Prediction of extremal precipitation by quantile regression forests: from SNU Multiscale Team

This paper considers the problem of spatio-temporal extreme value prediction of precipitation data. This work presents some methods that predict monthly extremes over the next 20 years corresponding to 0.998 quantile at several stations over a certain region. The proposed methods are based on a novel combination of quantile regression forests and circular transformation. As the core of the methodology, quantile regression forests by combining many decorrelated bootstrapping trees may improve prediction performance, and circular transformation is used for building circular transformed predictors of months, which are put into the quantile regression forests model for prediction. The empirical performance of the proposed methods are evaluated through real data analysis, which demonstrates promising results of the proposed approaches.     

Analyzing return asymmetry and quantiles through stochastic volatility models using asymmetric Laplace error via uniform scale mixtures

This paper proposes a new approach to analyze stock return asymmetry and quantiles. We also present a new scale mixture of uniform (SMU) representation for the asymmetric Laplace distribution (ALD). The use of the SMU for a probability distribution is a data augmentation technique that simplifies the Gibbs sampler of the Bayesian Markov chain Monte Carlo algorithms. We consider a stochastic volatility (SV) model with an ALD error distribution. With the SMU representation, the full conditional distribution for some parameters is shown to have closed form. It is also known that the ALD can be used to obtain the coefficients of quantile regression models. This paper also considers a quantile SV model by fixing the skew parameter of the ALD at specific quantile level. Simulation study shows that the proposed methodology works well in both SV and quantile SV models using Bayesian approach. In the empirical study, we analyze index returns of the stock markets in Australia, Japan, Hong Kong, Thailand, and the UK and study the effect of S&P 500 on these returns. The results show the significant return asymmetry in some markets and the influence by S&P 500 in all markets at all quantile levels. Copyright © 2014 John Wiley & Sons, Ltd.     

基于Monte Carlo模拟比较K近邻和局部线性分位数回归

局部线性分位数回归是目前比较流行的非参数分位数回归,其潜在假定待估函数线性光滑.K近邻分位数回归也是非参数分位数回归的重要组成部分,其具有不需待估函数光滑和不同分位点的回归曲线不相交等优点.通过Monte Carlo模拟,比较了两者的估计,得到当待估函数的跳跃点或突变点比较多时,K近邻分位数回归的拟合效果优于局部线性回归.其中模拟的函数是Blocks、Bumps和HeaviSine的函数,它们分别代表跳跃性、波动性、斜率突变性的函数.     

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.

     

Modal Regression Based on Nonparametric Quantile Estimator

Modal regression based on nonparametric quantile estimator is given. Unlike the traditional mean and median regression, modal regression uses mode but not mean or median to represent the center of a conditional distribution, which helps the model to be more robust for outliers, asymmetric or heavy-taileddistribution. Most of solutions for modal regression are based on kernel estimation of density. This paper studies a new solution for modal regression by means of nonparametric quantile estimator. This method builds on the fact that the distribution function is the inverse of the quantile function, then the flexibility of nonparametric quantile estimator is utilized to improve the estimation of modal function. The simulations and application show that the new model outperforms the modal regression model via linear quantile function estimation.     

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.     

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．     

Two-step generalised empirical likelihood inference for semiparametric models

This paper shows how the generalised empirical likelihood method can be used to obtain valid asymptotic inference for the finite dimensional component of semiparametric models defined by a set of moment conditions. The results of the paper are illustrated using three well-known semiparametric regression models: partially linear single index, linear transformation with random censoring, and quantile regression with random censoring. Monte Carlo simulations suggest that some of the proposed test statistics have competitive finite sample properties. The results of the paper are applied to test for functional misspecification in a hedonic price model of a housing market.     

Estimating value at risk with semiparametric support vector quantile regression

Value at Risk (VaR) has been used as an important tool to measure the market risk under normal market. Usually the VaR of log returns is calculated by assuming a normal distribution. However, log returns are frequently found not normally distributed. This paper proposes the estimation approach of VaR using semiparametric support vector quantile regression (SSVQR) models which are functions of the one-step-ahead volatility forecast and the length of the holding period, and can be used regardless of the distribution. We find that the proposed models perform better overall than the variance-covariance and linear quantile regression approaches for return data on S&P 500, NIKEI 225 and KOSPI 200 indices.     

The quantile process under random censoring

In this paper we discuss the asymptotic properties of quantile processes under random censoring. In contrast to most work in this area we prove weak convergence of an appropriately standardized quantile process under the assumption that the quantile regression model is only linear in the region, where the process is investigated. Additionally, we also discuss properties of the quantile process in sparse regression models including quantile processes obtained from the Lasso and adaptive Lasso. The results are derived by a combination of modern empirical process theory, classical martingale methods and a recent result of Kato (2009).     

Quantile regression with group lasso for classification

Applications of regression models for binary response are very common and models specific to these problems are widely used. Quantile regression for binary response data has recently attracted attention and regularized quantile regression methods have been proposed for high dimensional problems. When the predictors have a natural group structure, such as in the case of categorical predictors converted into dummy variables, then a group lasso penalty is used in regularized methods. In this paper, we present a Bayesian Gibbs sampling procedure to estimate the parameters of a quantile regression model under a group lasso penalty for classification problems with a binary response. Simulated and real data show a good performance of the proposed method in comparison to mean-based approaches and to quantile-based approaches which do not exploit the group structure of the predictors.     

Forecasting daily supermarket sales using exponentially weighted quantile regression

Inventory control systems typically require the frequent updating of forecasts for many different products. In addition to point predictions, interval forecasts are needed to set appropriate levels of safety stock. The series considered in this paper are characterised by high volatility and skewness, which are both time-varying. These features motivate the consideration of forecasting methods that are robust with regard to distributional assumptions. The widespread use of exponential smoothing for point forecasting in inventory control motivates the development of the approach for interval forecasting. In this paper, we construct interval forecasts from quantile predictions generated using exponentially weighted quantile regression. The approach amounts to exponential smoothing of the cumulative distribution function, and can be viewed as an extension of generalised exponential smoothing to quantile forecasting. Empirical results are encouraging, with improvements over traditional methods being particularly apparent when the approach is used as the basis for robust point forecasting.     

On weighted and locally polynomial directional quantile regression

The article deals with certain quantile regression methods for vector responses. In particular, it describes weighted and locally polynomial extensions to the projectional quantile regression, discusses their properties, addresses their computational side, compares their outcome with recent analogous generalizations of the competing multiple-output directional quantile regression, demonstrates a link between the two competing methodologies, complements the results already available in the literature, illustrates the concepts with a few simulated and insightful examples illustrating some of their features, and shows their application to a real financial data set, namely to Forex 1M exchange rates. The real-data example strongly indicates that the presented methods might have a huge impact on the analysis of multivariate time series consisting of two to four dimensional observations.     

A Frisch-Newton Algorithm for Sparse Quantile Regression

Recent experience has shown that interior-point methods using a log barrier approach are far superior to classical simplex methods for computing solutions to large parametric quantile regression problems. In many large empirical applications, the design matrix has a very sparse structure. A typical example is the classical fixed-effect model for panel data where the parametric dimension of the model can be quite large, but the number of non-zero elements is quite small. Adopting recent developments in sparse linear algebra we introduce a modified version of the Prisch-Newton algorithm for quantile regression described in Portnoy and Koenker~([28]). The new algorithm substantially reduces the storage (memory) requirements and increases computational speed. The modified algorithm also facilitates the development of nonparametric quantile regression methods. The pseudo design matrices employed in nonparametric quantile regression smoothing are inherently sparse in both the fidelity and roughness penalty components. Exploiting the sparse structure of these problems opens up a whole range of new possibilities for multivariate smoothing on large data sets via ANOVA-type decomposition and partial linear models.     

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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.     

Multi-stage nested classification credibility quantile regression model

In insurance (or in finance) practice, in a regression setting, there are cases where the error distribution is not normal and other cases where the set of data is contaminated due to outlier events. In such cases the classical credibility regression models lead to an unsatisfactory behavior of credibility estimators, and it is more appropriate to use quantile regression instead of the ordinary least squares estimation. However, these quantile credibility models cannot perform effectively when the set of data has nested (hierarchical) structure. This paper develops credibility models for regression quantiles with nested classification as an alternative to Norberg’s (1986) approach of random coefficient regression model with multi-stage nested classification. This paper illustrates two types of applications, one with insurance data and one with Fama/French financial data.     

Bayesian quantile regression model for claim count data

Quantile regression model estimates the relationship between the quantile of a response distribution and the regression parameters, and has been developed for linear models with continuous responses. In this paper, we apply Bayesian quantile regression model for the Malaysian motor insurance claim count data to study the effects of change in the estimates of regression parameters (or the rating factors) on the magnitude of the response variable (or the claim count). We also compare the results of quantile regression models from the Bayesian and frequentist approaches and the results of mean regression models from the Poisson and negative binomial. Comparison from Poisson and Bayesian quantile regression models shows that the effects of vehicle year decrease as the quantile increases, suggesting that the rating factor has lower risk for higher claim counts. On the other hand, the effects of vehicle type increase as the quantile increases, indicating that the rating factor has higher risk for higher claim counts.     

Quantile regression for longitudinal data

The penalized least squares interpretation of the classical random effects estimator suggests a possible way forward for quantile regression models with a large number of “fixed effects”. The introduction of a large number of individual fixed effects can significantly inflate the variability of estimates of other covariate effects. Regularization, or shrinkage of these individual effects toward a common value can help to modify this inflation effect. A general approach to estimating quantile regression models for longitudinal data is proposed employing ?₁ regularization methods. Sparse linear algebra and interior point methods for solving large linear programs are essential computational tools.     

Interquantile Shrinkage in Regression Models

Conventional analysis using quantile regression typically focuses on fitting the regression model at different quantiles separately. However, in situations where the quantile coefficients share some common feature, joint modeling of multiple quantiles to accommodate the commonality often leads to more efficient estimation. One example of common features is that a predictor may have a constant effect over one region of quantile levels but varying effects in other regions. To automatically perform estimation and detection of the interquantile commonality, we develop two penalization methods. When the quantile slope coefficients indeed do not change across quantile levels, the proposed methods will shrink the slopes toward constant and thus improve the estimation efficiency. We establish the oracle properties of the two proposed penalization methods. Through numerical investigations, we demonstrate that the proposed methods lead to estimations with competitive or higher efficiency than the standard quantile regression estimation in finite samples. Supplementary materials for the article are available online.     

Joint modeling for mixed-effects quantile regression of longitudinal data with detection limits and covariates measured with error,with application to AIDS studies

It is very common in AIDS studies that response variable (e.g., HIV viral load) may be subject to censoring due to detection limits while covariates (e.g., CD4 cell count) may be measured with error. Failure to take censoring in response variable and measurement errors in covariates into account may introduce substantial bias in estimation and thus lead to unreliable inference. Moreover, with non-normal and/or heteroskedastic data, traditional mean regression models are not robust to tail reactions. In this case, one may find it attractive to estimate extreme causal relationship of covariates to a dependent variable, which can be suitably studied in quantile regression framework. In this paper, we consider joint inference of mixed-effects quantile regression model with right-censored responses and errors in covariates. The inverse censoring probability weighted method and the orthogonal regression method are combined to reduce the biases of estimation caused by censored data and measurement errors. Under some regularity conditions, the consistence and asymptotic normality of estimators are derived. Finally, some simulation studies are implemented and a HIV/AIDS clinical data set is analyzed to to illustrate the proposed procedure.