In this paper we consider the problem of bootstrapping a class of spatial regression models when the sampling sites are generated by a (possibly nonuniform) stochastic design and are irregularly spaced. It is shown that the natural extension of the existing block bootstrap methods for grid spatial data does not work for irregularly spaced spatial data under nonuniform stochastic designs. A variant of the blocking mechanism is proposed. It is shown that the proposed block bootstrap method provides a valid approximation to the distribution of a class of M-estimators of the spatial regression parameters. Finite sample properties of the method are investigated through a moderately large simulation study and a real data example is given to illustrate the methodology.
We propose a novel resampling-based method to construct an asymptotically exact test for any subset of hypotheses on coefficients in high-dimensional linear regression. It can be embedded into any multiple testing procedure to make confidence statements on relevant predictor variables. The method constructs permutation test statistics for any individual hypothesis by means of repeated splits of the data and a variable selection technique; then it defines a test for any subset by suitably aggregating its variables' test statistics. The resulting procedure is extremely flexible, as it allows dif…
We investigate the validity of two resampling techniques when carrying out inference on the underlying unknown copula using a recently proposed class of smooth, possibly data-adaptive nonparametric estimators that contains empirical Bernstein copulas (and thus the empirical beta copula). Following \cite{KirSegTsu21}, the first resampling technique is based on drawing samples from the smooth estimator and can only can be used in the case of independent observations. The second technique is a smooth extension of the so-called sequential dependent multiplier bootstrap and can thus be used in a ti…
In non-linear estimations, it is common to assess sampling uncertainty by bootstrap inference. For complex models, this can be computationally intensive. This paper combines optimization with resampling: turning stochastic optimization into a fast resampling device. Two methods are introduced: a resampled Newton-Raphson (rNR) and a resampled quasi-Newton (rqN) algorithm. Both produce draws that can be used to compute consistent estimates, confidence intervals, and standard errors in a single run. The draws are generated by a gradient and Hessian (or an approximation) computed from batches of d…
We consider a resampling scheme for parameters estimates in nonlinear regression models. We provide an estimation procedure which recycles, via random weighting, the relevant parameters estimates to construct consistent estimates of the sampling distribution of the various estimates. We establish the asymptotic normality of the resampled estimates and demonstrate the applicability of the recycling approach in a small simulation study and via example.
We study prediction intervals based on leave-one-out residuals in a linear regression model where the number of explanatory variables can be large compared to sample size. We establish uniform asymptotic validity (conditional on the training sample) of the proposed interval under minimal assumptions on the unknown error distribution and the high dimensional design. Our intervals are generic in the sense that they are valid for a large class of linear predictors used to obtain a point forecast, such as robust M-estimators, James-Stein type estimators and penalized estimators like the LASSO. The…
We consider goodness-of-fit tests for the distribution of the composed error in Stochastic Frontier Models. The proposed test statistic utilizes the characteristic function of the composed error term, and is formulated as a weighted integral of properly standardized data. The new test statistic is shown to be consistent and computationally convenient. Simulation results are presented whereby resampling versions of the new tests are compared to classical goodness-of-fit methods.
We consider nonsynchronous sampling of parameterized stochastic regression models, which contain stochastic differential equations. Constructing a quasi-likelihood function, we prove that the quasi-maximum likelihood estimator and the Bayes type estimator are consistent and asymptotically mixed normal when the sampling frequency of the nonsynchronous data becomes large.
In this paper we consider the problem of constructing confidence intervals for coefficients of martingale regression models (in particular, time series models) after variable selection. Although constructing confidence intervals are common practice in statistical analysis, it is challenging in our framework due to the data-dependence of the selected model and the correlation among the variables being selected and not selected. We first introduce estimators for the selected coefficients and show that it is consistent under martingale regression model, in which the observations can be dependent …
Linear regression is perhaps one of the most popular statistical concepts, which permeates almost every scientific field of study. Due to the technical simplicity and wide applicability of linear regression, attention is almost always quickly directed to the algorithmic or computational side of linear regression. In particular, the underlying mathematics of stochastic linear regression itself as an entity usually gets either a peripheral treatment or a relatively in-depth but ad hoc treatment depending on the type of concerned problems; in other words, compared to the extensiveness of the stud…
Nonparametric density and regression estimators commonly depend on a bandwidth. The asymptotic properties of these estimators have been widely studied when bandwidths are nonstochastic. In practice, however, in order to improve finite sample performance of these estimators, bandwidths are selected by data driven methods, such as cross-validation or plug-in procedures. As a result nonparametric estimators are usually constructed using stochastic bandwidths. In this paper we establish the asymptotic equivalence in probability of local polynomial regression estimators under stochastic and nonstoc…
Logistic regression is a well-known statistical model which is commonly used in the situation where the output is a binary random variable. It has a wide range of applications including machine learning, public health, social sciences, ecology and econometry. In order to estimate the unknown parameters of logistic regression with data streams arriving sequentially and at high speed, we focus our attention on a recursive stochastic algorithm. More precisely, we investigate the asymptotic behavior of a new stochastic Newton algorithm. It enables to easily update the estimates when the data arriv…
In this paper we prove large deviations principles for the averaged stochastic approximation method for the estimation of a regression function introduced by A. Mokkadem et al. [Revisiting Révész's stochastic approximation method for the estimation of a regression function, ALEA Lat. Amm. J. Probab. Math. Stat. 6 (2009), 63-114]. We show that the averaged stochastic approximation algorithm constructed using the weight sequence which minimize the asymptotic variance gives the same pointwise LDP as the Nadaraya-Watson kernel estimator. Moreover, we give a moderate deviations principle for these …
Nonparametric resampling methods such as Direct Sampling are powerful tools to simulate new datasets preserving important data features such as spatial patterns from observed datasets while using only minimal assumptions. However, such methods cannot generate extreme events beyond the observed range of data values. We here propose using tools from extreme value theory for stochastic processes to extrapolate observed data towards yet unobserved high quantiles. Original data are first enriched with new values in the tail region, and then classical resampling algorithms are applied to enriched da…
Non linear regression models are a standard tool for modeling real phenomena, with several applications in machine learning, ecology, econometry... Estimating the parameters of the model has garnered a lot of attention during many years. We focus here on a recursive method for estimating parameters of non linear regressions. Indeed, these kinds of methods, whose most famous are probably the stochastic gradient algorithm and its averaged version, enable to deal efficiently with massive data arriving sequentially. Nevertheless, they can be, in practice, very sensitive to the case where the eigen…
In this article we study the estimation of the location of jump points in the first derivative (referred to as kinks) of a regression function μin two random design models with different long-range dependent (LRD) structures. The method is based on the zero-crossing technique and makes use of high-order kernels. The rate of convergence of the estimator is contingent on the level of dependence and the smoothness of the regression function μ. In one of the models, the convergence rate is the same as the minimax rate for kink estimation in the fixed design scenario with i.i.d. errors which sugges…
We present a practical approach for computing the sandwich variance estimator in two-stage regression model settings. As a motivating example for two-stage regression, we consider regression calibration, a popular approach for addressing covariate measurement error. The sandwich variance approach has been rarely applied in regression calibration, despite that it requires less computation time than popular resampling approaches for variance estimation, specifically the bootstrap. This is likely due to requiring specialized statistical coding. In practice, a simple bootstrap approach with Wald c…
We propose a new procedure for inference on optimal treatment regimes in the model-free setting, which does not require to specify an outcome regression model. Existing model-free estimators for optimal treatment regimes are usually not suitable for the purpose of inference, because they either have nonstandard asymptotic distributions or do not necessarily guarantee consistent estimation of the parameter indexing the Bayes rule due to the use of surrogate loss. We first study a smoothed robust estimator that directly targets the parameter corresponding to the Bayes decision rule for optimal t…
In a pioneer work, Révész (1973) introduces the stochastic approximation method to build up a recursive kernel estimator of the regression function $x\mapsto E(Y|X=x)$. However, according to Révész (1977), his estimator has two main drawbacks: on the one hand, its convergence rate is smaller than that of the nonrecursive Nadaraya-Watson's kernel regression estimator, and, on the other hand, the required assumptions on the density of the random variable $X$ are stronger than those usually needed in the framework of regression estimation. We first come back on the study of the convergence rate o…
In this paper, a new family of resampling-based penalization procedures for model selection is defined in a general framework. It generalizes several methods, including Efron's bootstrap penalization and the leave-one-out penalization recently proposed by Arlot (2008), to any exchangeable weighted bootstrap resampling scheme. In the heteroscedastic regression framework, assuming the models to have a particular structure, these resampling penalties are proved to satisfy a non-asymptotic oracle inequality with leading constant close to 1. In particular, they are asympotically optimal. Resampling…
We present a new family of model selection algorithms based on the resampling heuristics. It can be used in several frameworks, do not require any knowledge about the unknown law of the data, and may be seen as a generalization of local Rademacher complexities and $V$-fold cross-validation. In the case example of least-square regression on histograms, we prove oracle inequalities, and that these algorithms are naturally adaptive to both the smoothness of the regression function and the variability of the noise level. Then, interpretating $V$-fold cross-validation in terms of penalization, we e…
We are concerned with nonparametric hypothesis testing of time series functionals. It is known that the popular autoregressive sieve bootstrap is, in general, not valid for statistics whose (asymptotic) distribution depends on moments of order higher than two, irrespective of whether the data come from a linear time series or a nonlinear one. Inspired by nonlinear system theory we circumvent this non-validity by introducing a higher-order bootstrap scheme based on the Volterra series representation of the process. In order to estimate coefficients of such a representation efficiently, we rely …
This paper introduces and analyzes a stochastic search method for parameter estimation in linear regression models in the spirit of Beran and Millar (1987). The idea is to generate a random finite subset of a parameter space which will automatically contain points which are very close to an unknown true parameter. The motivation for this procedure comes from recent work of Duembgen, Samworth and Schuhmacher (2011) on regression models with log-concave error distributions.
We present a flexible framework for learning predictive models that approximately satisfy the equalized odds notion of fairness. This is achieved by introducing a general discrepancy functional that rigorously quantifies violations of this criterion. This differentiable functional is used as a penalty driving the model parameters towards equalized odds. To rigorously evaluate fitted models, we develop a formal hypothesis test to detect whether a prediction rule violates this property, the first such test in the literature. Both the model fitting and hypothesis testing leverage a resampled vers…
In this paper we study the consistency of different bootstrap procedures for constructing confidence intervals (CIs) for the unique jump discontinuity (change-point) in an otherwise smooth regression function in a stochastic design setting. This problem exhibits nonstandard asymptotics and we argue that the standard bootstrap procedures in regression fail to provide valid confidence intervals for the change-point. We propose a version of smoothed bootstrap, illustrate its remarkable finite sample performance in our simulation study, and prove the consistency of the procedure. The $m$ out of $n…
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 present new stochastic geometry theorems that give bounds on the probability that $m$ random data classes all contain a point in common in their convex hulls. We apply these stochastic separation theorems to obtain bounds on the probability of existence of maximum likelihood estimators in multinomial logistic regression. We also discuss connections to condition numbers for analysis of steepest descent algorithms in logistic regression and to the computation of centerpoints of data clouds.
We consider a regression modeling of the quantiles of residual life, remaining lifetime at a specific time. We propose a smoothed induced version of the existing non-smooth estimating equations approaches for estimating regression parameters. The proposed estimating equations are smooth in regression parameters, so solutions can be readily obtained via standard numerical algorithms. Moreover, the smoothness in the proposed estimating equations enables one to obtain a robust sandwich-type covariance estimator of regression estimators aided by an efficient resampling method. To handle data subje…
SZTAKI: Institute for Computer Science and Control（SZTAKI：计算机科学与控制研究所）；Eötvös Loránd Research Network（厄特沃什·罗兰研究网络）
In this paper we suggest two statistical hypothesis tests for the regression function of binary classification based on conditional kernel mean embeddings. The regression function is a fundamental object in classification as it determines both the Bayes optimal classifier and the misclassification probabilities. A resampling based framework is presented and combined with consistent point estimators of the conditional kernel mean map, in order to construct distribution-free hypothesis tests. These tests are introduced in a flexible manner allowing us to control the exact probability of type I e…
The University of Texas at Austin（得克萨斯大学奥斯汀分校）；Rice University（莱斯大学）
We present a novel statistical inference framework for convex empirical risk minimization, using approximate stochastic Newton steps. The proposed algorithm is based on the notion of finite differences and allows the approximation of a Hessian-vector product from first-order information. In theory, our method efficiently computes the statistical error covariance in $M$-estimation, both for unregularized convex learning problems and high-dimensional LASSO regression, without using exact second order information, or resampling the entire data set. We also present a stochastic gradient sampling s…
In order to overcome multicollinearity, we propose a stochastic restricted Liu-type max- imum likelihood estimator by incorporating Liu-type maximum likelihood estimator (Inan and Erdo- gan, 2013) to the logistic regression model when the linear restrictions are stochastic. We also discuss the properties of the new estimator. Moreover, we give a method to choose the biasing parameter in the new estimator. Finally, a simulation study is given to show the performance of the new estimator.
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 …
We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. This is challenging due to the non-smooth nature of the quantile regression (QR) loss function, which invalidates the use of existing methodology. The difficulties are resolved through a double-smoothing approach that is applied to the local (at each data source) and global objective functions. Despite the reliance on a delicate combination of local and global smoothing parameters, the quantile regression model is fully parametric, thereby facilitating interpretat…
Simulation study to evaluate when Plasmode simulation is superior to parametric simulation in estimating the mean squared error of the least squares estimator in linear regression
We consider the problem of inference in a linear regression model in which the relative ordering of the input features and output labels is not known. Such datasets naturally arise from experiments in which the samples are shuffled or permuted during the protocol. In this work, we propose a framework that treats the unknown permutation as a latent variable. We maximize the likelihood of observations using a stochastic expectation-maximization (EM) approach. We compare this to the dominant approach in the literature, which corresponds to hard EM in our framework. We show on synthetic data that …
Université de Toulouse（图卢兹大学）；INRA（法国国家农业研究院）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Univ. Lyon（里昂大学）；ENS de Lyon（里昂高等师范学院）
We report on an empirical study of the main strategies for quantile regression in the context of stochastic computer experiments. To ensure adequate diversity, six metamodels are presented, divided into three categories based on order statistics, functional approaches, and those of Bayesian inspiration. The metamodels are tested on several problems characterized by the size of the training set, the input dimension, the signal-to-noise ratio and the value of the probability density function at the targeted quantile. The metamodels studied reveal good contrasts in our set of experiments, enablin…
This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary Gaussian process. Two methods are developed based on 1) taking the limit of statistical linear regression of the discretised process and 2) minimising an upper bound to a cost functional. Their difference is manifested in the diffusion of the approximate processes. This in turn gives novel derivations of pre-existing Gaussian smoothers when Method 1 is used and…
The bootstrap, based on resampling, has, for several decades, been a widely used method for computing confidence intervals for applications where no exact method is available and when sample sizes are not large enough to be able to rely on easy-to-compute large-sample approximate methods, such a Wald (normal-approximation) confidence intervals. Simulation based bootstrap intervals have been proven useful in that their actual coverage probabilities are close to the nominal confidence level in small samples. Small samples analytical approximations such as the Wald method, however, tend to have c…
NTT DATA Mathematical Systems Inc.（NTT数据数学系统公司）；The University of Tokyo（东京大学）；RIKEN（理化学研究所）
We develop new stochastic gradient methods for efficiently solving sparse linear regression in a partial attribute observation setting, where learners are only allowed to observe a fixed number of actively chosen attributes per example at training and prediction times. It is shown that the methods achieve essentially a sample complexity of $O(1/\varepsilon)$ to attain an error of $\varepsilon$ under a variant of restricted eigenvalue condition, and the rate has better dependency on the problem dimension than existing methods. Particularly, if the smallest magnitude of the non-zero components o…
We study the problem of online generalized linear regression in the stochastic setting, where the label is generated from a generalized linear model with possibly unbounded additive noise. We provide a sharp analysis of the classical follow-the-regularized-leader (FTRL) algorithm to cope with the label noise. More specifically, for $σ$-sub-Gaussian label noise, our analysis provides a regret upper bound of $O(σ^2 d \log T) + o(\log T)$, where $d$ is the dimension of the input vector, $T$ is the total number of rounds. We also prove a $Ω(σ^2d\log(T/d))$ lower bound for stochastic online linear …
We consider the problem of choosing between several models in least-squares regression with heteroscedastic data. We prove that any penalization procedure is suboptimal when the penalty is a function of the dimension of the model, at least for some typical heteroscedastic model selection problems. In particular, Mallows' Cp is suboptimal in this framework. On the contrary, optimal model selection is possible with data-driven penalties such as resampling or $V$-fold penalties. Therefore, it is worth estimating the shape of the penalty from data, even at the price of a higher computational cost.…
Asymptotic Optimism of Random-Design Linear and Kernel Regression Models
Department of Statistics, Rice University（里士大学统计系）；Lawrence Berkeley National Laboratory（伯克利国家实验室）；Amazon（亚马逊公司）；Department of Statistics, the Ohio State University（俄亥俄州立大学统计系）
We design a numerical scheme for solving a Dynamic Programming equation with Malliavin weights arising from the time-discretization of backward stochastic differential equations with the integration by parts-representation of the $Z$-component by (Ann. Appl. Probab. 12 (2002) 1390-1418). When the sequence of conditional expectations is computed using empirical least-squares regressions, we establish, under general conditions, tight error bounds as the time-average of local regression errors only (up to logarithmic factors). We compute the algorithm complexity by a suitable optimization of the …
We consider a multivariate time series model which represents a high dimensional vector process as a sum of three terms: a linear regression of some observed regressors, a linear combination of some latent and serially correlated factors, and a vector white noise. We investigate the inference without imposing stationary conditions on the target multivariate time series, the regressors and the underlying factors. Furthermore we deal with the endogeneity that there exist correlations between the observed regressors and the unobserved factors. We also consider the model with nonlinear regression …
In this article, the analysis of misspecification was extended to the recently introduced stochastic restricted biased estimators when multicollinearity exists among the explanatory variables. The Stochastic Restricted Ridge Estimator (SRRE), Stochastic Restricted Almost Unbiased Ridge Estimator (SRAURE), Stochastic Restricted Liu Estimator (SRLE), Stochastic Restricted Almost Unbiased Liu Estimator (SRAULE), Stochastic Restricted Principal Component Regression Estimator (SRPCR), Stochastic Restricted r-k class estimator (SRrk) and Stochastic Restricted r-d class estimator (SRrd) were examined…
Random forest (RF) is one of the most popular methods for estimating regression functions. The local nature of the RF algorithm, based on intra-node means and variances, is ideal when errors are i.i.d. For dependent error processes like time series and spatial settings where data in all the nodes will be correlated, operating locally ignores this dependence. Also, RF will involve resampling of correlated data, violating the principles of bootstrap. Theoretically, consistency of RF has been established for i.i.d. errors, but little is known about the case of dependent errors. We propose RF-GLS,…
