Single Index Quantile Regression Research Articles

Empirical likelihood in single-index quantile regression with high dimensional and missing observations

Based on empirical likelihood method, we investigate statistical inference in partially linear single-index quantile regression with high dimensional linear and single-index parameters when the observations are missing at random, which allows the response or covariates or response and covariates simultaneously missing. In particular, applying B-spline approximation to the unknown link function, we establish asymptotic normality of bias-corrected empirical likelihood ratio function and maximum empirical likelihood estimators of the parameters. Variable selection is considered by using the SCAD penalty. Meanwhile, we propose a penalized empirical likelihood ratio statistic to test hypothesis, and prove its asymptotically chi-square distribution under the null hypothesis. Also, simulation study and a real data analysis are conducted to evaluate the performance of the proposed methods.

Semiparametric estimation of expected shortfall and its application in finance

AbstractMeasuring risk effectively is crucial for managing risk in financial markets. The expected shortfall has become an increasingly popular metric for risk in recent years. How to estimate it is important in statistics and financial econometrics. Based on the single index quantile regression, we introduce a new semiparametric approach, namely, weighted single index quantile regression. We assess the performance of the proposed expected shortfall estimator with backtesting. Our simulation results indicate that the estimator has a good finite sample performance and often outperforms existing methods. By applying the new method to both a market index and individual stocks, we show that it not only exhibits the best performance but also reveals an insight about the effect of the COVID pandemic, that is, the pandemic increases the market risk.

Single-Index Quantile Regression with Left Truncated Data

Single-Index Quantile Regression with Left Truncated Data

Bayesian analysis in single-index quantile regression with missing observation

Based on Bayesian method, we investigate single-index quantile regression with missing observation. In particular, using spline approximation for the link function, we construct quasi-posterior distribution of the index vector based on asymmetric Laplace likelihood with missing observation, and establish asymptotically normality of the posterior estimator of the index parameters. At the same time, we use a hierarchical model based on spike and slab Gaussian priors to do variable selection and study consistency of the variable selection. Finite sample performance of the proposed methods is analyzed via simulation and real data too.

No-Crossing Single-Index Quantile Regression Curve Estimation

Single-index quantile regression (QR) models can avoid the curse of dimensionality in nonparametric problems by assuming that the response is only related to a single linear combination of the covariates. Like the standard parametric or nonparametric QR whose estimated curves may cross, the single-index QR can also suffer quantile crossing, leading to an invalid distribution for the response. This issue has attracted considerable attention in the literature in the recent year. In this article, we consider single-index models, develop methods for QR that guarantee noncrossing quantile curves, and extend the methods and results to composite quantile regression. The asymptotic properties of the proposed estimators are derived and their advantages over existing methods are explained. Simulation studies and a real data application are conducted to illustrate the finite sample performance of the proposed methods.

Variable selection in the single-index quantile regression model with high-dimensional covariates

We study the effects of high-dimensional covariates in the single-index quantile regression model. An improved version of an estimation algorithm is proposed with variable selection. Finite sample performance is studied through an extensive simulation study which highlights the performance of the new procedure. A data analysis on US crime data is given that exhibits the usefulness of the proposed methodology.

Functional single-index quantile regression models

It is known that functional single-index regression models can achieve better prediction accuracy than functional linear models or fully nonparametric models, when the target is to predict a scalar response using a function-valued covariate. However, the performance of these models may be adversely affected by extremely large values or skewness in the response. In addition, they are not able to offer a full picture of the conditional distribution of the response. Motivated by using trajectories of $$\hbox {PM}_{{10}}$$ concentrations of last day to predict the maximum $$\hbox {PM}_{{10}}$$ concentration of the current day, a functional single-index quantile regression model is proposed to address those issues. A generalized profiling method is employed to estimate the model. Simulation studies are conducted to investigate the finite sample performance of the proposed estimator. We apply the proposed framework to predict the maximal value of $$\hbox {PM}_{{10}}$$ concentrations based on the intraday $$\hbox {PM}_{{10}}$$ concentrations of the previous day.

Central quantile subspace

Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. There is a great amount of work about linear and nonlinear QR models. Specifically, nonparametric estimation of the conditional quantiles received particular attention, due to its model flexibility. However, nonparametric QR techniques are limited in the number of covariates. Dimension reduction offers a solution to this problem by considering low-dimensional smoothing without specifying any parametric or nonparametric regression relation. Existing dimension reduction techniques focus on the entire conditional distribution. We, on the other hand, turn our attention to dimension reduction techniques for conditional quantiles and introduce a new method for reducing the dimension of the predictor X. The novelty of this paper is threefold. We start by considering a single index quantile regression model, which assumes that the conditional quantile depends on X through a single linear combination of the predictors, then extend to a multi index quantile regression model, and finally, generalize the proposed methodology to any statistical functional of the conditional distribution. The performance of the methodology is demonstrated through simulation examples and a real data application. Our results suggest that this method has a good finite sample performance and often outperforms existing methods.

New Bayesian Single Index Quantile Regression Based on Uniform Scale Mixture

To scale back the dimensionality while holding a lot of flexibility of a nonparametric model Wu, et al. (2010) proposed a single index conditional quantile regression model. In this paper, a new Bayesian lasso for single index quantile regression model is proposed based on a scale mixture uniform. In addition, we construct an efficient and sampling Gibbs algorithm for posterior inference based on a uniform scale mixture representation for Laplace distribution. Simulation study have considered to evaluate our proposed method compare to the existing methods. The results of simulations indicate that the new Bayesian algorithm performs well.

Estimation of value-at-risk using single index quantile regression

ABSTRACTValue-at-Risk (VaR) is one of the best known and most heavily used measures of financial risk. In this paper, we introduce a non-iterative semiparametric model for VaR estimation called the single index quantile regression time series (SIQRTS) model. To test its performance, we give an application to four major US market indices: the S&P 500 Index, the Russell 2000 Index, the Dow Jones Industrial Average, and the NASDAQ Composite Index. Our results suggest that this method has a good finite sample performance and often outperforms a number of commonly used methods.

Single index quantile regression for censored data

Quantile regression (QR) has become a popular method of data analysis, especially when the error term is heteroscedastic. It is particularly relevant for the analysis of censored survival data as an alternative to proportional hazards and the accelerated failure time models. Such data occur frequently in biostatistics, environmental sciences, social sciences and econometrics. There is a large body of work for linear/nonlinear QR models for censored data, but it is only recently that the single index quantile regression (SIQR) model has received some attention. However, the only existing method for fitting the SIQR model for censored data uses an iterative algorithm and no asymptotic theory for the resulting estimator of the parametric component is given. We propose a non-iterative estimation algorithm and derive the asymptotic distribution of the proposed estimator under heteroscedasticity. Results from simulation studies evaluating the finite sample performance of the proposed estimator are reported.

Variable selection in heteroscedastic single-index quantile regression

ABSTRACTWe propose a new algorithm for simultaneous variable selection and parameter estimation for the single-index quantile regression (SIQR) model . The proposed algorithm, which is non iterative , consists of two steps. Step 1 performs an initial variable selection method. Step 2 uses the results of Step 1 to obtain better estimation of the conditional quantiles and , using them, to perform simultaneous variable selection and estimation of the parametric component of the SIQR model. It is shown that the initial variable selection method consistently estimates the relevant variables , and the estimated parametric component derived in Step 2 satisfies the oracle property.

Estimation and variable selection for quantile partially linear single-index models

Partially linear single-index models are flexible dimension reduction semiparametric tools yet still retain ease of interpretability as linear models. This paper is concerned with the estimation and variable selection for partially linear single-index quantile regression models. Polynomial splines are used to estimate the unknown link function. We first establish the asymptotic properties of the quantile regression estimators. For feature selection, we adopt the smoothly clipped absolute deviation penalty (SCAD) approach to select simultaneously single-index variables and partially linear variables. We show that the regularized variable selection estimators are consistent and possess oracle properties. The consistency and oracle properties are also established under the proposed linear approximation of the nonparametric link function that facilitates fast computation. Furthermore, we show that the proposed SCAD tuning parameter selectors via the Schwarz information criterion can consistently identify the true model. Monte Carlo studies and an application to Boston Housing price data are presented to illustrate the proposed approach.

GEE analysis for longitudinal single-index quantile regression

We consider a single-index quantile regression model for longitudinal data. Based on generalized estimating equations, an estimation procedure is proposed by taking into account the correlation within subject. Under mild assumptions, we derive the convergence rate of the estimator of the unknown link function and the asymptotic normality of estimator of the index parameter using the “projection” technique. Since the estimating equations are non-continuous, we further adopt the smoothing approach and show that estimators obtained from the smoothed estimating equations are asymptotically equivalent to that from the unsmoothed estimating equations. It is also shown that the estimator is more efficient when the correlation is correctly specified. Finally, we present numerical examples including simulations and analysis of a lung function data.

Single index quantile regression for heteroscedastic data

Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. Linear and nonlinear QR models have been studied extensively, while recent research focuses on the single index quantile regression (SIQR) model. Compared to the single index mean regression (SIMR) problem, the fitting and the asymptotic theory of the SIQR model are more complicated due to the lack of closed form expressions for estimators of conditional quantiles. Consequently, the proposed methods are necessarily iterative. We propose a non-iterative estimation algorithm, and derive the asymptotic distribution of the proposed estimator under heteroscedasticity. For identifiability, we use a parametrization that sets the first coefficient to 1 instead of the typical condition which restricts the norm of the parametric component. This distinction is more than simply cosmetic as it affects, in a critical way, the correspondence between the estimator derived and the asymptotic theory.

Inference for single-index quantile regression models with profile optimization

Single index models offer greater flexibility in data analysis than linear models but retain some of the desirable properties such as the interpretability of the coefficients. We consider a pseudo-profile likelihood approach to estimation and testing for single-index quantile regression models. We establish the asymptotic normality of the index coefficient estimator as well as the optimal convergence rate of the nonparametric function estimation. Moreover, we propose a score test for the index coefficient based on the gradient of the pseudo-profile likelihood, and employ a penalized procedure to perform consistent model selection and model estimation simultaneously. We also use Monte Carlo studies to support our asymptotic results, and use an empirical example to illustrate the proposed method.

Bayesian elastic net single index quantile regression

ABSTRACTSingle index model conditional quantile regression is proposed in order to overcome the dimensionality problem in nonparametric quantile regression. In the proposed method, the Bayesian elastic net is suggested for single index quantile regression for estimation and variables selection. The Gaussian process prior is considered for unknown link function and a Gibbs sampler algorithm is adopted for posterior inference. The results of the simulation studies and numerical example indicate that our propose method, BENSIQReg, offers substantial improvements over two existing methods, SIQReg and BSIQReg. The BENSIQReg has consistently show a good convergent property, has the least value of median of mean absolute deviations and smallest standard deviations, compared to the other two methods.

Regularized Single Index Quantile Regression Model

This article proposes a new approach for variable selection in the single index quantile regression model. Compared to existing methods, the new approach produce sparse solutions for the index vector. Performance of the new method is enhanced by a fully adaptive penalty function. Finite sample performance is studied through a simulation study that compares the proposed method with existing work under several criteria. A data analysis is given which highlights the usefulness of the proposed methodology.

Non-iterative Estimation and Variable Selection in the Single-index Quantile Regression Model

A new estimation procedure is proposed for the single-index quantile regression model. Compared to existing work, this approach is non-iterative and hence, computationally efficient. The proposed method not only estimates the index parameter and the link function but also selects variables simultaneously. The performance of the variable selection is enhanced by a fully adaptive penalty function motivated by the sliced inverse regression technique. Finite sample performance is studied through a simulation study that compares the proposed method with existing work under several criteria. A data analysis is given that highlights the usefulness of the proposed methodology.

A SINGLE-INDEX QUANTILE REGRESSION MODEL AND ITS ESTIMATION

Models with single-index structures are among the many existing popular semiparametric approaches for either the conditional mean or the conditional variance. This paper focuses on a single-index model for the conditional quantile. We propose an adaptive estimation procedure and an iterative algorithm which, under mild regularity conditions, is proved to converge with probability 1. The resulted estimator of the single-index parametric vector is root-n consistent, asymptotically normal, and based on simulation study, is more efficient than the average derivative method in Chaudhuri, Doksum, and Samarov (1997, Annals of Statistics 19, 760–777). The estimator of the link function converges at the usual rate for nonparametric estimation of a univariate function. As an empirical study, we apply the single-index 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.