Conditional expectiles are becoming an increasingly important tool in finance as well as in other areas of applications. We analyse a support vector machine type approach for estimating conditional expectiles and establish learning rates that are minimax optimal modulo a logarithmic factor if Gaussian RBF kernels are used and the desired expectile is smooth in a Besov sense. As a special case, our learning rates improve the best known rates for kernel-based least squares regression in this scenario. Key ingredients of our statistical analysis are a general calibration inequality for the asymmetric least squares loss, a corresponding variance bound as well as an improved entropy number bound for Gaussian RBF kernels.
Despite the renewed interest in the Newey and Powell (1987) concept of expectiles in fields such as econometrics, risk management, and extreme value theory, expectile regression---or, more generally, M-quantile regression---unfortunately remains limited to single-output problems. To improve on this, we introduce hyperplane-valued multivariate M-quantiles that show strong advantages, for instance in terms of equivariance, over the various point-valued multivariate M-quantiles available in the literature. Like their competitors, our multivariate M-quantiles are directional in nature and provide …
High-dimensional data can often display heterogeneity due to heteroscedastic variance or inhomogeneous covariate effects. Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity in high-dimensional data. The former is computationally challenging due to the non-smooth nature of the check loss, and the latter is sensitive to heavy-tailed error distributions. In this paper, we propose and study (penalized) robust expectile regression (retire), with a focus on iteratively reweighted $\ell_1$-penalization which reduces the estimation bias from $\ell_1$-pe…
Renewable estimation in linear expectile regression models with streaming data sets
School of Economics and Management, Beihang University（北京航空航天大学经济管理学院）；MOE Key Laboratory of Complex System Analysis and Management Decision, Beihang University（北京航空航天大学复杂系统分析与管理决策教育部重点实验室）
The identification of influential observations is an important part of data analysis that can prevent erroneous conclusions drawn from biased estimators. However, in high dimensional data, this identification is challenging. Classical and recently-developed methods often perform poorly when there are multiple influential observations in the same dataset. In particular, current methods can fail when there is masking several influential observations with similar characteristics, or swamping when the influential observations are near the boundary of the space spanned by well-behaved observations.…
Neural networks are becoming an increasingly important tool in applications. However, neural networks are not widely used in statistical genetics. In this paper, we propose a new neural networks method called expectile neural networks. When the size of parameter is too large, the standard maximum likelihood procedures may not work. We use sieve method to constrain parameter space. And we prove its consistency and normality under nonparametric regression framework.
Expectiles were introduced by Newey and Powell (1987) in the context of linear regression models. Recently, Bellini et al. (2014) revealed that expectiles can also be seen as reasonable law-invariant risk measures. In this article, we show that the corresponding statistical functionals are continuous w.r.t. the $1$-weak topology and suitably functionally differentiable. By means of these regularity results we can derive several properties as consistency, asymptotic normality, bootstrap consistency, and qualitative robustness of the corresponding estimators in nonparametric and parametric stati…
National Technical University of Athens（雅典国立技术大学）；Hellenic Air Force（希腊空军）；World Meteorological Organisation (WMO)（世界气象组织）；University of the West of England（西英格兰大学）
The present article is devoted to the semi-parametric estimation of multivariate expectiles for extreme levels. The considered multivariate risk measures also include the possible conditioning with respect to a functional covariate, belonging to an infinite-dimensional space. By using the first order optimality condition, we interpret these expectiles as solutions of a multidimensional nonlinear optimum problem. Then the inference is based on a minimization algorithm of gradient descent type, coupled with consistent kernel estimations of our key statistical quantities such as conditional quant…
The curse of dimensionality is a recognized challenge in nonparametric estimation. This paper develops a new L0-norm regularization approach to the convex quantile and expectile regressions for subset variable selection. We show how to use mixed integer programming to solve the proposed L0-norm regularization approach in practice and build a link to the commonly used L1-norm regularization approach. A Monte Carlo study is performed to compare the finite sample performances of the proposed L0-penalized convex quantile and expectile regression approaches with the L1-norm regularization approache…
Expectile regression is a nice tool for investigating conditional distributions beyond the conditional mean. It is well-known that expectiles can be described with the help of the asymmetric least square loss function, and this link makes it possible to estimate expectiles in a non-parametric framework by a support vector machine like approach. In this work we develop an efficient sequential-minimal-optimization-based solver for the underlying optimization problem. The behavior of the solver is investigated by conducting various experiments and the results are compared with the recent R-packag…
The genetic etiologies of common diseases are highly complex and heterogeneous. Classic statistical methods, such as linear regression, have successfully identified numerous genetic variants associated with complex diseases. Nonetheless, for most complex diseases, the identified variants only account for a small proportion of heritability. Challenges remain to discover additional variants contributing to complex diseases. Expectile regression is a generalization of linear regression and provides completed information on the conditional distribution of a phenotype of interest. While expectile r…
Bayesian optimisation (BO) is widely used to optimise stochastic black box functions. While most BO approaches focus on optimising conditional expectations, many applications require risk-averse strategies and alternative criteria accounting for the distribution tails need to be considered. In this paper, we propose new variational models for Bayesian quantile and expectile regression that are well-suited for heteroscedastic noise settings. Our models consist of two latent Gaussian processes accounting respectively for the conditional quantile (or expectile) and the scale parameter of an asymm…
We propose a M-quantile regression model for the analysis of multivariate, continuous, longitudinal data. M-quantile regression represents an appealing alternative to standard regression models, as it combines the robustness of quantile and the efficiency of expectile regression, providing a complete picture of the response variable distribution. Discrete, individual-specific, random parameters are used to account for both dependence within the same response recorded at different times and association between different responses observed on the same sample unit at a given time. A suitable para…
The paper focuses on the automatic selection of the grouped explanatory variables in an high-dimensional model, when the model errors are asymmetric. After introducing the model and notations, we define the adaptive group LASSO expectile estimator for which we prove the oracle properties: the sparsity and the asymptotic normality. Afterwards, the results are generalized by considering the asymmetric $L_q$-norm loss function. The theoretical results are obtained in several cases with respect to the number of variable groups. This number can be fixed or dependent on the sample size $n$, with the…
We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales.
Measuring interdependence between probabilities of default (PDs) in different industry sectors of an economy plays a crucial role in financial stress testing. Thereby, regression approaches may be employed to model the impact of stressed industry sectors as covariates on other response sectors. We identify vine copula based quantile regression as an eligible tool for conducting such stress tests as this method has good robustness properties, takes into account potential nonlinearities of conditional quantile functions and ensures that no quantile crossing effects occur. We illustrate its perfo…
Imputation is a popular technique for handling missing data. We consider a nonparametric approach to imputation using the kernel ridge regression technique and propose consistent variance estimation. The proposed variance estimator is based on a linearization approach which employs the entropy method to estimate the density ratio. The root-n consistency of the imputation estimator is established when a Sobolev space is utilized in the kernel ridge regression imputation, which enables us to develop the proposed variance estimator. Synthetic data experiments are presented to confirm our theory.
Modal linear regression (MLR) is a method for obtaining a conditional mode predictor as a linear model. We study kernel selection for MLR from two perspectives: "which kernel achieves smaller error?" and "which kernel is computationally efficient?". First, we show that a Biweight kernel is optimal in the sense of minimizing an asymptotic mean squared error of a resulting MLR parameter. This result is derived from our refined analysis of an asymptotic statistical behavior of MLR. Secondly, we provide a kernel class for which iteratively reweighted least-squares algorithm (IRLS) is guaranteed to…
This paper aims to solve a basic problem in distributed statistical inference: how many machines can we use in parallel computing? In kernel ridge regression, we address this question in two important settings: nonparametric estimation and hypothesis testing. Specifically, we find a range for the number of machines under which optimal estimation/testing is achievable. The employed empirical processes method provides a unified framework, that allows us to handle various regression problems (such as thin-plate splines and nonparametric additive regression) under different settings (such as univa…
Kernel smoothers are considered near the boundary of the interval. Kernels which minimize the expected mean square error are derived. These kernels are equivalent to using a linear weighting function in the local polynomial regression. It is shown that any kernel estimator that satisfies the moment conditions up to order $m$ is equivalent to a local polynomial regression of order $m$ with some non-negative weight function if and only if the kernel has at most $m$ sign changes. A fast algorithm is proposed for computing the kernel estimate in the boundary region for an arbitrary placement of da…
In this paper we propose a variable bandwidth kernel regression estimator for $i.i.d.$ observations in $\mathbb{R}^2$ to improve the classical Nadaraya-Watson estimator. The bias is improved to the order of $O(h_n^4)$ under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kerne…
We review recent advances in modal regression studies using kernel density estimation. Modal regression is an alternative approach for investigating relationship between a response variable and its covariates. Specifically, modal regression summarizes the interactions between the response variable and covariates using the conditional mode or local modes. We first describe the underlying model of modal regression and its estimators based on kernel density estimation. We then review the asymptotic properties of the estimators and strategies for choosing the smoothing bandwidth. We also discuss u…
Regularization is an essential element of virtually all kernel methods for nonparametric regression problems. A critical factor in the effectiveness of a given kernel method is the type of regularization that is employed. This article compares and contrasts members from a general class of regularization techniques, which notably includes ridge regression and principal component regression. We derive an explicit finite-sample risk bound for regularization-based estimators that simultaneously accounts for (i) the structure of the ambient function space, (ii) the regularity of the true regression…
We consider the smoothed version of sliced average variance estimation (SAVE) dimension reduction method for dealing with spatially dependent data that are observations of a strongly mixing random field. We propose kernel estimators for the interest matrix and the effective dimension reduction (EDR) space, and show their consistency.
We implement extensions of the partial least squares generalized linear regression (PLSGLR) due to Bastien et al. (2005) through its combination with logistic regression and linear discriminant analysis, to get a partial least squares generalized linear regression-logistic regression model (PLSGLR-log), and a partial least squares generalized linear regression-linear discriminant analysis model (PLSGLRDA). These two classification methods are then compared with classical methodologies like the k-nearest neighbours (KNN), linear discriminant analysis (LDA), partial least squares discriminant an…
In this paper we consider the nonparametric estimation of density and regression functions with non-negative support using a gamma kernel procedure introduced by Chen (2000). Strong uniform consistency and asymptotic normality of the corresponding estimators are established under a general ergodic assumption on the data generation process. Our results generalize those of Shi and Song (2016), obtained in the classic i.i.d. framework, and the works of Bouezmarni and Rombouts (2008, 2010b) and Gospodinov and Hirukawa (2007) for mixing time series.
In this paper we propose an automatic selection of the bandwidth of the semi-recursive kernel estimators of a regression function defined by the stochastic approximation algorithm. We showed that, using the selected bandwidth and some special stepsizes, the proposed semi-recursive estimators will be very competitive to the nonrecursive one in terms of estimation error but much better in terms of computational costs. We corroborated these theoretical results through simulation study and a real dataset.
The celebrated Nadaraya-Watson kernel estimator is among the most studied method for nonparametric regression. A classical result is that its rate of convergence depends on the number of covariates and deteriorates quickly as the dimension grows, which underscores the "curse of dimensionality" and has limited its use in high dimensional settings. In this article, we show that when the true regression function is single or multi-index, the effects of the curse of dimensionality may be mitigated for the Nadaraya-Watson kernel estimator. Specifically, we prove that with $K$-fold cross-validation,…
Estimating the effective dimension reduction (EDR) space, related to the semiparametric regression model introduced by Li \cite{sir}, is based on the estimation of the covariance matrix $Λ$ of the conditional expectation of the vector of predictors given the response. An estimator $\widehatΛ_n$ of $Λ$ based on kernel method was introduced by Zhu and Fang \cite{Asymptotics} who then derived, under some conditions, the asymptotic distribution of $\sqrt{n}\left(\widehatΛ_n-Λ\right)$, as $n\rightarrow +\infty$. In this paper, we obtain, under specified conditions, the almost sure convergence of $\…
Matrix factorization is a popular approach to solving matrix estimation problems based on partial observations. Existing matrix factorization is based on least squares and aims to yield a low-rank matrix to interpret the conditional sample means given the observations. However, in many real applications with skewed and extreme data, least squares cannot explain their central tendency or tail distributions, yielding undesired estimates. In this paper, we propose \emph{expectile matrix factorization} by introducing asymmetric least squares, a key concept in expectile regression analysis, into th…
We develop a multi-kernel based regression method for graph signal processing where the target signal is assumed to be smooth over a graph. In multi-kernel regression, an effective kernel function is expressed as a linear combination of many basis kernel functions. We estimate the linear weights to learn the effective kernel function by appropriate regularization based on graph smoothness. We show that the resulting optimization problem is shown to be convex and pro- pose an accelerated projected gradient descent based solution. Simulation results using real-world graph signals show efficiency…
The primary hyperparameter in kernel regression (KR) is the choice of kernel. In most theoretical studies of KR, one assumes the kernel is fixed before seeing the training data. Under this assumption, it is known that the optimal kernel is equal to the prior covariance of the target function. In this note, we consider KR in which the kernel may be updated after seeing the training data. We point out that an analogous choice of kernel using the posterior of the target function is optimal in this setting. Connections to the view of deep neural networks as data-dependent kernel learners are discu…
In recent years, there have been significant efforts on mitigating unethical demographic biases in machine learning methods. However, very little is done for kernel methods. In this paper, we propose a new fair kernel regression method via fair feature embedding (FKR-F$^2$E) in kernel space. Motivated by prior works on feature selection in kernel space and feature processing for fair machine learning, we propose to learn fair feature embedding functions that minimize demographic discrepancy of feature distributions in kernel space. Compared to the state-of-the-art fair kernel regression method…
We propose Bayesian extensions of two nonparametric regression methods which are kernel and mutual $k$-nearest neighbor regression methods. Derived based on Gaussian process models for regression, the extensions provide distributions for target value estimates and the framework to select the hyperparameters. It is shown that both the proposed methods asymptotically converge to kernel and mutual $k$-nearest neighbor regression methods, respectively. The simulation results show that the proposed methods can select proper hyperparameters and are better than or comparable to the former methods for…
Kernel regression is a popular non-parametric fitting technique. It aims at learning a function which estimates the targets for test inputs as precise as possible. Generally, the function value for a test input is estimated by a weighted average of the surrounding training examples. The weights are typically computed by a distance-based kernel function and they strongly depend on the distances between examples. In this paper, we first review the latest developments of sparse metric learning and kernel regression. Then a novel kernel regression method involving sparse metric learning, which is …
In this work, we demonstrate the advantage of the pGMM (``powered generalized min-max'') kernel in the context of (ridge) regression. In recent prior studies, the pGMM kernel has been extensively evaluated for classification tasks, for logistic regression, support vector machines, as well as deep neural networks. In this paper, we provide an experimental study on ridge regression, to compare the pGMM kernel regression with the ordinary ridge linear regression as well as the RBF kernel ridge regression. Perhaps surprisingly, even without a tuning parameter (i.e., $p=1$ for the power parameter o…
The objective of this study is to investigate the efficient determination of $C$ and $γ$ for Support Vector Regression with RBF or mahalanobis kernel based on numerical and statistician considerations, which indicates the connection between $C$ and kernels and demonstrates that the deviation of geometric distance of neighbour observation in mapped space effects the predict accuracy of $ε$-SVR. We determinate the arrange of $γ$ & $C$ and propose our method to choose their best values.
Tree based ensembles such as Breiman's random forest (RF) and Gradient Boosted Trees (GBT) can be interpreted as implicit kernel generators, where the ensuing proximity matrix represents the data-driven tree ensemble kernel. Kernel perspective on the RF has been used to develop a principled framework for theoretical investigation of its statistical properties. Recently, it has been shown that the kernel interpretation is germane to other tree-based ensembles e.g. GBTs. However, practical utility of the links between kernels and the tree ensembles has not been widely explored and systematically…
Rochester Institute of Technology（罗切斯特理工学院）；School of Mathematical Sciences（数学科学学院）
In this paper, the flexibility, versatility and predictive power of kernel regression are combined with now lavishly available network data to create regression models with even greater predictive performances. Building from previous work featuring generalized linear models built in the presence of network cohesion data, we construct a kernelized extension that captures subtler nonlinearities in extremely high dimensional spaces and also produces far better predictive performances. Applications of seamless yet substantial adaptation to simulated and real-life data demonstrate the appeal and st…
