Nonparametric quantile regression with multivariate covariates is a difficult estimation problem due to the curse of dimensionality. Single index models offer greater flexibility in data analysis than linear. Quantile regression is an extension of linear regression used when the. In this paper, we propose a new full iteration estimation method for quantile regression qr of the singleindex model sim. Most previous studies on plsim concentrated on mean regression, based on least square or likelihood approach. Weighted composite quantile regression for single index model. Therefore, we could use quantile regression koenker, 2005 to obtain estimators for the censored singleindex models by extending the methods, e. Bayesian quantile regression for singleindex models bayesian quantile regression for singleindex models hu, yuao. School of mathematics and statistics, zhengzhou university. The quantile regression estimator for quantile q minimizes the objective function q q xn i. Regularized quantile regression and robust feature screening. Published by canadian center of science and education.
Penalized regression methods for linear models in sasstat. Keywords single index partial linear quantile regression asymptotic. Repeat step 1 and step 2 until convergence, then we obtain the. Thus, the precision of parameter estimation as well as the accuracy of prediction will be e. This paper focuses on the quantile regression qr estimation for the partially linear singleindex models where the data are right censored and the censoring indicators are missing at random.
Mar 17, 2012 bayesian quantile regression for single index models bayesian quantile regression for single index models hu, yuao. The asymptotic properties of the proposed estimator are derived. Penalized regression methods penalized regression methods keep all the predictor variables in the model but constrain regularize the regression coef. As an empirical study, we apply the singleindex quantile regression model to. When it comes to nonparametric specifications of the conditional quantile or more generally tail curve one faces, as in mean regression, a dimensionality problem. I proposed a multiple imputation mi based weighted enet miwenet method based. Quantile regression and variable selection for the single. But, unfortunately, none of the lq penalty family satis. Mar 30, 2014 partial linear single index model plsim is a flexible and applicable model when investigating the underlying relationship between the response and the multivariate covariates. Partial linear singleindex model plsim is a flexible and applicable model when investigating the underlying relationship between the response and the multivariate covariates. The first research topic is to address missing data problem for variable selection models including elastic net enet method and sparse partial least squares spls. Semiparametric modeling and estimation of heteroscedasticity in regression analysis of crosssectional data van keilegom.
Then we present example 2 for case 1 and additional simulation results for case 2 in the simulation studies. Composite quantile regression and variable selection 765 where the values of. For single index coefficient models, 20 considered a new estimation pro. Since ordinary quantile regression is not consistent in this case, we consider. Although its computation requires linear programming methods, the quantile regression estimator is asymptotically normally distributed. This is an advancement from the existing quantile regression methods for the highdimensional sparse model. Weighted composite quantile regression for single index. Distance correlation, penalized quantile regression, single index models, sure screening property, ultrahigh dimensionality. Fit a nonconvex penalized quantile regression model. The 1penalized quantile regression has been considered in 18 under small.
Classical least squares regression ma ybe view ed as a natural w a y of extending the idea of estimating an unconditio nal mean parameter to the problem of estimating conditional mean functions. Mar 27, 2019 this paper considers weighted composite quantile estimation of the single index model with missing covariates at random. Regularized quantile regression and robust feature. In this paper, we propose a robust variable selection technique for the sisvr model by using vital method to identify and minimize the. A full range of quantile analysis can provide more complete. We propose both a penalized quantile regression and an independence. One noteworthy point is that similar to the result in. Highdimensional varying index coefficient quantile regression model.
In this paper, we propose a robust variable selection technique for the sisvr model by using vital method to identify and minimize the effects of outliers in the data set. Quantile regression is in the focus of many estimation techniques and is an important tool in data analysis. Under some regularity conditions, we establish the large sample properties of the estimated index parameters and link function. As an empirical study, we apply the singleindex quantile regression model to boston housing data. By considering different levels of quantile, we explore how the covariates, of either social or environmental nature, could have different effects on individuals targeting the low, the median, and the high end of the housing market. This paper considers weighted composite quantile estimation of the singleindex model with missing covariates at random.
Penalized quantile regression 83 in this paper, we consider quantile regression in highdimensional sparse models hdsms. International journal of statistics and probability 3 3, 96, 2014. In this paper, we propose a new full iteration estimation method for quantile regression qr of the single index model sim. Rubinfeld 1978 based on a linear regression analysis. Whereas the method of least squares estimates the conditional mean of the response variable across values of the predictor variables, quantile regression estimates the conditional median or other quantiles of the response variable.
Penalized quantile regression with semiparametric correlated effects. Supplement to inference for singleindex quantile regression models with profile optimization. Quantile regression and variable selection of partial linear singleindex model. Variable selection of single index quantile regression. Therefore, we could use quantile regression koenker, 2005 to obtain estimators for the censored single index models by extending the methods, e. Pdf using an asymmetric laplace distribution, which provides a mechanism for bayesian. Introduction single index regression models are widely assumed to avoid the curse of dimensionality. Introduction plm stata semipar heteroskedasticity endogeneity heterogeneity mfx single index semiparametric regression in stata vincenzo verardi 20 uk stata users group meeting london, uk september 20 vincenzo verardi semiparametric regression 120920 1 66. This paper focuses on the quantile regression qr estimation for the partially linear single index models where the data are right censored and the censoring indicators are missing at random. Let y be a response variable and x be the associated covariate vector. Quantile regression of partial linear singleindex model 377 posite quantile regression cqr, proposed by zou and yuan 2008 of sim. Parametric component detection and variable selection in. The regularization path is computed for the nonconvex penalties at a grid of values for the tuning parameter lambda.
Local linear approximation approach used by wang, wu and li. In contrast to this method, in this paper, we propose minimizing average check loss. As an empirical study, we apply the single index quantile regression model to boston housing data. We propose weighted qr estimators of unknown parameters and link function based on the regression calibration, imputation and inverse probability. Inference for single index quantile regression models with profile optimization ma, shujie and he, xuming, the annals of statistics, 2016. Index 47 rqpenpackage penalized quantile regression for lasso, scad, and mcp penalty functions including group penalties description this package provides functions to. The quantile level is often denoted by the greek letter. Quantile regression and variable selection for partially. Regularized quantile regression and robust feature screening for single index models wei zhong, liping zhu, runze li and hengjian cui xiamen university, shanghai university of finance and economics pennsylvania state university and capital normal university abstract.
In real data analysis, the covariates of model 1 may include many irrelevant. Pdf bayesian quantile regression for singleindex models. Variable selection of singleindex quantile regression. After its inception in koenker and bassett 1978, quantile regression has become an important and widely used technique to study the whole conditional distribution of a response variable and grown into an important tool of applied statistics over the last three decades. If lambda is unselected than an iterative algorithm is used to find a maximum lambda such that the penalty is large enough to produce an intercept only model.
Kong and xia 25 combined quantile regression and a penalty function to develop an adaptive quantile estimation algorithm. High dimentional bic for quantile regression model qbic and cross validation will be used as criterion on the tuning parameter searching. In contrast to this method, in this paper, we propose minimizing average check loss estimation macle. The partial linear singleindex model assumes that the. A better exploitation of the available information.
Applications with heterogeneous preferences this paper proposes new 1penalized quantile regression estimators for panel data, which explicitly allows for individual heterogeneity associated with covariates. Bayesian quantile regression for singleindex models. Quantile regression and variable selection of partial linear. Highdimensional varying index coefficient quantile. Quantile regression and variable selection of singleindex. Quantile regression is a type of regression analysis used in statistics and econometrics. After obtaining the estimate, for any inner point u on the tight support of xt, gu can be estimated by g. Penalized variable selection and quantile regression in. We present several lemmas that will be used in the proof of the main theorems, and the proof of equation a. Fit a nonconvex penalized quantile model via qicd algorithm. Quantile regression and variable selection of partial linear single. International journal of statistics and probability 2, 1230. Penalized quantile regression with semiparametric correlated. We consider median regression and, more generally, a possibly infinite collection of quantile regressions in highdimensional sparse models.
Penalized singleindex quantile regression alkenani. On the predictive risk in misspecified quantile regression. Distance correlation, penalized quantile regression, single index models, sure screening property. Nonconcave penalty, quantile regression, varying index coefficient model. Inference for singleindex quantile regression models with. The large sample properties of the parametric part show that the estimator with estimated selection probability have a smaller limiting. Then range of lambdas goes from the maximum lambda found to eps on the log scale. The focus of this dissertation is to develop statistical methods, under the framework of penalized regressions, to handle three different problems.
Journal of computational and graphical statistics, 935939. Composite quantile regression for the singleindex model. We propose a projection based single index model specification. Fit a nonconvex penalized quantile regression model in. Variable selection in quantile regression 5 unbiasedness, sparsity, and continuity. Penalized likelihood for logisticnormal mixture models with unequal variances. By using adaptive weights for penalizing different coefficients in the lasso penalty, zou 2006 introduced. Penalized spline estimation for partially linear singleindex mod.
The single index support vector regression model sisvr is a useful regression technique used to alleviate the problem of highdimensionality. A simulation study comparing knot selection methods with equally spaced knots in a penalized regression spline. Quantile regression and variable selection of partial. Median regression is more robust to outliers than least squares.
To reduce the dimensionality while still retaining the flexibility of a nonparametric model, we propose modeling the conditional quantile by a singleindex function g 0 x t. To remedy this problem, fan and li 2001 studied the scad penalty in the penalized likelihood setting, which achieves these three desirable properties. Journal of computational and graphical statistics, vol. The quantile level is the probability or the proportion of the population that is associated with a quantile. If lambda is unselected than an iterative algorithm is used to. Composite quantile regression and the oracle model. Furthermore, we consider an adaptive lasso penalized variable. 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. Published by canadian center of science and education penalized singleindex quantile regression. However, it is possible that there are many unimportant. Some statistical methods for dimension reduction a thesis submitted for degree of. Produces penalized quantile regression models for a range of lambdas and penalty of choice.
Quantile regression in regional frequency analysis. If the amount of shrinkage is large enough, these methods can also perform variable selection by shrinking some coef. Partially linear additive quantile regression in ultrahigh dimension sherwood, ben and wang, lan, the annals of statistics, 2016. Furthermore, we propose a variable selection procedure for the qr of sim by combining the estimation method with the adaptive lasso penalized method to.