Akifumi Okuno, Keisuke Yano
This paper discusses a design-dependent nature of variance in nonparametric link regression aiming at predicting a mean outcome at a link, i.e., a pair of nodes, based on currently observed data comprising covariates at nodes and outcomes at links.
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We analyze the statistical properties of nonparametric regression estimators using covariates which are not directly observable, but have be estimated from data in a preliminary step. These so-called generated covariates appear in numerous applications, including two-stage nonparametric regression, estimation of simultaneous equation models or censored regression models. Yet so far there seems to be no general theory for their impact on the final estimator's statistical properties. Our paper provides such results. We derive a stochastic expansion that characterizes the influence of the generat…
We consider testing the significance of a subset of covariates in a nonparametric regression. These covariates can be continuous and/or discrete. We propose a new kernel-based test that smoothes only over the covariates appearing under the null hypothesis, so that the curse of dimensionality is mitigated. The test statistic is asymptotically pivotal and the rate of which the test detects local alternatives depends only on the dimension of the covariates under the null hypothesis. We show the validity of wild bootstrap for the test. In small samples, our test is competitive compared to existing…
This paper examines the problem of nonparametric testing for the no-effect of a random covariate (or predictor) on a functional response. This means testing whether the conditional expectation of the response given the covariate is almost surely zero or not, without imposing any model relating response and covariate. The covariate could be univariate, multivariate or functional. Our test statistic is a quadratic form involving univariate nearest neighbor smoothing and the asymptotic critical values are given by the standard normal law. When the covariate is multidimensional or functional, a pr…
We consider a bandit problem which involves sequential sampling from two populations (arms). Each arm produces a noisy reward realization which depends on an observable random covariate. The goal is to maximize cumulative expected reward. We derive general lower bounds on the performance of any admissible policy, and develop an algorithm whose performance achieves the order of said lower bound up to logarithmic terms. This is done by decomposing the global problem into suitably "localized" bandit problems. Proofs blend ideas from nonparametric statistics and traditional methods used in the ban…
Nonparametric local polynomial regression for functional covariates
本文研究无限维希尔伯特空间中函数型协变量的非参数回归，提出局部多项式估计方法，并在适当调节与回归函数超光滑条件下证明其可获得通常被认为不可达到的多项式收敛速度。
Uniform convergence rates and automatic variable selection in nonparametric regression with functional and categorical covariates
本文研究了具有函数型和分类协变量的非参数预测方法的渐近性质，证明了一致收敛速度，并表明数据驱动的最小二乘交叉验证能自动移除不相关的噪声变量。
The main purpose is to estimate the regression function of a real random variable with functional explanatory variable by using a recursive nonparametric kernel approach. The mean square error and the almost sure convergence of a family of recursive kernel estimates of the regression function are derived. These results are established with rates and precise evaluation of the constant terms. Also, a central limit theorem for this class of estimators is established. The method is evaluated on simulations and real data set studies.
A Minimax Theory of Nonparametric Regression Under Covariate Shift
本文提出了一种非参数回归在协变量偏移下的极小极大理论，引入转移函数并展示了其域性质对收敛速率的影响，覆盖了协变量无界支持的情况。
In interval censored models with current status observations, the variables are indicators of the presence of individuals on observation intervals and covariates. When several individuals share the same observation interval, a simple procedure provides new estimators for the distribution of the observation times and their intensity, in a closed form. They are $n^{1/2}$-consistent for piece-wise constant covariates. Estimators of the sample-sizes are deduced and asymptotic $χ^2$ tests for independence of the observations on consecutive intervals and for independence between consecutive classes …
Nonparametric Bayesian intensity estimation for covariate-driven inhomogeneous point processes
本文研究协变量驱动的非齐次泊松点过程强度函数的非参数贝叶斯估计，利用协变量实现跨空间位置的信息借用，并推导出全局与逐点损失下的最优后验收缩速率。
This paper studies the problem of nonparametric testing for the effect of a random functional covariate on a real-valued error term. The covariate takes values in $L^2[0,1]$, the Hilbert space of the square-integrable real-valued functions on the unit interval. The error term could be directly observed as a response or \emph{estimated} from a functional parametric model, like for instance the functional linear regression. Our test is based on the remark that checking the no-effect of the functional covariate is equivalent to checking the nullity of the conditional expectation of the error term…
The accuracy of a diagnostic test is typically characterised using the receiver operating characteristic (ROC) curve. Summarising indexes such as the area under the ROC curve (AUC) are used to compare different tests as well as to measure the difference between two populations. Often additional information is available on some of the covariates which are known to influence the accuracy of such measures. We propose nonparametric methods for covariate adjustment of the AUC. Models with normal errors and non-normal errors are discussed and analysed separately. Nonparametric regression is used for…
The varying-coefficient model is an important nonparametric statistical model that allows us to examine how the effects of covariates vary with exposure variables. When the number of covariates is big, the issue of variable selection arrives. In this paper, we propose and investigate marginal nonparametric screening methods to screen variables in ultra-high dimensional sparse varying-coefficient models. The proposed nonparametric independence screening (NIS) selects variables by ranking a measure of the nonparametric marginal contributions of each covariate given the exposure variable. The sur…
Phase Transition in Nonparametric Minimax Rates for Covariate Shifts on Approximate Manifolds
研究结构化协变量偏移下的非参数回归，提出局部多项式回归估计器与自适应流程，在目标分布近似流形的假设下达到最优极小化率，并揭示了由距离与维度等因素控制的相变现象。
In this paper, a nonparametric estimator is proposed for estimating the L1-median for multivariate conditional distribution when the covariates take values in an infinite dimensional space. The multivariate case is more appropriate to predict the components of a vector of random variables simultaneously rather than predicting each of them separately. While estimating the conditional L1-median function using the well-known Nadarya-Waston estimator, we establish the strong consistency of this estimator as well as the asymptotic normality. We also present some simulations and provide how to built…
We study covariate shift in the context of nonparametric regression. We introduce a new measure of distribution mismatch between the source and target distributions that is based on the integrated ratio of probabilities of balls at a given radius. We use the scaling of this measure with respect to the radius to characterize the minimax rate of estimation over a family of Hölder continuous functions under covariate shift. In comparison to the recently proposed notion of transfer exponent, this measure leads to a sharper rate of convergence and is more fine-grained. We accompany our theory with …
We consider nonparametric estimation of a regression curve when the data are observed with multiplicative distortion which depends on an observed confounding variable. We suggest several estimators, ranging from a relatively simple one that relies on restrictive assumptions usually made in the literature, to a sophisticated piecewise approach that involves reconstructing a smooth curve from an estimator of a constant multiple of its absolute value, and which can be applied in much more general scenarios. We show that, although our nonparametric estimators are constructed from predictors of the…
Semiparametric single-index assumptions are convenient and widely used dimen\-sion reduction approaches that represent a compromise between the parametric and fully nonparametric models for regressions or conditional laws. In a mean regression setup, the SIM assumption means that the conditional expectation of the response given the vector of covariates is the same as the conditional expectation of the response given a scalar projection of the covariate vector. In a conditional distribution modeling, under the SIM assumption the conditional law of a response given the covariate vector coincide…
A nonparametric Bayesian analysis of independent and identically distributed observations of covariate-driven Poisson processes
本文提出了一种基于多带宽高斯过程的非参数贝叶斯方法，用于分析协变量驱动的泊松过程的各向异性光滑性问题，并通过数值实验和野火数据集应用验证了方法的有效性。
This paper establishes that so-called instrumental variables enable the identification and the estimation of a fully nonparametric regression model with Berkson-type measurement error in the regressors. An estimator is proposed and proven to be consistent. Its practical performance and feasibility are investigated via Monte Carlo simulations as well as through an epidemiological application investigating the effect of particulate air pollution on respiratory health. These examples illustrate that Berkson errors can clearly not be neglected in nonlinear regression models and that the proposed m…
Nonparametric regression on random geometric graphs sampled from submanifolds
该研究针对协变量位于未知光滑紧子流形的非参数回归问题，基于图拉普拉斯特征基随机基展开的贝叶斯先验，证明了在Hölder光滑性假设下后验收缩率达极小极大最优。
Increasing domain asymptotics for covariate-based nonparametric Bayesian intensity estimation with Gaussian and Besov-Laplace priors
研究在递增域渐近性和协变量遍历性假设下，使用高斯先验和贝索-拉普拉斯先验的非参数贝叶斯方法估计协变量驱动点过程强度函数，并证明其达到极小极大最优后验收缩率。
This paper deals with nonparametric estimation of conditional den-sities in mixture models in the case when additional covariates are available. The proposed approach consists of performing a prelim-inary clustering algorithm on the additional covariates to guess the mixture component of each observation. Conditional densities of the mixture model are then estimated using kernel density estimates ap-plied separately to each cluster. We investigate the expected L 1 -error of the resulting estimates and derive optimal rates of convergence over classical nonparametric density classes provided the…
In this paper we prove large deviations principles for the Nadaraya-Watson estimator of the regression of a real-valued variable with a functional covariate. Under suitable conditions, we show pointwise and uniform large deviations theorems with good rate functions.
Asymptotic equivalence results for nonparametric regression experiments have always assumed that the variances of the observations are known. In practice, however the variance of each observation is generally considered to be an unknown nuisance parameter. We establish an asymptotic approximation to the nonparametric regression experiment when the value of the variance is an additional parameter to be estimated or tested. This asymptotically equivalent experiment has two components: the first contains all the information about the variance and the second has all the information about the mean.…
This paper presents a model selection technique of estimation in semiparametric regression models of the type Y_i=β^{\prime}\underbarX_i+f(T_i)+W_i, i=1,...,n. The parametric and nonparametric components are estimated simultaneously by this procedure. Estimation is based on a collection of finite-dimensional models, using a penalized least squares criterion for selection. We show that by tailoring the penalty terms developed for nonparametric regression to semiparametric models, we can consistently estimate the subset of nonzero coefficients of the linear part. Moreover, the selected estimator…
Covariate-Adaptive Randomization in Clinical Trials without Inflated Variances
本文提出了一种新的协变量自适应随机化方法，通过平衡协变量比率ρ:(1-ρ)来减少失衡方差，避免了传统方法中未指定协变量方差膨胀的问题。
Covariate shift in nonparametric regression with Markovian design
本文研究非参数回归中的协变量偏移问题，将独立同分布数据的收敛速率结果扩展到马尔可夫依赖结构，证明Nadaraya-Watson估计器的泛化风险收敛速率由源和目标马尔可夫链平稳分布的相似性决定，并推导了有限和谱隙马尔可夫链的精确速率。
Nonparametric Instrumental Variable Regression with Observed Covariates
本文提出了一种结合观测协变量的非参数工具变量回归方法，通过引入傅里叶测度和自适应核长度尺度，实现了对异质因果效应的高效估计，并建立了理论学习速率的上下界。
Consider a Gaussian nonparametric regression problem having both an unknown mean function and unknown variance function. This article presents a class of difference-based kernel estimators for the variance function. Optimal convergence rates that are uniform over broad functional classes and bandwidths are fully characterized, and asymptotic normality is also established. We also show that for suitable asymptotic formulations our estimators achieve the minimax rate.
Local Learning for Covariate Selection in Nonparametric Causal Effect Estimation with Latent Variables
针对潜变量存在下非参数因果效应估计的协变量选择问题，提出利用可观测变量间可检验关系进行局部学习的有效且完备的方法，并通过合成与真实数据实验验证其有效性。
We address the issue of variable selection in the regression model with very high ambient dimension, i.e., when the number of covariates is very large. The main focus is on the situation where the number of relevant covariates, called intrinsic dimension, is much smaller than the ambient dimension. Without assuming any parametric form of the underlying regression function, we get tight conditions making it possible to consistently estimate the set of relevant variables. These conditions relate the intrinsic dimension to the ambient dimension and to the sample size. The procedure that is provab…
Asymptotic equivalence for nonparametric additive regression
本文研究了非参数可加回归与高斯白噪声实验的渐近等价性，并探讨了在特定设定下算子的分解对逼近的影响。
Nonparametric estimation of a covariate-adjusted counterfactual treatment regimen response curve
本文针对个体化医疗中的条件价值函数，提出基于筛的非参数协变量调整治疗方案响应曲线估计量，并建立其有效推断、置信区间及最优规则估计性质。
We present a nonparametric family of estimators for the tail index of a Pareto-type distribution when covariate information is available. Our estimators are based on a weighted sum of the log-spacings between some selected observations. This selection is achieved through a moving window approach on the covariate domain and a random threshold on the variable of interest. Asymptotic normality is proved under mild regularity conditions and illustrated for some weight functions. Finite sample performances are presented on a real data study.
This paper is concerned with modeling the dependence structure of two (or more) time-series in the presence of a (possible multivariate) covariate which may include past values of the time series. We assume that the covariate influences only the conditional mean and the conditional variance of each of the time series but the distribution of the standardized innovations is not influenced by the covariate and is stable in time. The joint distribution of the time series is then determined by the conditional means, the conditional variances and the marginal distributions of the innovations, which …
Optimally tackling covariate shift in RKHS-based nonparametric regression
本文研究 RKHS 非参数回归中的协变量偏移，证明有界似然比下 KRR 的 minimax 最优性，并对无界但二阶矩有限情形提出截断重加权 KRR。
Estimation of genewise variance arises from two important applications in microarray data analysis: selecting significantly differentially expressed genes and validation tests for normalization of microarray data. We approach the problem by introducing a two-way nonparametric model, which is an extension of the famous Neyman--Scott model and is applicable beyond microarray data. The problem itself poses interesting challenges because the number of nuisance parameters is proportional to the sample size and it is not obvious how the variance function can be estimated when measurements are correl…
In this paper, we develop new multiscale methods to test qualitative hypotheses about the regression function m in a nonparametric regression model with fixed design points and time series errors. In time series applications, m represents a nonparametric time trend. Practitioners are often interested in whether the trend m has certain shape properties. For example, they would like to know whether m is constant or whether it is increasing/decreasing in certain time regions. Our multiscale methods allow to test for such shape properties of the trend m. In order to perform the methods, we require…
One goal in survival analysis of right-censored data is to estimate the marginal survival function in the presence of dependent censoring. When many auxiliary covariates are sufficient to explain the dependent censoring, estimation based on either a semiparametric model or a nonparametric model of the conditional survival function can be problematic due to the high dimensionality of the auxiliary information. In this paper, we use two working models to condense these high-dimensional covariates in dimension reduction; then an estimate of the marginal survival function can be derived nonparamet…
This paper deals with the nonparametric density estimation of the regression error term assuming its independence with the covariate. The difference between the feasible estimator which uses the estimated residuals and the unfeasible one using the true residuals is studied. An optimal choice of the bandwidth used to estimate the residuals is given. We also study the asymptotic normality of the feasible kernel estimator and its rate-optimality.
Analysis of gradual changes in nonparametric regression based on a new optimization method in the non-unique case
针对一维协变量非参数回归模型，提出一种新的优化方法估计渐变点，定义并比较多种一致估计量，讨论收敛速度和基于渐变结构的回归函数估计。
We consider a wavelet thresholding approach to adaptive variance function estimation in heteroscedastic nonparametric regression. A data-driven estimator is constructed by applying wavelet thresholding to the squared first-order differences of the observations. We show that the variance function estimator is nearly optimally adaptive to the smoothness of both the mean and variance functions. The estimator is shown to achieve the optimal adaptive rate of convergence under the pointwise squared error simultaneously over a range of smoothness classes. The estimator is also adaptively within a log…
Nonparametric Goodness-of-fit Testing under Covariate Shift
本文针对协变量偏移场景，提出结合截断重要性加权核岭回归与乘数自助法的非参数拟合优度检验方法，经理论证明与数值实验验证其有效性。
We consider a general nonparametric regression model called the compound model. It includes, as special cases, sparse additive regression and nonparametric (or linear) regression with many covariates but possibly a small number of relevant covariates. The compound model is characterized by three main parameters: the structure parameter describing the "macroscopic" form of the compound function, the "microscopic" sparsity parameter indicating the maximal number of relevant covariates in each component and the usual smoothness parameter corresponding to the complexity of the members of the compo…
Adaptive estimation for nonparametric circular regression with errors in variables
针对变量含误差的圆回归问题，本文构建自适应估计器并分析其理论收敛速率，通过仿真与真实数据验证方法有效性，为圆型或线型协变量场景提供非参数估计方案。
In this paper we consider a regression model that allows for time series covariates as well as heteroscedasticity with a regression function that is modelled nonparametrically. We assume that the regression function changes at some unknown time $\lfloor ns_0\rfloor$, $s_0\in(0,1)$, and our aim is to estimate the (rescaled) change point $s_0$. The considered estimator is based on a Kolmogorov-Smirnov functional of the marked empirical process of residuals. We show consistency of the estimator and prove a rate of convergence of $O_P(n^{-1})$ which in this case is clearly optimal as there are onl…
The pervasive use of prevalent cohort studies on disease duration, increasingly calls for appropriate methodologies to account for the biases that invariably accompany samples formed by such data. It is well-known, for example, that subjects with shorter lifetime are less likely to be present in such studies. Moreover, certain covariate values could be preferentially selected into the sample, being linked to the long-term survivors. The existing methodology for estimation of the propensity score using data collected on prevalent cases requires the correct conditional survival/hazard function g…
In this paper, we develop nonparametric inference on spatial regression models as an extension of Lu and Tj\ostheim(2014), which develops nonparametric inference on density functions of stationary spatial processes under domain expanding and infill (DEI) asymptotics. In particular, we derive multivariate central limit theorems of mean and variance functions of nonparametric spatial regression models. Built upon those results, we propose a method to construct confidence bands for mean and variance functions.
Symmetry: a General Structure in Nonparametric Regression
本文提出非参数回归中的对称性框架，推广协变量稀疏性，利用已知或可估计的对称性获得更快收敛速率，并构造部分对称化算子和两步M估计器，在合成数据上验证了有限样本性能。
This article investigates nonparametric estimation of variance functions for functional data when the mean function is unknown. We obtain asymptotic results for the kernel estimator based on squared residuals. Similar to the finite dimensional case, our asymptotic result shows the smoothness of the unknown mean function has an effect on the rate of convergence. Our simulaton studies demonstrate that estimator based on residuals performs much better than that based on conditional second moment of the responses.
Variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived. We are particularly interested in the effect of the unknown mean on the estimation of the variance function. Our results indicate that, contrary to the common practice, it is not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean when the mean function is not smooth. Instead it is more desirable to use estimators of the mean with minimal bias. On the other hand, when the mean function is very smooth, our numerical re…
We consider heteroscedastic nonparametric regression models, when both the mean function and variance function are unknown and to be estimated with nonparametric approaches. We derive convergence rates of posterior distributions for this model with different priors, including splines and Gaussian process priors. The results are based on the general ones on the rates of convergence of posterior distributions for independent, non-identically distributed observations, and are established for both of the cases with random covariates, and deterministic covariates. We also illustrate that the result…
In many epidemiological contexts, disease occurrences and their rates are naturally modelled by counting processes and their intensities, allowing an analysis based on martingale methods. These methods lend themselves to extensions of nested case-control sampling designs where general methods of control selection can be easily incorporated. This same methodology allows for extensions of the Mantel-Haenszel estimator in two main directions. First, a variety of new sampling designs can be incorporated which can yield substantial efficiency gains over simple random sampling. Second, the extension…
Partial heteroscedastic deconvolution estimation in nonparametric regression
本文提出了一种非参数回归中处理部分异方差测量误差的核估计方法，证明其在合适条件下具有最优收敛速率，并通过模拟验证了其有效性。
On a synergistic learning phenomenon in nonparametric domain adaptation
本文研究了非参数领域自适应中源数据与目标数据协同学习现象，发现联合使用数据可提升回归函数估计的收敛速率。
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,…
Adaptive thresholding for wavelet-based nonparametric heteroskedastic variance estimation on the sphere
研究单位球面上空间变化的异方差函数非参数估计问题，提出基于局部球面框架、结合硬阈值协议和样本分割方案的全数据驱动多分辨率估计器，证明其在标准损失函数下达到极小极大最优收敛率，有空间适应性，在多领域有潜在应用。
Local convergence rates of the nonparametric least squares estimator with applications to transfer learning
本文研究了非参数最小二乘估计在加权一致范数下的局部收敛速率，证明其极小极大最优性，并将所得速率应用于协变量偏移下的迁移学习分析。
This paper considers multiple regression procedures for analyzing the relationship between a response variable and a vector of covariates in a nonparametric setting where both tuning parameters and the number of covariates need to be selected. We introduce an approach which handles the dilemma that with high dimensional data the sparsity of data in regions of the sample space makes estimation of nonparametric curves and surfaces virtually impossible. This is accomplished by abandoning the goal of trying to estimate true underlying curves and instead estimating measures of dependence that can d…
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