Bayesian analysis of quantile regression for censored dynamic panel data

This paper develops a Bayesian approach to analyzing quantile regression models for censored dynamic panel data. We employ a likelihood-based approach using the asymmetric Laplace error distribution and introduce lagged observed responses into the conditional quantile function. We also deal with the initial conditions problem in dynamic panel data models by introducing correlated random effects into the model. For posterior inference, we propose a Gibbs sampling algorithm based on a location-scale mixture representation of the asymmetric Laplace distribution. It is shown that the mixture representation provides fully tractable conditional posterior densities and considerably simplifies existing estimation procedures for quantile regression models. In addition, we explain how the proposed Gibbs sampler can be utilized for the calculation of marginal likelihood and the modal estimation. Our approach is illustrated with real data on medical expenditures.     

Pyramid Quantile Regression

We describe a Bayesian model for simultaneous linear quantile regression at several specified quantile levels. More specifically, we propose to model the conditional distributions by using random probability measures, known as quantile pyramids, introduced by Hjort and Walker. Unlike many existing approaches, this framework allows us to specify meaningful priors on the conditional distributions, while retaining the flexibility afforded by the nonparametric error distribution formulation. Simulation studies demonstrate the flexibility of the proposed approach in estimating diverse scenarios, generally outperforming other competitive methods. We also provide conditions for posterior consistency. The method is particularly promising for modeling the extremal quantiles. Applications to extreme value analysis and in higher dimensions are also explored through data examples. Supplemental material for this article is available online.     

Adaptive local linear quantile regression

In this paper we propose a new method of local linear adaptive smoothing for nonparametric conditional quantile regression. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on a simulated example and compare it with other methods. The simulation results demonstrate a reasonable performance of our method proposed especially in situations when the underlying image is piecewise linear or can be approximated by such images. Generally speaking, our method outperforms most other existing methods in the sense of the mean square estimation (MSE) and mean absolute estimation (MAE) criteria. The procedure is very stable with respect to increasing noise level and the algorithm can be easily applied to higher dimensional situations.     

Semi-parametric quantile estimation for double threshold autoregressive models with heteroskedasticity

Compared to the conditional mean or median, conditional quantiles provide a more comprehensive picture of a variable in various scenarios. A semi-parametric quantile estimation method for a double threshold auto-regression with exogenous regressors and heteroskedasticity is considered, allowing representation of both asymmetry and volatility clustering. As such, GARCH dynamics with nonlinearity are added to a nonlinear time series regression model. An adaptive Bayesian Markov chain Monte Carlo scheme, exploiting the link between the quantile loss function and the asymmetric-Laplace distribution, is employed for estimation and inference, simultaneously estimating and accounting for nonlinear heteroskedasticity plus unknown threshold limits and delay lags. A simulation study illustrates sampling properties of the method. Two data sets are considered in the empirical applications: modelling daily maximum temperatures in Melbourne, Australia; and exploring dynamic linkages between financial markets in the US and Hong Kong.     

Semiparametric quantile regression with random censoring

This paper considers estimation and inference in semiparametric quantile regression models when the response variable is subject to random censoring. The paper considers both the cases of independent and dependent censoring and proposes three iterative estimators based on inverse probability weighting, where the weights are estimated from the censoring distribution using the Kaplan–Meier, a fully parametric and the conditional Kaplan–Meier estimators. The paper proposes a computationally simple resampling technique that can be used to approximate the finite sample distribution of the parametric estimator. The paper also considers inference for both the parametric and nonparametric components of the quantile regression model. Monte Carlo simulations show that the proposed estimators and test statistics have good finite sample properties. Finally, the paper contains a real data application, which illustrates the usefulness of the proposed methods.

     

Adaptive Local Linear Quantile Regression

In this paper we propose a new method of local linear adaptive smoothing for nonparametric conditional quantile regression. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on a simulated example and compare it with other methods. The simulation results demonstrate a reasonable performance of our method proposed especially in situations when the underlying image is piecewise linear or can be approximated by such images. Generally speaking, our ...     

A New Approach to Censored Quantile Regression Estimation

Quantile regression provides an attractive tool to the analysis of censored responses, because the conditional quantile functions are often of direct interest in regression analysis, and moreover, the quantiles are often identifiable while the conditional mean functions are not. Existing methods of estimation for censored quantiles are mostly limited to singly left- or right-censored data, with some attempts made to extend the methods to doubly censored data. In this article, we propose a new and unified approach, based on a variation of the data augmentation algorithm, to censored quantile regression estimation. The proposed method adapts easily to different forms of censoring including doubly censored and interval censored data, and somewhat surprisingly, the resulting estimates improve on the performance of the best known estimators with singly censored data. Supplementary material for this article is available online.     

Portfolio optimization with a copula-based extension of conditional value-at-risk

The paper presents a copula-based extension of Conditional Value-at-Risk and its application to portfolio optimization. Copula-based conditional value-at-risk (CCVaR) is a scalar risk measure for multivariate risks modeled by multivariate random variables. It is assumed that the univariate risk components are perfect substitutes, i.e., they are expressed in the same units. CCVaR is a quantile risk measure that allows one to emphasize the consequences of more pessimistic scenarios. By changing the level of a quantile, the measure permits to parameterize prudent attitudes toward risk ranging from the extreme risk aversion to the risk neutrality. In terms of definition, CCVaR is slightly different from popular and well-researched CVaR. Nevertheless, this small difference allows one to efficiently solve CCVaR portfolio optimization problems based on the full information carried by a multivariate random variable by employing column generation algorithm.     

An effective method to reduce the computational complexity of composite quantile regression

In this article, we aim to reduce the computational complexity of the recently proposed composite quantile regression (CQR). We propose a new regression method called infinitely composite quantile regression (ICQR) to avoid the determination of the number of uniform quantile positions. Unlike the composite quantile regression, our proposed ICQR method allows combining continuous and infinite quantile positions. We show that the proposed ICQR criterion can be readily transformed into a linear programming problem. Furthermore, the computing time of the ICQR estimate is far less than that of the CQR, though it is slightly larger than that of the quantile regression. The oracle properties of the penalized ICQR are also provided. The simulations are conducted to compare different estimators. A real data analysis is used to illustrate the performance.     

Efficient estimation and variable selection for infinite variance autoregressive models

In this paper, a self-weighted composite quantile regression estimation procedure is developed to estimate unknown parameter in an infinite variance autoregressive (IVAR) model. The proposed estimator is asymptotically normal and more efficient than a single quantile regression estimator. At the same time, the adaptive least absolute shrinkage and selection operator (LASSO) for variable selection are also suggested. We show that the adaptive LASSO based on the self-weighted composite quantile regression enjoys the oracle properties. Simulation studies and a real data example are conducted to examine the performance of the proposed approaches.     

Bayesian lasso binary quantile regression

In this paper, a Bayesian hierarchical model for variable selection and estimation in the context of binary quantile regression is proposed. Existing approaches to variable selection in a binary classification context are sensitive to outliers, heteroskedasticity or other anomalies of the latent response. The method proposed in this study overcomes these problems in an attractive and straightforward way. A Laplace likelihood and Laplace priors for the regression parameters are proposed and estimated with Bayesian Markov Chain Monte Carlo. The resulting model is equivalent to the frequentist lasso procedure. A conceptional result is that by doing so, the binary regression model is moved from a Gaussian to a full Laplacian framework without sacrificing much computational efficiency. In addition, an efficient Gibbs sampler to estimate the model parameters is proposed that is superior to the Metropolis algorithm that is used in previous studies on Bayesian binary quantile regression. Both the simulation studies and the real data analysis indicate that the proposed method performs well in comparison to the other methods. Moreover, as the base model is binary quantile regression, a much more detailed insight in the effects of the covariates is provided by the approach. An implementation of the lasso procedure for binary quantile regression models is available in the R-package bayesQR.     

Conditional quantiles with varying Gaussians

In this paper we study conditional quantile regression by learning algorithms generated from Tikhonov regularization schemes associated with pinball loss and varying Gaussian kernels. Our main goal is to provide convergence rates for the algorithm and illustrate differences between the conditional quantile regression and the least square regression. Applying varying Gaussian kernels improves the approximation ability of the algorithm. Bounds for the sample error are achieved by using a projection operator, a variance-expectation bound derived from a condition on conditional distributions and a tight bound for the covering numbers involving the Gaussian kernels.     

Descent Direction Stochastic Approximation Algorithm with Adaptive Step Sizes

A stochastic approximation (SA) algorithm with new adaptive step sizes for solving unconstrained minimization problems in noisy environment is proposed. New adaptive step size scheme uses ordered statistics of fixed number of previous noisy function values as a criterion for accepting good and rejecting bad steps. The scheme allows the algorithm to move in bigger steps and avoid steps proportional to $1/k$ when it is expected that larger steps will improve the performance. An algorithm with the new adaptive scheme is defined for a general descent direction. The almost sure convergence is established. The performance of new algorithm is tested on a set of standard test problems and compared with relevant algorithms. Numerical results support theoretical expectations and verify efficiency of the algorithm regardless of chosen search direction and noise level. Numerical results on problems arising in machine learning are also presented. Linear regression problem is considered using real data set. The results suggest that the proposed algorithm shows promise.     

Unified Noncrossing Multiple Quantile Regressions Tree

In this article, we consider the estimation problem of a tree model for multiple conditional quantile functions of the response. Using the generalized, unbiased interaction detection and estimation algorithm, the quantile regression tree (QRT) method has been developed to construct a tree model for an individual quantile function. However, QRT produces different tree models across quantile levels because it estimates several QRT models separately. Furthermore, the estimated quantile functions from QRT often cross each other and consequently violate the basic properties of quantiles. This undesirable phenomenon reduces prediction accuracy and makes it difficult to interpret the resulting tree models. To overcome such limitations, we propose the unified noncrossing multiple quantile regressions tree (UNQRT) method, which constructs a common tree structure across all interesting quantile levels for better data visualization and model interpretation. Furthermore, the UNQRT estimates noncrossing multiple quantile functions simultaneously by enforcing noncrossing constraints, resulting in the improvement of prediction accuracy. The numerical results are presented to demonstrate the competitive performance of the proposed UNQRT over QRT. Supplementary materials for this article are available online.     

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多元非参数分位数回归常常是难于估计的, 为了降低维数同时保持非参数估计的灵活性, 人们常常用单指标的方法模拟响应变量的条件分位数. 本文主要研究单指标分位数回归的变量选择. 以最小化平均损失估计为基础, 我们通过最小化具有SCAD惩罚项的平均损失进行变量选择和参数估计. 在正则条件下, 得到了单指标分位数回归SCAD变量选择的Oracle性质, 给出了SCAD变量选择的计算方法, 并通过模拟研究说明了本文所提方法变量选择的样本性质.     

Weighted composite quantile estimation and variable selection method for censored regression model

This paper considers the weighted composite quantile (WCQ) regression for linear model with random censoring. The adaptive penalized procedure for variable selection in this model is proposed, and the consistency, asymptotic normality and oracle property of the resulting estimators are also derived. The simulation studies and the analysis of an acute myocardial infarction data set are conducted to illustrate the finite sample performance of the proposed method.     

左截断相依数据下条件分位数的双核局部线性估计

本文在左截断相依数据下,利用局部线性估计的方法,先提出了条件分布函数的双核估计;然后利用该估计导出了条件分位数的双核局部线性估计,并建立了这些估计的渐近正态性结果;最后,通过模拟显示该估计在偏移和边界点调节上要比一般的核估计更好.     

Variable selection via quantile regression with the process of Ornstein-Uhlenbeck type

Based on the data-cutoff method,we study quantile regression in linear models,where the noise process is of Ornstein-Uhlenbeck type with possible jumps.In single-level quantile regression,we allow the noise process to be heteroscedastic,while in composite quantile regression,we require that the noise process be homoscedastic so that the slopes are invariant across quantiles.Similar to the independent noise case,the proposed quantile estimators are root-n consistent and asymptotic normal.Furthermore,the adaptive least absolute shrinkage and selection operator(LASSO)is applied for the purpose of variable selection.As a result,the quantile estimators are consistent in variable selection,and the nonzero coefficient estimators enjoy the same asymptotic distribution as their counterparts under the true model.Extensive numerical simulations are conducted to evaluate the performance of the proposed approaches and foreign exchange rate data are analyzed for the illustration purpose.     

失混合效应模型的分位回归及变量选择

纵向数据常常用正态混合效应模型进行分析.然而,违背正态性的假定往往会导致无效的推断.与传统的均值回归相比较,分位回归可以给出响应变量条件分布的完整刻画,对于非正态误差分布也可以给稳健的估计结果．本文主要考虑右删失响应下纵向混合效应模型的分位回归估计和变量选择问题．首先,逆删失概率加权方法被用来得到模型的参数估计．其次,结合逆删失概率加权和LASSO惩罚变量选择方法考虑了模型的变量选择问题.蒙特卡洛模拟显示所提方法要比直接删除删失数据的估计方法更具优势．最后,分析了一组艾滋病数据集来展示所提方法的实际应用效果．     

A robust, locally interpretable algorithm for Lyapunov exponents

An enhanced version of the well known Wolf algorithm for the estimation of the Lyapunov characteristic exponents (LCEs) is proposed. It permits interpretation of the local behavior of non-linear flows. The new variant allows for reliable calculation of the non-uniformity-factors (NUFs). The NUFs can be interpreted as standard deviations of the LCEs. Since the latter can also be estimated by the Wolf algorithm, however, without local information on the flow, the new version ensures local interpretability and therefore allows the calculation of the NUFs. The local contributions to the LCEs which we call “local LCEs” can at least be calculated up to three dimensions. Application of the modified method to a hyperchaotic flow in four dimensions shows that an extension to many dimensions is possible and promises new insight into so far not fully understood high-dimensional non-linear systems.