Using and extending fractional order statistic theory, we characterize the $O(n^{-1})$ coverage probability error of the previously proposed confidence intervals for population quantiles using $L$-statistics as endpoints in Hutson (1999). We derive an analytic expression for the $n^{-1}$ term, which may be used to calibrate the nominal coverage level to get $O\bigl(n^{-3/2}[\log(n)]^3\bigr)$ coverage error. Asymptotic power is shown to be optimal. Using kernel smoothing, we propose a related method for nonparametric inference on conditional quantiles. This new method compares favorably with asymptotic normality and bootstrap methods in theory and in simulations. Code is available from the second author's website for both unconditional and conditional methods, simulations, and empirical examples.
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.
We discuss nonparametric tests for parametric specifications of regression quantiles. The test is based on the comparison of parametric and nonparametric fits of these quantiles. The nonparametric fit is a Nadaraya-Watson quantile smoothing estimator. An asymptotic treatment of the test statistic requires the development of new mathematical arguments. An approach that makes only use of plugging in a Bahadur expansion of the nonparametric estimator is not satisfactory. It requires too strong conditions on the dimension and the choice of the bandwidth. Our alternative mathematical approach requi…
In several different fields, there is interest in analyzing the upper or lower tail quantile of the underlying distribution rather than mean or center quantile. However, the investigation of the tail quantile is difficult because of data sparsity. In this paper, we attempt to develop nonparametric quantile regression for the extremal quantile level. In extremal quantile regression, there are two types of technical conditions of the order of convergence of the quantile level: intermediate order or extreme order. For the intermediate order quantile, the ordinary nonparametric estimator is used. …
Based on the novel concept of multivariate center-outward quantiles introduced recently in Chernozhukov et al. (2017) and Hallin et al. (2021), we are considering the problem of nonparametric multiple-output quantile regression. Our approach defines nested conditional center-outward quantile regression contours and regions with given conditional probability content irrespective of the underlying distribution; their graphs constitute nested center-outward quantile regression tubes. Empirical counterparts of these concepts are constructed, yielding interpretable empirical regions and contours wh…
The issues of bias-correction and robustness are crucial in the strategy of divide-and-conquer (DC), especially for asymmetric nonparametric models with massive data. It is known that quantile-based methods can achieve the robustness, but the quantile estimation for nonparametric regression has non-ignorable bias when the error distribution is asymmetric. This paper explores a global bias-corrected DC by quantile-matched composite for nonparametric regressions with general error distributions. The proposed strategies can achieve the bias-correction and robustness, simultaneously. Unlike common…
The R package quantreg.nonpar implements nonparametric quantile regression methods to estimate and make inference on partially linear quantile models. quantreg.nonpar obtains point estimates of the conditional quantile function and its derivatives based on series approximations to the nonparametric part of the model. It also provides pointwise and uniform confidence intervals over a region of covariate values and/or quantile indices for the same functions using analytical and resampling methods. This paper serves as an introduction to the package and displays basic functionality of the functio…
We investigate nonparametric regression methods based on spatial depth and quantiles when the response and the covariate are both functions. As in classical quantile regression for finite dimensional data, regression techniques developed here provide insight into the influence of the functional covariate on different parts, like the center as well as the tails, of the conditional distribution of the functional response. Depth and quantile based nonparametric regressions are useful to detect heteroscedasticity in functional regression. We derive the asymptotic behaviour of nonparametric depth a…
We address the issue of lack-of-fit testing for a parametric quantile regression. We propose a simple test that involves one-dimensional kernel smoothing, so that the rate at which it detects local alternatives is independent of the number of covariates. The test has asymptotically gaussian critical values, and wild bootstrap can be applied to obtain more accurate ones in small samples. Our procedure appears to be competitive with existing ones in simulations. We illustrate the usefulness of our test on birthweight data.
A collection of quantile curves provides a complete picture of conditional distributions. Properly centered and scaled versions of estimated curves at various quantile levels give rise to the so-called quantile regression process (QRP). In this paper, we establish weak convergence of QRP in a general series approximation framework, which includes linear models with increasing dimension, nonparametric models and partial linear models. An interesting consequence is obtained in the last class of models, where parametric and non-parametric estimators are shown to be asymptotically independent. App…
The Hong Kong Polytechnic University（香港理工大学）；The Chinese University of Hong Kong（香港中文大学）
We propose a nonparametric quantile regression method using deep neural networks with a rectified linear unit penalty function to avoid quantile crossing. This penalty function is computationally feasible for enforcing non-crossing constraints in multi-dimensional nonparametric quantile regression. We establish non-asymptotic upper bounds for the excess risk of the proposed nonparametric quantile regression function estimators. Our error bounds achieve optimal minimax rate of convergence for the Holder class, and the prefactors of the error bounds depend polynomially on the dimension of the pr…
We consider the problem of nonparametric quantile regression for twice censored data. Two new estimates are presented, which are constructed by applying concepts of monotone rearrangements to estimates of the conditional distribution function. The proposed methods avoid the problem of crossing quantile curves. Weak uniform consistency and weak convergence is established for both estimates and their finite sample properties are investigated by means of a simulation study. As a by-product, we obtain a new result regarding the weak convergence of the Beran estimator for right censored data on the…
Quantile regression is a field with steadily growing importance in statistical modeling. It is a complementary method to linear regression, since computing a range of conditional quantile functions provides a more accurate modelling of the stochastic relationship among variables, especially in the tails. We introduce a non-restrictive and highly flexible nonparametric quantile regression approach based on C- and D-vine copulas. Vine copulas allow for separate modeling of marginal distributions and the dependence structure in the data, and can be expressed through a graph theoretical model give…
In this paper, we establish a uniform error rate of a Bahadur representation for local polynomial estimators of quantile regression functions. The error rate is uniform over a range of quantiles, a range of evaluation points in the regressors, and over a wide class of probabilities for observed random variables. Most of the existing results on Bahadur representations for local polynomial quantile regression estimators apply to the fixed data generating process. In the context of testing monotonicity where the null hypothesis is of a complex composite hypothesis, it is particularly relevant to …
We consider the problem of testing significance of predictors in multivariate nonparametric quantile regression. A stochastic process is proposed, which is based on a comparison of the responses with a nonparametric quantile regression estimate under the null hypothesis. It is demonstrated that under the null hypothesis this process converges weakly to a centered Gaussian process and the asymptotic properties of the test under fixed and local alternatives are also discussed. In particular we show, that - in contrast to the nonparametric approach based on estimation of $L^2$-distances - the new…
Quantile regression (QR) is a principal regression method for analyzing the impact of covariates on outcomes. The impact is described by the conditional quantile function and its functionals. In this paper we develop the nonparametric QR-series framework, covering many regressors as a special case, for performing inference on the entire conditional quantile function and its linear functionals. In this framework, we approximate the entire conditional quantile function by a linear combination of series terms with quantile-specific coefficients and estimate the function-valued coefficients from t…
This paper proposes a model-free nonparametric estimator of conditional quantile of a time series regression model where the covariate vector is repeated many times for different values of the response. This type of data is abound in climate studies. To tackle such problems, our proposed method exploits the replicated nature of the data and improves on restrictive linear model structure of conventional quantile regression. Relevant asymptotic theory for the nonparametric estimators of the mean and variance function of the model are derived under a very general framework. We provide a detailed …
Many panel data have the latent subgroup effect on individuals, and it is important to correctly identify these groups since the efficiency of resulting estimators can be improved significantly by pooling the information of individuals within each group. However, the currently assumed parametric and semiparametric relationship between the response and predictors may be misspecified, which leads to a wrong grouping result, and the nonparametric approach hence can be considered to avoid such mistakes. Moreover, the response may depend on predictors in different ways at various quantile levels, a…
Quantile regression is an increasingly important empirical tool in economics and other sciences for analyzing the impact of a set of regressors on the conditional distribution of an outcome. Extremal quantile regression, or quantile regression applied to the tails, is of interest in many economic and financial applications, such as conditional value-at-risk, production efficiency, and adjustment bands in (S,s) models. In this paper we provide feasible inference tools for extremal conditional quantile models that rely upon extreme value approximations to the distribution of self-normalized quan…
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.
This paper considers the problem of nonparametric quantile regression under the assumption that the target conditional quantile function is a composition of a sequence of low-dimensional functions. We study the nonparametric quantile regression estimator using deep neural networks to approximate the target conditional quantile function. For convenience, we shall refer to such an estimator as a deep quantile regression (DQR) estimator. We show that the DQR estimator achieves the nonparametric optimal convergence rate up to a logarithmic factor determined by the intrinsic dimension of the underl…
The partial copula provides a method for describing the dependence between two random variables $X$ and $Y$ conditional on a third random vector $Z$ in terms of nonparametric residuals $U_1$ and $U_2$. This paper develops a nonparametric test for conditional independence by combining the partial copula with a quantile regression based method for estimating the nonparametric residuals. We consider a test statistic based on generalized correlation between $U_1$ and $U_2$ and derive its large sample properties under consistency assumptions on the quantile regression procedure. We demonstrate thro…
We address the estimation of conditional quantiles when the covariate is functional and when the order of the quantiles converges to one as the sample size increases. In a first time, we investigate to what extent these large conditional quantiles can still be estimated through a functional kernel estimator of the conditional survival function. Sufficient conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed estimators. In a second time, basing on these result, a functional Weissman estimator is derived, permitting to estimate la…
Truncated conditional expectation functions are objects of interest in a wide range of economic applications, including income inequality measurement, financial risk management, and impact evaluation. They typically involve truncating the outcome variable above or below certain quantiles of its conditional distribution. In this paper, based on local linear methods, a novel, two-stage, nonparametric estimator of such functions is proposed. In this estimation problem, the conditional quantile function is a nuisance parameter that has to be estimated in the first stage. The proposed estimator is …
This paper, investigates the conditional quantile estimation of a scalar random response and a functional random covariate (i.e. valued in some infinite-dimensional space) whenever {\it functional stationary ergodic data with random censorship} are considered. We introduce a kernel type estimator of the conditional quantile function. We establish the strong consistency with rate of this estimator as well as the asymptotic normality which induces a confidence interval that is usable in practice since it does not depend on any unknown quantity. An application to electricity peak demand interval …
We address the estimation of "extreme" conditional quantiles i.e. when their order converges to one as the sample size increases. Conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed kernel estimators. A Weissman-type estimator and kernel estimators of the conditional tail-index are derived, permitting to estimate extreme conditional quantiles of arbitrary order.
In this article, we propose a penalized high dimensional semiparametric model average quantile prediction approach that is robust for forecasting the conditional quantile of the response. We consider a two-step estimation procedure. In the first step, we use a local linear regression approach to estimate the individual marginal quantile functions, and approximate the conditional quantile of the response by an affine combination of one-dimensional marginal quantile regression functions. In the second step, based on the nonparametric kernel estimates of the marginal quantile regression functions…
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…
In this paper, we develop uniform inference methods for the conditional mode based on quantile regression. Specifically, we propose to estimate the conditional mode by minimizing the derivative of the estimated conditional quantile function defined by smoothing the linear quantile regression estimator, and develop two bootstrap methods, a novel pivotal bootstrap and the nonparametric bootstrap, for our conditional mode estimator. Building on high-dimensional Gaussian approximation techniques, we establish the validity of simultaneous confidence rectangles constructed from the two bootstrap met…
Deep learning has enjoyed tremendous success in a variety of applications but its application to quantile regressions remains scarce. A major advantage of the deep learning approach is its flexibility to model complex data in a more parsimonious way than nonparametric smoothing methods. However, while deep learning brought breakthroughs in prediction, it often lacks interpretability due to the black-box nature of multilayer structure with millions of parameters, hence it is not well suited for statistical inference. In this paper, we leverage the advantages of deep learning to apply it to quan…
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…
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 …
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…
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.
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…
We propose three novel consistent specification tests for quantile regression models which generalize former tests in three ways. First, we allow the covariate effects to be quantile-dependent and nonlinear. Second, we allow parameterizing the conditional quantile functions by appropriate basis functions, rather than parametrically. We are hence able to test for functional forms beyond linearity, while retaining the linear effects as special cases. In both cases, the induced class of conditional distribution functions is tested with a Cramér-von Mises type test statistic for which we derive th…
Conditional Diffusion for Nonparametric Instrumental Variable Quantile Regression
Institute of Data Science and Statistics, Shanghai University of Finance and Economics（上海财经大学数据科学与统计研究院）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；School of Artificial Intelligence, Wuhan University（武汉大学人工智能学院）
In this paper, we consider estimation of the conditional mode of an outcome variable given regressors. To this end, we propose and analyze a computationally scalable estimator derived from a linear quantile regression model and develop asymptotic distributional theory for the estimator. Specifically, we find that the pointwise limiting distribution is a scale transformation of Chernoff's distribution despite the presence of regressors. In addition, we consider analytical and subsampling-based confidence intervals for the proposed estimator. We also conduct Monte Carlo simulations to assess the…
We propose two types of Quantile Graphical Models (QGMs) --- Conditional Independence Quantile Graphical Models (CIQGMs) and Prediction Quantile Graphical Models (PQGMs). CIQGMs characterize the conditional independence of distributions by evaluating the distributional dependence structure at each quantile index. As such, CIQGMs can be used for validation of the graph structure in the causal graphical models (\cite{pearl2009causality, robins1986new, heckman2015causal}). One main advantage of these models is that we can apply them to large collections of variables driven by non-Gaussian and non…
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 study quantile trend filtering, a recently proposed method for nonparametric quantile regression with the goal of generalizing existing risk bounds known for the usual trend filtering estimators which perform mean regression. We study both the penalized and the constrained version (of order $r \geq 1$) of univariate quantile trend filtering. Our results show that both the constrained and the penalized version (of order $r \geq 1$) attain the minimax rate up to log factors, when the $(r-1)$th discrete derivative of the true vector of quantiles belongs to the class of bounded variation signal…
In this paper the nonparametric quantile regression model is considered in a location-scale context. The asymptotic properties of the empirical independence process based on covariates and estimated residuals are investigated. In particular an asymptotic expansion and weak convergence to a Gaussian process are proved. The results can, on the one hand, be applied to test for validity of the location-scale model. On the other hand, they allow to derive various specification tests in conditional quantile location-scale models. In detail a test for monotonicity of the conditional quantile curve is…
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…
Motivated by applications in biological science, we propose a novel test to assess the conditional mean dependence of a response variable on a large number of covariates. Our procedure is built on the martingale difference divergence recently proposed in Shao and Zhang (2014), and it is able to detect a certain type of departure from the null hypothesis of conditional mean independence without making any specific model assumptions. Theoretically, we establish the asymptotic normality of the proposed test statistic under suitable assumption on the eigenvalues of a Hermitian operator, which is c…
This paper considers quantile regression for a wide class of time series models including ARMA models with asymmetric GARCH (AGARCH) errors. The classical mean-variance models are reinterpreted as conditional location-scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location-scale time series models under mild conditions. In the application of this result to ARMA-AGARCH models, more primitive conditions are deduced to obtain the asymptotic prope…
