Efang Kong, Yingcun Xia
The single-index model is one of the most popular semiparametric models in Econometrics. In this paper, we define a quantile regression single-index model, which includes the single-index structure for conditional mean and for conditional variance.
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Estimating the structures at high or low quantiles has become an important subject and attracted increasing attention across numerous fields. However, due to data sparsity at tails, it usually is a challenging task to obtain reliable estimation, especially for high-dimensional data. This paper suggests a flexible parametric structure to tails, and this enables us to conduct the estimation at quantile levels with rich observations and then to extrapolate the fitted structures to far tails. The proposed model depends on some quantile indices and hence is called the quantile index regression. Mor…
As a competitive alternative to least squares regression, quantile regression is popular in analyzing heterogenous data. For quantile regression model specified for one single quantile level $τ$, major difficulties of semiparametric efficient estimation are the unavailability of a parametric efficient score and the conditional density estimation. In this paper, with the help of the least favorable submodel technique, we first derive the semiparametric efficient scores for linear quantile regression models that are assumed for a single quantile level, multiple quantile levels and all the quanti…
Comment: Monitoring Networked Applications With Incremental Quantile Estimation [arXiv:0708.0302]
Quantile-respectful density estimation based on the Harrell-Davis quantile estimator
针对传统密度与分位数估计不一致的问题，提出一种与 Harrell-Davis 分位数估计量天然一致的新型平滑密度估计量，并通过 jittering 支持离散-连续混合分布。
Comment: Monitoring Networked Applications With Incremental Quantile Estimation [arXiv:0708.0302]
Rejoinder: Monitoring Networked Applications With Incremental Quantile Estimation [arXiv:0708.0302]
We consider a semiparametric generalized linear model and study estimation of both marginal and quantile effects in this model. We propose an approximate maximum likelihood estimator, and rigorously establish the consistency, the asymptotic normality, and the semiparametric efficiency of our method in both the marginal effect and the quantile effect estimation. Simulation studies are conducted to illustrate the finite sample performance, and we apply the new tool to analyze a Swiss non-labor income data and discover a new interesting predictor.
Kink model is developed to analyze the data where the regression function is twostage linear but intersects at an unknown threshold. In quantile regression with longitudinal data, previous work assumed that the unknown threshold parameters or kink points are heterogeneous across different quantiles. However, the location where kink effect happens tend to be the same across different quantiles, especially in a region of neighboring quantile levels. Ignoring such homogeneity information may lead to efficiency loss for estimation. In view of this, we propose a composite estimator for the common k…
Estimation of Quantile Functionals in Linear Model
本文研究线性模型中分位数泛函的非参数估计问题，提出将平均两步回归分位数与斜率R估计量相结合的新估计方法。
This paper introduces a novel quantile approach to harness the high-frequency information and improve the daily conditional quantile estimation. Specifically, we model the conditional standard deviation as a realized GARCH model and employ conditional standard deviation, realized volatility, realized quantile, and absolute overnight return as innovations in the proposed dynamic quantile models. We devise a two-step estimation procedure to estimate the conditional quantile parameters. The first step applies a quasi-maximum likelihood estimation procedure, with the realized volatility as a proxy…
Spline Autoregression Method for Estimation of Quantile Spectrum
本文提出样条自回归方法用于估计时间序列的量纲谱，通过量纲离散傅里叶变换和时间域量纲序列，利用平滑样条建模量纲水平与自回归系数的关系，验证了该方法在非线性动态中的有效性。
Spatial dependent data frequently occur in many fields such as spatial econometrics and epidemiology. To deal with the dependence of variables and estimate quantile-specific effects by covariates, spatial quantile autoregressive models (SQAR models) are introduced. Conventional quantile regression only focuses on the fitting models but ignores the examination of multiple conditional quantile functions, which provides a comprehensive view of the relationship between the response and covariates. Thus, it is necessary to study the different regression slopes at different quantiles, especially in …
Estimating conditional quantiles of financial time series is essential for risk management and many other applications in finance. It is well-known that financial time series display conditional heteroscedasticity. Among the large number of conditional heteroscedastic models, the generalized autoregressive conditional heteroscedastic (GARCH) process is the most popular and influential one. So far, feasible quantile regression methods for this task have been confined to a variant of the GARCH model, the linear GARCH model, owing to its tractable conditional quantile structure. This paper consid…
M-quantile regression is a general form of quantile-like regression which usually utilises the Huber influence function and corresponding tuning constant. Estimation requires a nuisance scale parameter to ensure the M-quantile estimates are scale invariant, with several scale estimators having previously been proposed. In this paper we assess these scale estimators and evaluate their suitability, as well as proposing a new scale estimator based on the method of moments. Further, we present two approaches for estimating data-driven tuning constant selection for M-quantile regression. The tuning…
We give methods for the construction of designs for linear models, when the purpose of the investigation is the estimation of the conditional quantile function and the estimation method is quantile regression. The designs are robust against misspecified response functions, and against unanticipated heteroscedasticity. The methods are illustrated by example, and in a case study in which they are applied to growth charts.
We propose a new method for estimating the extreme quantiles for a function of several dependent random variables. In contrast to the conventional approach based on extreme value theory, we do not impose the condition that the tail of the underlying distribution admits an approximate parametric form, and, furthermore, our estimation makes use of the full observed data. The proposed method is semiparametric as no parametric forms are assumed on all the marginal distributions. But we select appropriate bivariate copulas to model the joint dependence structure by taking the advantage of the recen…
We prove an almost sure weak limit theorem for simple linear rank statistics for samples with continuous distributions functions. As a corollary the result extends to samples with ties, and the vector version of an a.s. central limit theorem for vectors of linear rank statistics. Moreover, we derive such a weak convergence result for some quadratic forms. These results are then applied to quantile estimation, and to hypothesis testing for nonparametric statistical designs, here demonstrated by the c-sample problem, where the samples may be dependent. In general, the method is known to be compa…
Interpolated Quantile Estimation: A Unified Framework Bridging Quantiles and the Mean
提出三类在经典分位数与样本均值之间连续插值的估计量，基于平滑L1损失构建统一M估计框架，证明一致性和渐近正态性，并揭示轻尾和重尾分布下的不同效率特性。
Strong uniform consistency of nonparametric estimation for quantile-based entropy function under length-biased sampling
本文研究长度偏差抽样下基于分位数的 Shannon 熵的非参数估计，证明两种估计量的强一致相合性，并通过模拟和真实数据验证其有效性。
This paper proposes a maximum-likelihood approach to jointly estimate marginal conditional quantiles of multivariate response variables in a linear regression framework. We consider a slight reparameterization of the Multivariate Asymmetric Laplace distribution proposed by Kotz et al (2001) and exploit its location-scale mixture representation to implement a new EM algorithm for estimating model parameters. The idea is to extend the link between the Asymmetric Laplace distribution and the well-known univariate quantile regression model to a multivariate context, i.e. when a multivariate depend…
Locally sparse estimation for simultaneous functional quantile regression
本文提出一种局部稀疏的同时函数分位数回归模型，用于分析温度对大豆产量的影响，通过稀疏斜率函数提升模型可解释性，并在模拟和实际数据中验证其有效性。
Quantile Fourier Transform, Quantile Series, and Nonparametric Estimation of Quantile Spectra
本文提出非参数方法估计分位数谱及交叉谱，基于分位数离散傅里叶变换和分位数序列，利用滞后窗方法构建自协方差函数估计谱，通过平滑技术减少统计波动，通过模拟研究评估方法性能。
Our comments are in two parts. First, we make some observations regarding the methodology in Chambers et al. [arXiv:0708.0302]. Second, we briefly describe another interesting network monitoring problem that arises in the context of assessing quality of service, such as loss rates and delay distributions, in packet-switched networks.
Asymmetric power GARCH models have been widely used to study the higher order moments of financial returns, while their quantile estimation has been rarely investigated. This paper introduces a simple monotonic transformation on its conditional quantile function to make the quantile regression tractable. The asymptotic normality of the resulting quantile estimators is established under either stationarity or non-stationarity. Moreover, based on the estimation procedure, new tests for strict stationarity and asymmetry are also constructed. This is the first try of the quantile estimation for no…
We introduce a new measure of interdependence among the components of a random vector along the main diagonal of the vector copula, i.e. along the line $u_{1}=\ldots=u_{J}$, for $\left(u_{1},\ldots,u_{J}\right)\in\left[0,1\right]^{J}$. Our measure is related to the Shannon entropy of a discrete random variable, hence we call it an "entropy index". This entropy index is invariant with respect to marginal non-decreasing transformations and can be used to quantify the intensity of the vector components association in arbitrary dimensions. We show the applicability of our entropy index by an examp…
We propose a bivariate quantile regression method for the bivariate varying coefficient model through a directional approach. The varying coefficients are approximated by the B-spline basis and an $L_{2}$ type penalty is imposed to achieve desired smoothness. We develop a multistage estimation procedure based the Propagation-Separation~(PS) approach to borrow information from nearby directions. The PS method is capable of handling the computational complexity raised by simultaneously considering multiple directions to efficiently estimate varying coefficients while guaranteeing certain smoothn…
Empirical relationships are derived for the expected sampling error of quantile estimations using Monte Carlo experiments for two frequency distributions frequently encountered in climate sciences. The relationships found are expressed as a scaling factor times the standard error of the mean; these give a quick tool to estimate the uncertainty of quantiles for a given finite sample size.
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 fo…
Smoothed Quantile Estimation via Interpolation to the Mean
本研究提出一类可在经验分位数与样本均值间连续插值的平滑分位数估计量，证明其统计性质，揭示不同尾部分布下的效率-稳健性权衡规律，为平衡分位数估计的稳健性与效率提供了新方法。
Design-based Estimation and Inference on Quantile Exposure Effect under General Interference
针对网络干扰下分位数暴露效应，提出基于设计的估计与推断框架，解决非光滑性与有限总体挑战，并通过模拟和教育干预应用揭示平均效应遗漏的尾部异质影响。
Networked applications have software components that reside on different computers. Email, for example, has database, processing, and user interface components that can be distributed across a network and shared by users in different locations or work groups. End-to-end performance and reliability metrics describe the software quality experienced by these groups of users, taking into account all the software components in the pipeline. Each user produces only some of the data needed to understand the quality of the application for the group, so group performance metrics are obtained by combini…
The Multi-Kink Quantile Regression (MKQR) model is an important tool for analyzing data with heterogeneous conditional distributions, especially when quantiles of response variable are of interest, due to its robustness to outliers and heavy-tailed errors in the response. It assumes different linear quantile regression forms in different regions of the domain of the threshold covariate but are still continuous at kink points. In this paper, we investigate parameter estimation, kink point detection and statistical inference in MKQR models. We propose an iterative segmented quantile regression a…
The paper introduces a Bayesian estimation method for quantile regression in univariate ordinal models. Two algorithms are presented that utilize the latent variable inferential framework of Albert and Chib (1993) and the normal-exponential mixture representation of the asymmetric Laplace distribution. Estimation utilizes Markov chain Monte Carlo simulation - either Gibbs sampling together with the Metropolis-Hastings algorithm or only Gibbs sampling. The algorithms are employed in two simulation studies and implemented in the analysis of problems in economics (educational attainment) and poli…
Data integration has become increasingly popular owing to the availability of multiple data sources. This study considered quantile regression estimation when a key covariate had multiple proxies across several datasets. In a unified estimation procedure, the proposed method incorporates multiple proxies that have various relationships with the unobserved covariates. The proposed approach allows the inference of both the quantile function and unobserved covariates. Moreover, it does not require the quantile function's linearity and, simultaneously, accommodates both the linear and nonlinear pr…
Quantile Super Learning for independent and online settings with application to solar power forecasting
本文提出Quantile Super Learning集成方法，通过交叉验证分位数损失组合多个候选算法进行条件分位数估计，并提供独立同分布与在线场景的理论保证，最终在钙钛矿材料性质预测和太阳辐照度在线预报中验证其有效性。
A Semiparametric Quantile Single-Index Model for Zero-Inflated Outcomes
针对微生物组零膨胀数据，提出半参数单指数分位数回归模型，放宽参数假设，适应零比例变化，并证明其拟合性能优越。
In this paper, a new estimation method is introduced for the quantile spectrum, which uses a parametric form of the autoregressive (AR) spectrum coupled with nonparametric smoothing. The method begins with quantile periodograms which are constructed by trigonometric quantile regression at different quantile levels, to represent the serial dependence of time series at various quantiles. At each quantile level, we approximate the quantile spectrum by a function in the form of an ordinary AR spectrum. In this model, we first compute what we call the quantile autocovariance function (QACF) by the …
Local polynomial estimation of quantile density functions
研究提出基于局部多项式估计分位数密度函数及导数的新方法，该估计器在边界处性质更优，证明其渐近正态性并与其他估计器比较偏差、方差及边界性质。
This paper studies estimation in functional linear quantile regression in which the dependent variable is scalar while the covariate is a function, and the conditional quantile for each fixed quantile index is modeled as a linear functional of the covariate. Here we suppose that covariates are discretely observed and sampling points may differ across subjects, where the number of measurements per subject increases as the sample size. Also, we allow the quantile index to vary over a given subset of the open unit interval, so the slope function is a function of two variables: (typically) time an…
We address the estimation of quantiles from heavy-tailed distributions when functional covariate information is available and in the case where the order of the quantile converges to one as the sample size increases. Such "extreme" quantiles can be located in the range of the data or near and even beyond the boundary of the sample, depending on the convergence rate of their order to one. Nonparametric estimators of these functional extreme quantiles are introduced, their asymptotic distributions are established and their finite sample behavior is investigated.
This paper develops a framework for quantile regression in binary longitudinal data settings. A novel Markov chain Monte Carlo (MCMC) method is designed to fit the model and its computational efficiency is demonstrated in a simulation study. The proposed approach is flexible in that it can account for common and individual-specific parameters, as well as multivariate heterogeneity associated with several covariates. The methodology is applied to study female labor force participation and home ownership in the United States. The results offer new insights at the various quantiles, which are of …
MM Algorithms for Statistical Estimation in Quantile Regression
本文针对线性分位数回归的统计估计问题构建并测试易于编码的 MM 算法，并通过模拟比较与多组真实数据应用验证其有效性。
Moment Estimator-Based Extreme Quantile Estimation with Erroneous Observations: Application to Elliptical Extreme Quantile Region Estimation
针对极值理论中关注变量含观测误差的问题，量化误差对基于矩的极值指数及极端分位数估计的影响，推导误差渐近可忽略条件，并将方法应用于多元椭圆分布下仿射等变极端分位数区域估计。
We present a new non-parametric estimator of the conditional density of the kernel type. It is based on an efficient transformation of the data by quantile transform. By use of the copula representation, it turns out to have a remarkable product form. We study its asymptotic properties and compare its bias and variance to competitors based on nonparametric regression.
Sequential quantile estimation refers to incorporating observations into quantile estimates in an incremental fashion thus furnishing an online estimate of one or more quantiles at any given point in time. Sequential quantile estimation is also known as online quantile estimation. This area is relevant to the analysis of data streams and to the one-pass analysis of massive data sets. Applications include network traffic and latency analysis, real time fraud detection and high frequency trading. We introduce new techniques for online quantile estimation based on Hermite series estimators in the…
We show that the full-sample bootstrap is asymptotically valid for constructing confidence intervals for high-quantiles, tail probabilities, and other tail parameters of a univariate distribution. This resolves the doubts that have been raised about the validity of such bootstrap methods. In our extensive simulation study, the overall performance of the bootstrap method was better than that of the standard asymptotic method, indicating that the bootstrap method is at least as good, if not better than, the asymptotic method for inference. This paper also lays the foundation for developing boots…
In this study, we develop a novel estimation method for quantile treatment effects (QTE) under rank invariance and rank stationarity assumptions. Ishihara (2020) explores identification of the nonseparable panel data model under these assumptions and proposes a parametric estimation based on the minimum distance method. However, when the dimensionality of the covariates is large, the minimum distance estimation using this process is computationally demanding. To overcome this problem, we propose a two-step estimation method based on the quantile regression and minimum distance methods. We then…
This paper investigates pooling strategies for tail index and extreme quantile estimation from heavy-tailed data. To fully exploit the information contained in several samples, we present general weighted pooled Hill estimators of the tail index and weighted pooled Weissman estimators of extreme quantiles calculated through a nonstandard geometric averaging scheme. We develop their large-sample asymptotic theory across a fixed number of samples, covering the general framework of heterogeneous sample sizes with different and asymptotically dependent distributions. Our results include optimal ch…
In this paper, we propose a class of low-rank panel quantile regression models which allow for unobserved slope heterogeneity over both individuals and time. We estimate the heterogeneous intercept and slope matrices via nuclear norm regularization followed by sample splitting, row- and column-wise quantile regressions and debiasing. We show that the estimators of the factors and factor loadings associated with the intercept and slope matrices are asymptotically normally distributed. In addition, we develop two specification tests: one for the null hypothesis that the slope coefficient is a co…
This paper presents two results concerning uniform confidence intervals for the tail index and the extreme quantile. First, we show that it is impossible to construct a length-optimal confidence interval satisfying the correct uniform coverage over a local non-parametric family of tail distributions. Second, in light of the impossibility result, we construct honest confidence intervals that are uniformly valid by incorporating the worst-case bias in the local non-parametric family. The proposed method is applied to simulated data and a real data set of National Vital Statistics from National C…
Statistical Estimation for Covariance Structures with Tail Estimates using Nodewise Quantile Predictive Regression Models
本文针对带尾部估计的协方差结构，提出用分位数预测回归估计VaR-CoVaR风险矩阵，还构建了无调优参数、具有一致性的FOCE变量选择算法，经真实与模拟数据集验证可用于网络系统性风险建模。
We propose a novel multi-task method for quantile forecasting with shared Linear layers. Our method is based on the Implicit quantile learning approach, where samples from the Uniform distribution $\mathcal{U}(0, 1)$ are reparameterized to quantile values of the target distribution. We combine the implicit quantile and input time series representations to directly forecast multiple quantile estimations for multiple horizons jointly. Prior works have adopted a Linear layer for the direct estimation of all forecasting horizons in a multi-task learning setup. We show that following similar intuit…
We propose an estimation method for the conditional mode when the conditioning variable is high-dimensional. In the proposed method, we first estimate the conditional density by solving quantile regressions multiple times. We then estimate the conditional mode by finding the maximum of the estimated conditional density. The proposed method has two advantages in that it is computationally stable because it has no initial parameter dependencies, and it is statistically efficient with a fast convergence rate. Synthetic and real-world data experiments demonstrate the better performance of the prop…
This paper considers a nonlinear quantile model with change-points. The quantile estimation method, which as a particular case includes median model, is more robust with respect to other traditional methods when model errors contain outliers. Under relatively weak assumptions, the convergence rate and asymptotic distribution of change-point and of regression parameter estimators are obtained. Numerical study by Monte Carlo simulations shows the performance of the proposed method for nonlinear model with change-points.
In real applications of the linear model, the explanatory variables are very often naturally grouped, the most common example being the multivariate variance analysis. In the present paper, a quantile model with structure group is considered, the number of groups can diverge with sample size. We introduce and study the adaptive elastic-net group estimator, for improving the parameter estimation accuracy. This method allows automatic selection, with a probability converging to one, of significant groups and further the non zero parameter estimators are asymptotically normal. The convergence rat…
In this paper, we present a statistical framework for modeling conditional quantiles of spatial processes assumed to be strongly mixing in space. We establish the $L_1$ consistency and the asymptotic normality of the kernel conditional quantile estimator in the case of random fields. We also define a nonparametric spatial predictor and illustrate the methodology used with some simulations.
L-Estimation of Population Quantiles Using Ranked Set Sampling
本文提出两种基于秩集抽样的L估计方法，用于估计人口分位数，通过分解秩集设计构造了可扩展的计算方法，并在模拟和实际数据中验证了其效率优势。
In this paper, we develop a new censored quantile instrumental variable (CQIV) estimator and describe its properties and computation. The CQIV estimator combines Powell (1986) censored quantile regression (CQR) to deal with censoring, with a control variable approach to incorporate endogenous regressors. The CQIV estimator is obtained in two stages that are non-additive in the unobservables. The first stage estimates a non-additive model with infinite dimensional parameters for the control variable, such as a quantile or distribution regression model. The second stage estimates a non-additive …
Partial frontiers are not quantiles
本文证明部分前沿估计并非分位数，提出凸化阶-α及保序CQR/CER等替代估计量，并通过蒙特卡洛模拟和电厂数据验证其性能优劣。
Quantile regression with generalized multiquadric loss function
本文提出基于广义多二次损失函数的分位数回归方法，通过构造平滑损失函数解决非可导问题，利用梯度下降法实现高效估计，并通过理论分析和模拟验证其有效性。
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