Related papers: Sparsity oracle inequalities for the Lasso
In this manuscript, we study quantile regression in partial functional linear model where response is scalar and predictors include both scalars and multiple functions. Wavelet basis are adopted to better approximate functional slopes while…
Variance estimation in the linear model when $p > n$ is a difficult problem. Standard least squares estimation techniques do not apply. Several variance estimators have been proposed in the literature, all with accompanying asymptotic…
We point out some pitfalls related to the concept of an oracle property as used in Fan and Li (2001, 2002, 2004) which are reminiscent of the well-known pitfalls related to Hodges' estimator. The oracle property is often a consequence of…
In this paper we present new theoretical results for the Dantzig and Lasso estimators of the drift in a high dimensional Ornstein-Uhlenbeck model under sparsity constraints. Our focus is on oracle inequalities for both estimators and error…
Partial least squares (PLS) regression combines dimensionality reduction and prediction using a latent variable model. Since partial least squares regression (PLS-R) does not require matrix inversion or diagonalization, it can be applied to…
The least trimmed squares (LTS) estimator is a renowned robust alternative to the classic least squares estimator and is popular in location, regression, machine learning, and AI literature. Many studies exist on LTS, including its…
We consider a linear regression problem in a high dimensional setting where the number of covariates $p$ can be much larger than the sample size $n$. In such a situation, one often assumes sparsity of the regression vector, \textit i.e.,…
We discuss the fundamental issue of identification in linear instrumental variable (IV) models with unknown IV validity. With the assumption of the "sparsest rule", which is equivalent to the plurality rule but becomes operational in…
Censored data are quite common in statistics and have been studied in depth in the last years. In this paper we consider censored high-dimensional data. High-dimensional models are in some way more complex than their low-dimensional…
We construct an objective function that consists of a quadratic approximation term and a penalty term. Thanks to the quadratic approximation, we can deal with various kinds of loss functions into a unified way, and by taking advantage of…
We consider the optimization of a quadratic objective function whose gradients are only accessible through a stochastic oracle that returns the gradient at any given point plus a zero-mean finite variance random error. We present the first…
Solving linear regression problems based on the total least-squares (TLS) criterion has well-documented merits in various applications, where perturbations appear both in the data vector as well as in the regression matrix. However,…
We consider least squares estimation in a general nonparametric regression model. The rate of convergence of the least squares estimator (LSE) for the unknown regression function is well studied when the errors are sub-Gaussian. We find…
The paper considers a linear regression model in high-dimension for which the predictive variables can change the influence on the response variable at unknown times (called change-points). Moreover, the particular case of the heavy-tailed…
The problem of prediction in functional linear regression is conventionally addressed by reducing dimension via the standard principal component basis. In this paper we show that an alternative basis chosen through weighted least-squares,…
We consider the problem of automatic variable selection in a linear model with asymmetric or heavy-tailed errors when the number of explanatory variables diverges with the sample size. For this high-dimensional model, the penalized least…
In a recent work (arXiv:0910.2517), for nonlinear models with sparse underlying linear structures, we studied the error bounds of $\ell_0$-regularized estimation. In this note, we show that $\ell_1$-regularized estimation in some important…
We consider a regression framework where the design points are deterministic and the errors possibly non-i.i.d. and heavy-tailed (with a moment of order $p$ in $[1,2]$). Given a class of candidate regression functions, we propose a…
Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the…
Misspecified models often provide useful information about the true data generating distribution. For example, if $y$ is a non-linear function of $x$ the least squares estimator $\hat{\beta}$ is an estimate of $\beta$, the slope of the best…