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65
Nonasymptotic Oracle Inequalities for the Lasso and Group Lasso in high dimensional logistic model
, 2012
"... We consider the problem of estimating a function f0 in logistic regression model. We propose to estimate this function f0 by a sparse approximation build as a linear combination of elements of a given dictionary of p functions. This sparse approximation is selected by the Lasso or Group Lasso proced ..."
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We consider the problem of estimating a function f0 in logistic regression model. We propose to estimate this function f0 by a sparse approximation build as a linear combination of elements of a given dictionary of p functions. This sparse approximation is selected by the Lasso or Group Lasso procedure. In this context, we state non asymptotic oracle inequalities for Lasso and Group Lasso under restricted eigenvalues assumption as introduced in [1]. Those theoretical results are illustrated through a simulation study.
INFERENCE IN HIGH DIMENSIONAL PANEL MODELS WITH AN APPLICATION TO GUN CONTROL
"... Abstract. We consider estimation and inference in panel data models with additive unobserved individual specific heterogeneity in a high dimensional setting. The setting allows the number of time varying regressors to be larger than the sample size. To make informative estimation and inference feas ..."
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Abstract. We consider estimation and inference in panel data models with additive unobserved individual specific heterogeneity in a high dimensional setting. The setting allows the number of time varying regressors to be larger than the sample size. To make informative estimation and inference feasible, we require that the overall contribution of the time varying variables after eliminating the individual specific heterogeneity can be captured by a relatively small number of the available variables whose identities are unknown. This restriction allows the problem of estimation to proceed as a variable selection problem. Importantly, we treat the individual specific heterogeneity as fixed effects which allows this heterogeneity to be related to the observed time varying variables in an unspecified way and allows that this heterogeneity may be nonzero for all individuals. Within this framework, we provide procedures that give uniformly valid inference over a fixed subset of parameters in the canonical linear fixed effects model and over coefficients on a fixed vector of endogenous variables in panel data instrumental variables models with fixed effects and many instruments. An input to developing the properties of our proposed procedures is the use of a variant of the Lasso estimator that allows for a grouped data structure where data across groups are independent and dependence within groups is unrestricted. We provide formal conditions within this structure under which the proposed Lasso variant selects a sparse model with good approximation properties. We present simulation results in support of the theoretical developments and illustrate the use of the methods in an application aimed at estimating the effect of gun prevalence on crime rates. Key Words: panel data, fixed effects, partially linear model, instrumental variables, high dimensionalsparse regression, inference under imperfect model selection, uniformly valid inference after model selection, clustered standard errors 1.
Reducedrank Regression in Sparse Multivariate VaryingCoefficient Models with Highdimensional Covariates
, 2013
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Supplement to “Nonparametric regression with the scale depending on auxiliary variable.” DOI:10.1214/13AOS1126SUPP
, 2013
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Variable Selection in Nonparametric and Semiparametric Regression Models∗
, 2012
"... This chapter reviews the literature on variable selection in nonparametric and semiparametric regression models via shrinkage. We highlight recent developments on simultaneous variable selection and estimation through the methods of least absolute shrinkage and selection operator (Lasso), smoothly ..."
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This chapter reviews the literature on variable selection in nonparametric and semiparametric regression models via shrinkage. We highlight recent developments on simultaneous variable selection and estimation through the methods of least absolute shrinkage and selection operator (Lasso), smoothly clipped absolute deviation (SCAD) or their variants, but restrict our attention to nonparametric and semiparametric regression models. In particular, we consider variable selection in additive models, partially linear models, functional/varying coefficient models, single index models, general nonparametric regression models, and semiparametric/nonparametric quantile regression models.
Flexible Shrinkage Estimation in HighDimensional Varying Coefficient Models.” Working Paper
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Generalized Dynamic Semiparametric Factor Models for High Dimensional Nonstationary Time Series
, 2013
"... High dimensional nonstationary time series, which reveal both complex trends and stochastic behavior, occur in many scientific fields, e.g. macroeconomics, finance and neuroeconomics, etc. To address them, we propose a generalized dynamic semiparametric factor model with a twostep estimation proced ..."
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High dimensional nonstationary time series, which reveal both complex trends and stochastic behavior, occur in many scientific fields, e.g. macroeconomics, finance and neuroeconomics, etc. To address them, we propose a generalized dynamic semiparametric factor model with a twostep estimation procedure. After choosing smoothed functional principal components as space functions (factor loadings), we extract various temporal trends by employing variable selection techniques for the time basis (common factors), and establish its nonasymptotic statistical properties under the dependent scenario (βmixing and mdependent) with the weakly crosscorrelated error term, which is not built upon any specific forms of the time and space basis. At the second step, we obtain a detrended low dimensional stochastic process exhibiting the dynamics of the original high dimensional (stochastic) objects and further justify statistical inference based on it. Crucially required for pricing weather derivatives, an analysis of temperature dynamics in China is presented to illustrate its performance together with a simulation study designed to mimic it.
Variable Selection for Nonparametric Quantile Regression via Smoothing Spline ANOVA
, 2012
"... Quantile regression provides a more thorough view of the affect of covariates on a response. In many cases, assuming a parametric form for the conditional quantile can be overly restrictive. Nonparametric quantile regression has recently become a viable alternative. The problem of variable selection ..."
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Quantile regression provides a more thorough view of the affect of covariates on a response. In many cases, assuming a parametric form for the conditional quantile can be overly restrictive. Nonparametric quantile regression has recently become a viable alternative. The problem of variable selection for quantile regression is challenging, since important variables can influence various quantiles in different ways. We propose to tackle the problem using the approach of nonparametric quantile regression via regularization in the context of smoothing spline ANOVA models. By imposing the sum of the reproducing kernel Hilbert space norms on functions, the proposed sparse nonparametric quantile regression (SNQR) can identify variables which are important in either conditional mean or conditional variance, and provide flexible nonparametric estimates for quantiles. We develop an efficient algorithm to solve the optimization problem and contribute an R package. Our numerical study suggests the promising performance of the new procedure in variable selection for heteroscedastic data analysis.
a) Nonparametric Model Checking and Variable Selection
, 2012
"... Let X be a d dimensional vector of covariates and Y be the response variable. Under the nonparametric model Y = m(X) + σ(X) we develop an ANOVAtype test for the null hypothesis that a particular coordinate of X has no influence on the regression function. The asymptotic distribution of the test st ..."
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Let X be a d dimensional vector of covariates and Y be the response variable. Under the nonparametric model Y = m(X) + σ(X) we develop an ANOVAtype test for the null hypothesis that a particular coordinate of X has no influence on the regression function. The asymptotic distribution of the test statistic, using residuals based on NadarayaWatson type kernel estimator and d ≤ 4, is established under the null hypothesis and local alternatives. Simulations suggest that under a sparse model, the applicability of the test extends to arbitrary d through sufficient dimension reduction. Using pvalues from this test, a variable selection method based on multiple testing ideas is proposed. The proposed test outperforms existing procedures, while additional simulations reveal that the proposed variable selection method performs competitively against well established procedures. A real data set is analyzed.
LASSO ISOtone for High Dimensional Additive Isotonic Regression
, 2010
"... Additive isotonic regression attempts to determine the relationship between a multidimensional observation variable and a response, under the constraint that the estimate is the additive sum of univariate component effects that are monotonically increasing. In this article, we present a new method ..."
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Additive isotonic regression attempts to determine the relationship between a multidimensional observation variable and a response, under the constraint that the estimate is the additive sum of univariate component effects that are monotonically increasing. In this article, we present a new method for such regression called LASSO Isotone (LISO). LISO adapts ideas from sparse linear modelling to additive isotonic regression. Thus, it is viable in many situations with high dimensional predictor variables, where selection of significant versus insignificant variables are required. We suggest an algorithm involving a modification of the backfitting algorithm CPAV. We give a numerical convergence result, and finally examine some of its properties through simulations. We also suggest some possible extensions that improve performance, and allow calculation to be carried out when the direction of the monotonicity is unknown.