learning tools are commonly used to provide {\it predictions} of survival outcomes, but the estimated effect of a selected predictor suffers from confirmation bias unless the selection is taken into account. The new approach involves construction of semi-p…
Traditional variable selection methods could fail to be sign consistent when irrepresentable conditions are violated. This is especially critical in high-dimensional settings when the number of predictors exceeds the sample size. In this paper, we propose a new semi-standard partial covariance (SPAC) approach which is capable of reducing correlation effects from other covariates while fully capturing the magnitude of coefficients. The proposed SPAC is effective in choosing covariates which have direct effects on the response variable, while eliminating the predictors which are not directly ass…
Sparse model selection by structural risk minimization leads to a set of a few predictors, ideally a subset of the true predictors. This selection clearly depends on the underlying loss function $\tilde L$. For linear regression with square loss, the particular (functional) Gradient Boosting variant $L_2-$Boosting excels for its computational efficiency even for very large predictor sets, while still providing suitable estimation consistency. For more general loss functions, functional gradients are not always easily accessible or, like in the case of continuous ranking, need not even exist. T…
