Quantile regression of partial linear singleindex model 377 posite quantile regression cqr, proposed by zou and yuan 2008 of sim. Fit a nonconvex penalized quantile regression model. Quantile regression is an extension of linear regression used when the. By using adaptive weights for penalizing different coefficients in the lasso penalty, zou 2006 introduced. Spatially modeling the effects of meteorological drivers of pm2. Quantile regression and variable selection of partial linear singleindex model. Regularized quantile regression and robust feature. Penalized spline estimation for partially linear singleindex mod. Variable selection of singleindex quantile regression. Penalized quantile regression 83 in this paper, we consider quantile regression in highdimensional sparse models hdsms. Single index models offer greater flexibility in data analysis than linear models but retain.
Bayesian quantile regression for singleindex models. 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. We propose weighted qr estimators of unknown parameters and link function based on the regression calibration, imputation and inverse probability. In such models, the overall number of regressors p is very large, possibly much larger than the sample size n. However, it is possible that there are many unimportant. In this paper, we propose a new full iteration estimation method for quantile regression qr of the single index model sim. When it comes to nonparametric specifications of the conditional quantile or more generally tail curve one faces, as in mean regression, a dimensionality problem. Penalized variable selection and quantile regression in. Inference for single index quantile regression models with profile optimization ma, shujie and he, xuming, the annals of statistics, 2016. Parametric component detection and variable selection in varyingcoefficient partially linear models.
High dimentional bic for quantile regression model qbic and cross validation will be used as criterion on the tuning parameter searching. School of mathematics and statistics, zhengzhou university. On the predictive risk in misspecified quantile regression. Then range of lambdas goes from the maximum lambda found to eps on the log scale. The regularization path is computed for the nonconvex penalties at a grid of values for the tuning parameter lambda.
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. 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. The partial linear singleindex model assumes that the. After obtaining the estimate, for any inner point u on the tight support of xt, gu can be estimated by g. 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. 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. For single index coefficient models, 20 considered a new estimation pro. Distance correlation, penalized quantile regression, single index models, sure screening property, ultrahigh dimensionality. In contrast to this method, in this paper, we propose minimizing average check loss estimation macle. Partial linear singleindex model plsim is a flexible and applicable model when investigating the underlying relationship between the response and the multivariate covariates. Penalized quantile regression with semiparametric correlated. Published by canadian center of science and education penalized singleindex quantile regression. Composite quantile regression for the singleindex model.
Regularized quantile regression and robust feature screening. 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. Rubinfeld 1978 based on a linear regression analysis. The quantile level is often denoted by the greek letter. However, it is possible that there are many unimportant predictors within the original predictors. Nonconcave penalty, quantile regression, varying index coefficient model. International journal of statistics and probability 2, 1230. But, unfortunately, none of the lq penalty family satis. To remedy this problem, fan and li 2001 studied the scad penalty in the penalized likelihood setting, which achieves these three desirable properties. Variable selection in quantile regression 5 unbiasedness, sparsity, and continuity. It is important to note that the penalized quantile regression problem 2. Inference for singleindex quantile regression models with profile optimization. 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. In this paper, we propose a robust variable selection technique for the sisvr model by using vital method to identify and minimize the.
Penalized likelihood for logisticnormal mixture models with unequal variances. Quantile regression and variable selection for partially. As an empirical study, we apply the singleindex quantile regression model to boston housing data. Introduction single index regression models are widely assumed to avoid the curse of dimensionality. Partially linear additive quantile regression in ultrahigh dimension sherwood, ben and wang, lan, the annals of statistics, 2016. A full range of quantile analysis can provide more complete. Most previous studies on plsim concentrated on mean regression, based on least square or likelihood approach. The focus of this dissertation is to develop statistical methods, under the framework of penalized regressions, to handle three different problems. 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. 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. Fit a nonconvex penalized quantile model via qicd algorithm.
Pdf bayesian quantile regression for singleindex models. 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. Since ordinary quantile regression is not consistent in this case, we consider. Weighted composite quantile regression for single index. Applications with heterogeneous preferences this paper proposes new 1penalized quantile regression estimators for panel data, which explicitly allows for individual heterogeneity associated with covariates. Under some regularity conditions, we establish the large sample properties of the estimated index parameters and link function. Repeat step 1 and step 2 until convergence, then we obtain the. Penalized regressions for variable selection model, single.
Supplement to inference for singleindex quantile regression models with profile optimization. Bayesian quantile regression for singleindex models bayesian quantile regression for singleindex models hu, yuao. In contrast to this method, in this paper, we propose minimizing average check loss. Quantile regression and variable selection for the single. Kong and xia 25 combined quantile regression and a penalty function to develop an adaptive quantile estimation algorithm. A simulation study comparing knot selection methods with equally spaced knots in a penalized regression spline. Highdimensional varying index coefficient quantile. The aim of this paper is to propose a quantile regression qr model that gives directly the. Quantile regression and variable selection of partial linear. Mar 17, 2012 bayesian quantile regression for single index models bayesian quantile regression for single index models hu, yuao. Quantile regression minimizes a sum that gives asymmetric penalties 1 qjei jfor overprediction and qjei jfor underprediction. We present several lemmas that will be used in the proof of the main theorems, and the proof of equation a. 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.
Composite quantile regression and the oracle model. The large sample properties of the parametric part show that the estimator with estimated selection probability have a smaller limiting variance. This paper considers weighted composite quantile estimation of the singleindex model with missing covariates at random. Therefore, we could use quantile regression koenker, 2005 to obtain estimators for the censored singleindex models by extending the methods, e. In real data analysis, the covariates of model 1 may include many irrelevant. Although its computation requires linear programming methods, the quantile regression estimator is asymptotically normally distributed. Quantile regression is a type of regression analysis used in statistics and econometrics.
International journal of statistics and probability 3 3, 96, 2014. Highdimensional varying index coefficient quantile regression model. Thus, the precision of parameter estimation as well as the accuracy of prediction will be e. One noteworthy point is that similar to the result in. Local linear approximation approach used by wang, wu and li.
Composite quantile regression and variable selection 765 where the values of. Quantile regression in regional frequency analysis. Pdf using an asymmetric laplace distribution, which provides a mechanism for bayesian. Furthermore, we consider an adaptive lasso penalized variable. Penalized quantile regression with semiparametric correlated effects. Fit a nonconvex penalized quantile regression model in. The large sample properties of the parametric part show that the estimator with estimated selection probability have a smaller limiting. Journal of computational and graphical statistics, 935939. A better exploitation of the available information. Penalized regression methods for linear models in sasstat. If lambda is unselected than an iterative algorithm is used to.
The asymptotic properties of the proposed estimator are derived. The single index support vector regression model sisvr is a useful regression technique used to alleviate the problem of highdimensionality. As an empirical study, we apply the singleindex quantile regression model to. Five things you should know about quantile regression. The 1penalized quantile regression has been considered in 18 under small. 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.
Single index models offer greater flexibility in data analysis than linear. Penalized singleindex quantile regression alkenani. Semiparametric modeling and estimation of heteroscedasticity in regression analysis of crosssectional data van keilegom. Some statistical methods for dimension reduction a thesis submitted for degree of. Quantile regression and variable selection of partial. Therefore, we could use quantile regression koenker, 2005 to obtain estimators for the censored single index models by extending the methods, e. Variable selection of single index quantile regression. Furthermore, we propose a variable selection procedure for the qr of sim by combining the estimation method with the adaptive lasso penalized method to. Median regression is more robust to outliers than least squares. Nonparametric quantile regression with multivariate covariates is a difficult estimation problem due to the curse of dimensionality. 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. 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. Index 47 rqpenpackage penalized quantile regression for lasso, scad, and mcp penalty functions including group penalties description this package provides functions to. This is an advancement from the existing quantile regression methods for the highdimensional sparse model.
Quantile regression is in the focus of many estimation techniques and is an important tool in data analysis. Let y be a response variable and x be the associated covariate vector. In this paper, we propose a new full iteration estimation method for quantile regression qr of the singleindex model sim. We consider median regression and, more generally, a possibly infinite collection of quantile regressions in highdimensional sparse models. Then we present example 2 for case 1 and additional simulation results for case 2 in the simulation studies. As an empirical study, we apply the single index quantile regression model to boston housing data.
Weighted composite quantile regression for single index model. Journal of computational and graphical statistics, vol. We propose both a penalized quantile regression and an independence. Quantile regression and variable selection of partial linear single. We propose a projection based single index model specification. Quantile regression and variable selection of singleindex. Penalized regression methods penalized regression methods keep all the predictor variables in the model but constrain regularize the regression coef. The quantile regression estimator for quantile q minimizes the objective function q q xn i. The quantile level is the probability or the proportion of the population that is associated with a quantile. If the amount of shrinkage is large enough, these methods can also perform variable selection by shrinking some coef.
Keywords single index partial linear quantile regression asymptotic. Produces penalized quantile regression models for a range of lambdas and penalty of choice. Published by canadian center of science and education. I proposed a multiple imputation mi based weighted enet miwenet method based.
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