We study the distribution of the adaptive LASSO estimator (Zou (2006)) in finite samples as well as in the large-sample limit. The large-sample distributions are derived both for the case where the adaptive LASSO estimator is tuned to perform conservative model selection as well as for the case where the tuning results in consistent model selection. We show that the finite-sample as well as the large-sample distributions are typically highly non-normal, regardless of the choice of the tuning parameter. The uniform convergence rate is also obtained, and is shown to be slower than $n^{-1/2}$ in case the estimator is tuned to perform consistent model selection. In particular, these results question the statistical relevance of the `oracle' property of the adaptive LASSO estimator established in Zou (2006). Moreover, we also provide an impossibility result regarding the estimation of the distribution function of the adaptive LASSO estimator.The theoretical results, which are obtained for a regression model with orthogonal design, are complemented by a Monte Carlo study using non-orthogonal regressors.
In the context of multiple regression model, suppose that the vector parameter of interest βis subjected to lie in the subspace hypothesis Hβ= h, where this restriction is based on either additional information or prior knowledge. Then, the restricted estimator performs fairly well than the ordinary least squares one. In addition, when the number of variables is relatively large with respect to observations, the use of least absolute shrinkage and selection operator (LASSO) estimator is suggested for variable selection purposes. In this paper, we deffine a restricted LASSO estimator and config…
The paper considers a linear regression model with multiple change-points occurring at unknown times. The LASSO technique is very interesting since it allows the parametric estimation, including the change-points, and automatic variable selection simultaneously. The asymptotic properties of the LASSO-type (which has as particular case the LASSO estimator) and of the adaptive LASSO estimators are studied. For this last estimator the oracle properties are proved. In both cases, a model selection criterion is proposed. Numerical examples are provided showing the performances of the adaptive LASSO…
Inverse probability weighted estimators are the oldest and potentially most commonly used class of procedures for the estimation of causal effects. By adjusting for selection biases via a weighting mechanism, these procedures estimate an effect of interest by constructing a pseudo-population in which selection biases are eliminated. Despite their ease of use, these estimators require the correct specification of a model for the weighting mechanism, are known to be inefficient, and suffer from the curse of dimensionality. We propose a class of nonparametric inverse probability weighted estimato…
The least absolute shrinkage and selection operator (LASSO) is a popular technique for simultaneous estimation and model selection. There have been a lot of studies on the large sample asymptotic distributional properties of the LASSO estimator, but it is also well-known that the asymptotic results can give a wrong picture of the LASSO estimator's actual finite-sample behavior. The finite sample distribution of the LASSO estimator has been previously studied for the special case of orthogonal models. The aim in this work is to generalize the finite sample distribution properties of LASSO estim…
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 bounds with respect to several norms. In the context of the Lasso estimator our paper is strongly related to [11], who investigated the same problem under row sparsity. We improve their rates and also prove the restricted eigenvalue property solely under ergodicity assumption on the model. Finally, we demonstrate a numerical analysis to uncover the finite sa…
We derive expressions for the finite-sample distribution of the Lasso estimator in the context of a linear regression model in low as well as in high dimensions by exploiting the structure of the optimization problem defining the estimator. In low dimensions, we assume full rank of the regressor matrix and present expressions for the cumulative distribution function as well as the densities of the absolutely continuous parts of the estimator. Our results are presented for the case of normally distributed errors, but do not hinge on this assumption and can easily be generalized. Additionally, w…
In a linear regression model of fixed dimension $p \leq n$, we construct confidence regions for the unknown parameter vector based on the Lasso estimator that uniformly and exactly hold the prescribed in finite samples as well as in an asymptotic setup. We thereby quantify estimation uncertainty as well as the "post-model selection error" of this estimator. More concretely, in finite samples with Gaussian errors and asymptotically in the case where the Lasso estimator is tuned to perform conservative model selection, we derive exact formulas for computing the minimal coverage probability over …
The precision matrix that encodes conditional linear dependency relations among a set of variables forms an important object of interest in multivariate analysis. Sparse estimation procedures for precision matrices such as the graphical lasso (Glasso) gained popularity as they facilitate interpretability, thereby separating pairs of variables that are conditionally dependent from those that are independent (given all other variables). Glasso lacks, however, robustness to outliers. To overcome this problem, one typically applies a robust plug-in procedure where the Glasso is computed from a rob…
We define the group-lasso estimator for the natural parameters of the exponential families of distributions representing hierarchical log-linear models under multinomial sampling scheme. Such estimator arises as the solution of a convex penalized likelihood optimization problem based on the group-lasso penalty. We illustrate how it is possible to construct an estimator of the underlying log-linear model using the blocks of nonzero coefficients recovered by the group-lasso procedure. We investigate the asymptotic properties of the group-lasso estimator as a model selection method in a double-as…
We consider the linear regression problem, where the number $p$ of covariates is possibly larger than the number $n$ of observations $(x_{i},y_{i})_{i\leq i \leq n}$, under sparsity assumptions. On the one hand, several methods have been successfully proposed to perform this task, for example the LASSO or the Dantzig Selector. On the other hand, consider new values $(x_{i})_{n+1\leq i \leq m}$. If one wants to estimate the corresponding $y_{i}$'s, one should think of a specific estimator devoted to this task, referred by Vapnik as a "transductive" estimator. This estimator may differ from an e…
In this paper we study the properties of the Lasso estimator of the drift component in the diffusion setting. More specifically, we consider a multivariate parametric diffusion model $X$ observed continuously over the interval $[0,T]$ and investigate drift estimation under sparsity constraints. We allow the dimensions of the model and the parameter space to be large. We obtain an oracle inequality for the Lasso estimator and derive an error bound for the $L^2$-distance using concentration inequalities for linear functionals of diffusion processes. The probabilistic part is based upon elements …
We investigate the problem of estimating the drift parameter of a high-dimensional Lévy-driven Ornstein--Uhlenbeck process under sparsity constraints. It is shown that both Lasso and Slope estimators achieve the minimax optimal rate of convergence (up to numerical constants), for tuning parameters chosen independently of the confidence level, which improves the previously obtained results for standard Ornstein--Uhlenbeck processes. The results are nonasymptotic and hold both in probability and conditional expectation with respect to an event resembling the restricted eigenvalue condition.
This paper considers a multiple regression model and compares, under full model hypothesis, analytically as well as by simulation, the performance characteristics of some popular penalty estimators such as ridge regression, LASSO, adaptive LASSO, SCAD, and elastic net versus Least Squares Estimator, restricted estimator, preliminary test estimator, and Stein-type estimators when the dimension of the parameter space is smaller than the sample space dimension. We find that RR uniformly dominates LSE, RE, PTE, SE and PRSE while LASSO, aLASSO, SCAD, and EN uniformly dominates LSE only. Further, it…
This paper proposes a theory for $\ell_1$-norm penalized high-dimensional $M$-estimators, with nonconvex risk and unrestricted domain. Under high-level conditions, the estimators are shown to attain the rate of convergence $s_0\sqrt{\log(nd)/n}$, where $s_0$ is the number of nonzero coefficients of the parameter of interest. Sufficient conditions for our main assumptions are then developed and finally used in several examples including robust linear regression, binary classification and nonlinear least squares.
We derive the $l_{\infty}$ convergence rate simultaneously for Lasso and Dantzig estimators in a high-dimensional linear regression model under a mutual coherence assumption on the Gram matrix of the design and two different assumptions on the noise: Gaussian noise and general noise with finite variance. Then we prove that simultaneously the thresholded Lasso and Dantzig estimators with a proper choice of the threshold enjoy a sign concentration property provided that the non-zero components of the target vector are not too small.
The paper considers a linear model with grouped explanatory variables. If the model errors are not with zero mean and bounded variance or if model contains outliers, then the least squares framework is not appropriate. Thus, the quantile regression is an interesting alternative. In order to automatically select the relevant variable groups, we propose and study here the adaptive group LASSO quantile estimator. We establish the sparsity and asymptotic normality of the proposed estimator in two cases: fixed number and divergent number of variable groups. Numerical study by Monte Carlo simulation…
The LASSO is a variable subset selection procedure in statistical linear regression based on $\ell_1$ penalization of the least-squares operator. Uniqueness of the LASSO is an important issue, especially for the study of the LASSO path. The goal of the present paper is to provide a generic sufficient condition on the design matrix for the LASSO minimizer to be unique. Unlike previous works on the question of uniqueness, our condition only depends on the design matrix. Our study is based on a general position condition on the design matrix which holds with probability one for most experimental …
A new estimator, named S-LASSO, is proposed for the coefficient function of the Function-on-Function linear regression model. The S-LASSO estimator is shown to be able to increase the interpretability of the model, by better locating regions where the coefficient function is zero, and to smoothly estimate non-zero values of the coefficient function. The sparsity of the estimator is ensured by a \textit{functional LASSO penalty}, which pointwise shrinks toward zero the coefficient function, while the smoothness is provided by two roughness penalties that penalize the curvature of the final esti…
We show that the two-stage adaptive Lasso procedure (Zou, 2006) is consistent for high-dimensional model selection in linear and Gaussian graphical models. Our conditions for consistency cover more general situations than those accomplished in previous work: we prove that restricted eigenvalue conditions (Bickel et al., 2008) are also sufficient for sparse structure estimation.
In this article we study post-model selection estimators that apply ordinary least squares (OLS) to the model selected by first-step penalized estimators, typically Lasso. It is well known that Lasso can estimate the nonparametric regression function at nearly the oracle rate, and is thus hard to improve upon. We show that the OLS post-Lasso estimator performs at least as well as Lasso in terms of the rate of convergence, and has the advantage of a smaller bias. Remarkably, this performance occurs even if the Lasso-based model selection "fails" in the sense of missing some components of the "t…
High dimensional Poisson regression has become a standard framework for the analysis of massive counts datasets. In this work we estimate the intensity function of the Poisson regression model by using a dictionary approach, which generalizes the classical basis approach, combined with a Lasso or a group-Lasso procedure. Selection depends on penalty weights that need to be calibrated. Standard methodologies developed in the Gaussian framework can not be directly applied to Poisson models due to heteroscedasticity. Here we provide data-driven weights for the Lasso and the group-Lasso derived fr…
The Lasso has attracted the attention of many authors these last years. While many efforts have been made to prove that the Lasso behaves like a variable selection procedure at the price of strong (though unavoidable) assumptions on the geometric structure of these variables, much less attention has been paid to the analysis of the performance of the Lasso as a regularization algorithm. Our first purpose here is to provide a conceptually very simple result in this direction. We shall prove that, provided that the regularization parameter is properly chosen, the Lasso works almost as well as th…
The purpose of the present paper is to investigate on a class of spherical functional autoregressive processes in order to introduce and study LASSO (Least Absolute Shrinkage and Selection Operator) type estimators for the corresponding autoregressive kernels, defined in the harmonic domain by means of their spectral decompositions. Some crucial properties for these estimators are proved, in particular, consistency and oracle inequalities.
Recent results have proven the minimax optimality of LASSO and related algorithms for noisy linear regression. However, these results tend to rely on variance estimators that are inefficient or optimizations that are slower than LASSO itself. We propose an efficient estimator for the noise variance in high dimensional linear regression that is faster than LASSO, only requiring $p$ matrix-vector multiplications. We prove this estimator is consistent with a good rate of convergence, under the condition that the design matrix satisfies the Restricted Isometry Property (RIP). In practice, our esti…
In this paper, we derive non-asymptotic error bounds for the Lasso estimator when the penalty parameter for the estimator is chosen using $K$-fold cross-validation. Our bounds imply that the cross-validated Lasso estimator has nearly optimal rates of convergence in the prediction, $L^2$, and $L^1$ norms. For example, we show that in the model with the Gaussian noise and under fairly general assumptions on the candidate set of values of the penalty parameter, the estimation error of the cross-validated Lasso estimator converges to zero in the prediction norm with the $\sqrt{s\log p / n}\times \…
We propose an improved LASSO estimation technique based on Stein-rule. We shrink classical LASSO estimator using preliminary test, shrinkage, and positive-rule shrinkage principle. Simulation results have been carried out for various configurations of correlation coefficients ($r$), size of the parameter vector ($β$), error variance ($σ^2$) and number of non-zero coefficients ($k$) in the model parameter vector. Several real data examples have been used to demonstrate the practical usefulness of the proposed estimators. Our study shows that the risk ordering given by LSE $>$ LASSO $>$ Stein-ty…
In linear regression with fixed design, we propose two procedures that aggregate a data-driven collection of supports. The collection is a subset of the $2^p$ possible supports and both its cardinality and its elements can depend on the data. The procedures satisfy oracle inequalities with no assumption on the design matrix. Then we use these procedures to aggregate the supports that appear on the regularization path of the Lasso in order to construct an estimator that mimics the best Lasso estimator. If the restricted eigenvalue condition on the design matrix is satisfied, then this estimator…
A representation of the degrees of freedom akin to Stein's lemma is given for a class of estimators of a mean value parameter in $\mathbb{R}^n$. Contrary to previous results our representation holds for a range of discontinues estimators. It shows that even though the discontinuities form a Lebesgue null set, they cannot be ignored when computing degrees of freedom. Estimators with discontinuities arise naturally in regression if data driven variable selection is used. Two such examples, namely best subset selection and lasso-OLS, are considered in detail in this paper. For lasso-OLS the gener…
In this paper we introduce and study fused lasso nearly-isotonic signal approximation, which is a combination of fused lasso and generalized nearly-isotonic regression. We show how these three estimators relate to each other, derive solution to the general problem, show that it is computationally feasible and provides a trade-off between piecewise monotonicity, sparsity and goodness-of-fit. Also, we derive an unbiased estimator of the degrees of freedom of the approximator.
The LASSO estimator is an $\ell_1$-norm penalized least-squares estimator, which was introduced for variable selection in the linear model. When the design matrix satisfies, e.g. the Restricted Isometry Property, or has a small coherence index, the LASSO estimator has been proved to recover, with high probability, the support and sign pattern of sufficiently sparse regression vectors. Under similar assumptions, the LASSO satisfies adaptive prediction bounds in various norms. The present note provides a prediction bound based on a new index for measuring how favorable is a design matrix for the…
We address the issue of estimating the regression vector $β$ in the generic $s$-sparse linear model $y = Xβ+z$, with $β\in\R^{p}$, $y\in\R^{n}$, $z\sim\mathcal N(0,\sg^2 I)$ and $p> n$ when the variance $\sg^{2}$ is unknown. We study two LASSO-type methods that jointly estimate $β$ and the variance. These estimators are minimizers of the $\ell_1$ penalized least-squares functional, where the relaxation parameter is tuned according to two different strategies. In the first strategy, the relaxation parameter is of the order $\chσ \sqrt{\log p}$, where $\chσ^2$ is the empirical variance. %The res…
Many problems in signal processing require finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. When dealing with real-world data, the presence of outliers and impulsive noise must also be accounted for. In past decades, the vast majority of robust linear regression estimators has focused on robustness against rowwise contamination. Even so called `high breakdown' estimators rely on the assumption that a majority of rows of the regression matrix is not affected by outliers. Only very recently, the first cellwise robust regression estimation methods hav…
We study a group lasso estimator for the multivariate linear regression model that accounts for correlated error terms. A block coordinate descent algorithm is used to compute this estimator. We perform a simulation study with categorical data and multivariate time series data, typical settings with a natural grouping among the predictor variables. Our simulation studies show the good performance of the proposed group lasso estimator compared to alternative estimators. We illustrate the method on a time series data set of gene expressions.
The Lasso is a popular model selection and estimation procedure for linear models that enjoys nice theoretical properties. In this paper, we study the Lasso estimator for fitting autoregressive time series models. We adopt a double asymptotic framework where the maximal lag may increase with the sample size. We derive theoretical results establishing various types of consistency. In particular, we derive conditions under which the Lasso estimator for the autoregressive coefficients is model selection consistent, estimation consistent and prediction consistent. Simulation study results are repo…
We revisit the adaptive Lasso as well as the thresholded Lasso with refitting, in a high-dimensional linear model, and study prediction error, $\ell_q$-error ($q \in \{1, 2 \} $), and number of false positive selections. Our theoretical results for the two methods are, at a rather fine scale, comparable. The differences only show up in terms of the (minimal) restricted and sparse eigenvalues, favoring thresholding over the adaptive Lasso. As regards prediction and estimation, the difference is virtually negligible, but our bound for the number of false positives is larger for the adaptive Lass…
When a series of (related) linear models has to be estimated it is often appropriate to combine the different data-sets to construct more efficient estimators. We use $\ell_1$-penalized estimators like the Lasso or the Adaptive Lasso which can simultaneously do parameter estimation and model selection. We show that for a time-course of high-dimensional linear models the convergence rates of the Lasso and of the Adaptive Lasso can be improved by combining the different time-points in a suitable way. Moreover, the Adaptive Lasso still enjoys oracle properties and consistent variable selection. T…
The least absolute shrinkage and selection operator (LASSO) of Tibshirani (1996) is a prominent estimator which selects significant (under some sense) features and kills insignificant ones. Indeed the LASSO shrinks features lager than a noise level to zero. In this paper, we force LASSO to be shrunken more by proposing a Stein-type shrinkage estimator emanating from the LASSO, namely the Stein-type LASSO. The newly proposed estimator proposes good performance in risk sense numerically. Variants of this estimator have smaller relative MSE and prediction error, compared to the LASSO, in the anal…
We study the property of the Fused Lasso Signal Approximator (FLSA) for estimating a blocky signal sequence with additive noise. We transform the FLSA to an ordinary Lasso problem. By studying the property of the design matrix in the transformed Lasso problem, we find that the irrepresentable condition might not hold, in which case we show that the FLSA might not be able to recover the signal pattern. We then apply the newly developed preconditioning method -- Puffer Transformation [Jia and Rohe, 2012] on the transformed Lasso problem. We call the new method the preconditioned fused Lasso and …
This paper considers the penalized least squares estimator with arbitrary convex penalty. When the observation noise is Gaussian, we show that the prediction error is a subgaussian random variable concentrated around its median. We apply this concentration property to derive sharp oracle inequalities for the prediction error of the LASSO, the group LASSO and the SLOPE estimators, both in probability and in expectation. In contrast to the previous work on the LASSO type methods, our oracle inequalities in probability are obtained at any confidence level for estimators with tuning parameters tha…
Consider the case that we observe $n$ independent and identically distributed copies of a random variable with a probability distribution known to be an element of a specified statistical model. We are interested in estimating an infinite dimensional target parameter that minimizes the expectation of a specified loss function. In \cite{generally_efficient_TMLE} we defined an estimator that minimizes the empirical risk over all multivariate real valued cadlag functions with variation norm bounded by some constant $M$ in the parameter space, and selects $M$ with cross-validation. We referred to …
We consider a general high-dimensional additive hazard model in a non-asymptotic setting, including regression for censored-data. In this context, we consider a Lasso estimator with a fully data-driven $\ell_1$ penalization, which is tuned for the estimation problem at hand. We prove sharp oracle inequalities for this estimator. Our analysis involves a new "data-driven" Bernstein's inequality, that is of independent interest, where the predictable variation is replaced by the optional variation.
We consider the linear regression problem. We propose the S-Lasso procedure to estimate the unknown regression parameters. This estimator enjoys sparsity of the representation while taking into account correlation between successive covariates (or predictors). The study covers the case when $p\gg n$, i.e. the number of covariates is much larger than the number of observations. In the theoretical point of view, for fixed $p$, we establish asymptotic normality and consistency in variable selection results for our procedure. When $p\geq n$, we provide variable selection consistency results and sh…
A well-know drawback of l_1-penalized estimators is the systematic shrinkage of the large coefficients towards zero. A simple remedy is to treat Lasso as a model-selection procedure and to perform a second refitting step on the selected support. In this work we formalize the notion of refitting and provide oracle bounds for arbitrary refitting procedures of the Lasso solution. One of the most widely used refitting techniques which is based on Least-Squares may bring a problem of interpretability, since the signs of the refitted estimator might be flipped with respect to the original estimator.…
We investigate properties of estimators obtained by minimization of U-processes with the Lasso penalty in high-dimensional settings. Our attention is focused on the ranking problem that is popular in machine learning. It is related to guessing the ordering between objects on the basis of their observed predictors. We prove the oracle inequality for the excess risk of the considered estimator as well as the bound for the l1 distance between the estimator and the oracle.
The abundance of high-dimensional data in the modern sciences has generated tremendous interest in penalized estimators such as the lasso, scaled lasso, square-root lasso, elastic net, and many others. In this paper, we establish a general oracle inequality for prediction in high-dimensional linear regression with such methods. Since the proof relies only on convexity and continuity arguments, the result holds irrespective of the design matrix and applies to a wide range of penalized estimators. Overall, the bound demonstrates that generic estimators can provide consistent prediction with any …
The Adaptive Lasso(Alasso) was proposed by Zou [\textit{J. Amer. Statist. Assoc. \textbf{101} (2006) 1418-1429}] as a modification of the Lasso for the purpose of simultaneous variable selection and estimation of the parameters in a linear regression model. Zou (2006) established that the Alasso estimator is variable-selection consistent as well as asymptotically Normal in the indices corresponding to the nonzero regression coefficients in certain fixed-dimensional settings. In an influential paper, Minnier, Tian and Cai [\textit{J. Amer. Statist. Assoc. \textbf{106} (2011) 1371-1382}] propose…
We consider statistical inference for a single coordinate of regression coefficients in high-dimensional linear models. Recently, the debiased estimators are popularly used for constructing confidence intervals and hypothesis testing in high-dimensional models. However, some representative numerical experiments show that they tend to be biased for large coefficients, especially when the number of large coefficients dominates the number of small coefficients. In this paper, we propose a modified debiased Lasso estimator based on bootstrap. Let us denote the proposed estimator BS-DB for short. W…
Much of the theory for the lasso in the linear model $Y = X β^* + \varepsilon$ hinges on the quantity $2 \| X^\top \varepsilon \|_{\infty} / n$, which we call the lasso's effective noise. Among other things, the effective noise plays an important role in finite-sample bounds for the lasso, the calibration of the lasso's tuning parameter, and inference on the parameter vector $β^*$. In this paper, we develop a bootstrap-based estimator of the quantiles of the effective noise. The estimator is fully data-driven, that is, does not require any additional tuning parameters. We equip our estimator w…
We consider tests of significance in the setting of the graphical lasso for inverse covariance matrix estimation. We propose a simple test statistic based on a subsequence of the knots in the graphical lasso path. We show that this statistic has an exponential asymptotic null distribution, under the null hypothesis that the model contains the true connected components. Though the null distribution is asymptotic, we show through simulation that it provides a close approximation to the true distribution at reasonable sample sizes. Thus the test provides a simple, tractable test for the significa…
