Quantile regression uses an l 1loss function, and an optimal solution by means of linear programming. Regularized partially functional quantile regression. R package for highdimensional sparse penalized quantile regression authors. This is an advancement from the existing quantile regression methods for the highdimensional sparse model. A general framework for robust testing and con dence. In addition, the rank correlation screening method is used to accommodate ultra. Our analysis and results on the estimation error bound for quantile regression, interestingly, has the same order as that for regularized least squares regression for. Robust inference in highdimensional approximately sparse quantile. Greg ridgeway abstract random forests were introduced as a machine learning tool in breiman 2001 and have since proven to be very popular and powerful for highdimensional regression and classi. Penalized quantile regression is considered to accommodate models where the number of covariates is larger than the sample size. In this work, the consistency, asymptotic normality, and.
Oracle estimation of a change point in high dimensional. We present a quantile regression model where the response variable is scalar while the explanatory variables involves both in nitedimensional predictor processesviewed as functional data, and highdimensional scalar covariates. L1penalized quantile regression in highdimensional sparse models, arxiv 2009, annals of statistics 2011, with a. Debiasing and distributed estimation for highdimensional quantile. Quantile regression offers an alternative that is robust to outliers in the y direction and directly models heteroscedastic behavior.
Highdimensional varying index coefficient quantile regression model 09042018 by li jialiang, et al. To account for censoring data in high dimensional case, we employ effective dimension reduction and the ideas of informative subset idea. L regularized quantile regression with many regressors. Moreover, the oracle property with proper selection of tuning parameter for quantile regression under certain regularity conditions is also established. Indeed, in model selection problems,the statisticalefficiency criterionfavors theuse of theopenaltyfunctionsakaike 1 andschwarz 32, where the. Robust inference in high dimensional approximately sparse. Produces penalized quantile regression models for a range of lambdas and. Semiparametric quantile regression with high dimensional covariates liping zhu1, mian huang1 and runze li2 1shanghai university of finance and economics and 2pennsylvania state university abstract.
We consider median regression and, more generally, a possibly infinite collection of quantile regressions in high dimensional sparse models. Request pdf debiasing and distributed estimation for highdimensional quantile regression distributed and parallel computing is. Quantile regression is a useful method in this field with many applications. A general framework for robust testing and con dence regions. Second, any linear combination of the estimates is asymptotically normal with the same asymptotic variance as that of the oracle estimator. For computational purposes, it is important to note that the penalized quantile regression. We propose a twostep variable selection procedure for censored quantile regression with high dimensional predictors.
We present a quantile regression model where the response variable is scalar while the explanatory variables involves both in nite dimensional predictor processesviewed as functional data, and high dimensional scalar covariates. In order to understand how the covariate affects the response variable, a new tool is required. Statistical learning evolves quickly with more and more sophisticated models proposed to incorporate the complicated data structure from modern scientific and business problems. Since ultrahigh dimensional data often display heterogeneity, we advocate a quantileadaptive feature screening framework. Highdimensional sparse econometric models, an introduction,springer lecture notes 2009, with a. We show that the resulting estimator of the quantile. In the highdimensional settings, particularly when p n, quantile regression is generally not consistent, which motivates the use of penalization in order to remove all or at least nearly all regressors whose population coecients are zero, thereby possibly restoring con. Hypothesis tests in models whose dimension far exceeds the sample size can be formulated much like the classical studentized tests only after the initial bias of estimation is removed successfully.
It is important to correctly set the range of the tuning parameter in variable selection. The linear quantile regression problem has formal dual problem. Variable selection in censored quantile regression with high. High dimensional quantile inference plays a critical role in. In these models, the number of regressors p is very large, possibly larger than the sample size n, but only at most s regressors have a nonzero impact on each conditional quantile of the response variable, where s grows more slowly than n. This 1penalized quantile regression or quantile regression lasso has been considered by knight and fu 18 under the small. L1penalized quantile regression in high dimensional sparse. For quantile regression, belloni and chernozhukov applied an l 1 penalty for model selection and estimation in high dimensional sparse linear models, where the number of predictors is possibly larger than the sample size, but the number of predictors related to the response grows more slowly than the sample size. Selected problems for highdimensional data quantile and errorsinvariables regressions by seyoung park a dissertation submitted in partial ful llment of the requirements for the degree of doctor of philosophy statistics in the university of michigan 2016 doctoral committee. Quantileadaptive modelfree variable screening for high.
Much of the published work on high dimensional quantile regression is generally confined to model a single or multiple quantiles wang et al. Globally adaptive quantile regression with ultrahigh. High quantile regression for extreme events journal of. Analysis of highdimensional data poses many challenges for statisticians. In particular, the existing variable selection procedure for the linear quantile regression with high dimensional covariates wu and liu, 2009. Journal of computational and graphical statistics, 24, 676694. Penalized quantile regression 83 in this paper, we consider quantile regression in highdimensional sparse models hdsms. We propose a generalization of the linear panel quantile regression model to accommodate both sparse and dense parts. Sparse penalized quantile regression is a useful tool for variable selection, robust estimation, and heteroscedasticity detection in high dimensional data analysis. Belloni and chernozhukov, 2011 may be used to select significant variables for model.
Quantile regression for analyzing heterogeneity in ultrahigh dimension lan wang, yichao wu and runze li abstract ultrahigh dimensional data often display heterogeneity due to either heteroscedastic variance or other forms of nonlocationscale covariate e ects. Yuwen gu and hui zou the authors keep all the s of the code. Furthermore, while various quantile regression methods 7, 16, 9 can be to used to estimate a single quantile of a highdimensional distribution, extensions of those to estimate qquantiles is mostly nontrivial. The quantile regression technique is considered as an alternative to the classical ordinary least squares ols regression in case of outliers and heavy tailed errors existing in linear models. In such models, the overall number of regressors p is very large, possibly much larger than the sample size n. A notable advantage of quantile regression over leastsquares regression in high dimension is that the sparsity pattern is allowed to be. High dimensional sparse econometric models, an introduction,springer lecture notes 2009, with a. Quantile regression aims at modeling the conditional median and quantiles of a response variable given certain predictor variables. Our penalized estimators not only select covariates but also discriminate between a model with homogeneous sparsity and a model with a change. Dealing with high dimensional quantile regression with an unknown change point calls for a new proof technique since the quantile loss function is nonsmooth and furthermore the corresponding. Selected problems for highdimensional data quantile and. Quantile regression for analyzing heterogeneity in ultra. Highdimensional structured quantile regression vidyashankar sivakumar 1arindam banerjee abstract quantile regression aims at modeling the conditional median and quantiles of a response variable given certain predictor variables.
In this paper, we propose a new quantileadaptive, modelfree variable screening procedure, which is particularly appealing for analyzing high dimensional heterogeneous data and data with censored responses. Both challenges can be addressed by making simplifying assumptions, such as additivity or intrinsic lower dimensionality of the expensive objective. Second, we derive an almost sure debiased representation of the lassopenalized highdimensional misspeci. Lan wang research interests quantile regression, personalized optimal decision estimation, highdimensional nonparametric and semiparametric methods and statistical applications.
Multitask quantile regression under the transnormal model. Adaptive penalized quantile regression 2227 a naive pointwise approach for globally concerned quantile regression in the ultra high dimensional setting would perform an existing locally concerned penal ization method separately at each. In this work we consider the problem of linear quantile regression in high dimensions where the number of predictor variables is much higher than the number of samples available for parameter estimation. Variable selection in censored quantile regression with. For quantile regression, belloni and chernozhukov applied an l 1 penalty for model selection and estimation in highdimensional sparse linear models, where the number of predictors is possibly larger than the sample size, but the number of predictors related to. Quantile regression for additive coefficient models in high. Variable selection in highdimensional partially linear additive models for composite quantile regression. Belloni r program is here and matlab program is here. Due to space constraints, additional simulation results, the proofs of technical lemmas and corollaries, justification for the proposed grid approximation, and sample codes are relegated to the supplement zheng, peng and he 2015. Admm for highdimensional sparse penalized quantile. Second, we derive an almost sure debiased representation of the lassopenalized high dimensional misspeci. In particular, the existing variable selection procedure for the linear quantile regression with highdimensional covariates wu and liu, 2009. A third distinctive feature of the lrm is its normality assumption.
R package for highdimensional sparse penalized quantile. Sparse penalized quantile regression is a useful tool for variable selection, robust estimation, and heteroscedasticity detection in highdimensional data analysis. Regions in high dimensional quantile regression tianqi zhao mladen kolar y han liu march 16, 2015 abstract we propose a robust inferential procedure for assessing uncertainties of parameter estimation in high dimensional linear models, where the dimension p can grow exponentially fast with the sample size n. R package for high dimensional sparse penalized quantile regression authors. Their results apply to quantile regression without the sparseness assumption but require that p on. On highdimensional misspecified quantile regression. The nitesample performance of the proposed method is also examined. In this paper, we propose a weighted quantile regression method. The additive partial linear model is extended to the highdimensional case.
Unconditional quantile regression has quickly become popular after being introduced by firpo, fortin, and lemieux 2009, econometrica 77. Quantile regression is an appropriate tool for accomplishing this task. Monte carlo simulations demonstrate finite performance of the proposed estimator. We consider highdimensional models where the number of regressors potentially exceeds the sample size but a subset of them su. The acquisition function selects a new point to evaluate the blackbox function. For extreme events, estimation of high conditional quantiles for heavy tailed distributions is an important problem. Additionally, inference in quantile regression requires estimation of the so called. Nov 01, 2011 we consider median regression and, more generally, a possibly infinite collection of quantile regressions in high dimensional sparse models. In this paper, we extend the methodology and theory of quantile regression to ultra high dimension.
In this work we consider the problem of linear quantile regression in high dimensions where the num. In this paper, we consider a highdimensional quantile regression model where the sparsity structure may differ between two subpopulations. In addition, we allow for heteroskesdastic and nonnormal regression errors and stochastic covariates. Thus, the quantile regression provides an effective tool to reduce the dimension of the covariate in the presence of high dimensional covariates. Pdf although mean regression achieved its greatest diffusion in the. This property enables us to reduce substantially the computational cost of the backfitting algorithm wu, yu and yu, 2010.
Quantile regression for analyzing heterogeneity in ultrahigh. Supplement to globally adaptive quantile regression with ultrahigh dimensional data. Quantile regression for analyzing heterogeneity in ultra high dimension lan wang, yichao wu and runze li abstract ultra high dimensional data often display heterogeneity due to either heteroscedastic variance or other forms of nonlocationscale covariate e ects. We consider median regression and, more generally, a possibly infinite collection of quantile regressions in highdimensional sparse models. This paper is concerned with quantile regression for a semiparametric regression model, in which both the conditional mean and conditional variance. Users of this package please cite the following paper gu, y. Here, q canbeacomponentofx,andp ispotentiallymuch larger than the sample sizen. Highdimensional varying index coefficient quantile. Quantile regression methods for high dimensional data lan wang. Fellow, institute of mathematical statistics, elected 2017. Regions in highdimensional quantile regression tianqi zhao mladen kolar y han liu march 16, 2015 abstract we propose a robust inferential procedure for assessing uncertainties of parameter estimation in highdimensional linear models, where the dimension p can.
Highdimensional varying index coefficient quantile regression model. To accommodate heterogeneity, we advocate a more general interpretation of spar. High dimensional bayesian quantile regression prithwish bhaumik the university of texas at austin and. Admm for highdimensional sparse penalized quantile regression. In this paper, we consider a high dimensional quantile regression model where the sparsity structure may differ between two subpopulations. Professor xuming he, cochair assistant professor shuheng zhou. Highdimensional structured quantile regression proceedings of. Highdimensional bayesian optimization with projections. Quantile regression for additive coefficient models in. Pdf uniform inference for highdimensional quantile. Uniform inference for highdimensional quantile regression. Next, we perform a finite sample analysis of the newly proposed robust density. Regions in highdimensional quantile regression tianqi zhao mladen kolar y han liu march 16, 2015 abstract we propose a robust inferential procedure for assessing uncertainties of parameter estimation in high dimensional linear models, where the dimension p can grow exponentially fast with the sample size n.
Semiparametric quantile regression with highdimensional covariates liping zhu1, mian huang1 and runze li2 1shanghai university of finance and economics and 2pennsylvania state university abstract. Variable selection in highdimensional partially linear. Variable selection in highdimensional partially linear additive models for composite. Due to space constraints, additional simulation results, the proofs of technical lemmas and corollaries, justification for the proposed grid approximation, and sample codes are. Oracle estimation of a change point in highdimensional. This proce dure is computationally efficient, which is very appealing in high dimensional data analysis. For all that, quantile regression is a very useful statistical technology for a large diversity of disciplines. Honors fellow, american statistical association, elected 2018. L1penalized quantile regression in high dimensional. Furthermore, the proposed method strikes a good balance between robustness and e ciency, achieves the \oraclelike convergence rate, and provides the provable prediction interval under the highdimensional setting. Lan wang research interests quantile regression, personalized optimal decision estimation, high dimensional nonparametric and semiparametric methods and statistical applications. However, including highdimensional fixed effects in rifreg is quite burdensome and sometimes even impossible. Key challenges of bayesian optimization in high dimensions are both learning the response surface and optimizing an acquisition function.
62 1538 578 30 199 1234 461 431 1421 477 1123 245 1260 509 1216 346 386 1476 993 1514 29 1494 905 344 1502 455 819 35 224 938 1365 312 635