We consider the problem of predicting values of a random process or field satisfying a linear model $y(x)=θ^\top f(x) + \varepsilon(x)$, where errors $\varepsilon(x)$ are correlated. This is a common problem in kriging, where the case of discrete observations is standard. By focussing on the case of continuous observations, we derive expressions for the best linear unbiased predictors and their mean squared error. Our results are also applicable in the case where the derivatives of the process $y$ are available, and either a response or one of its derivatives need to be predicted. The theoretical results are illustrated by several examples in particular for the popular Matérn $3/2$ kernel.
In multivariate regression estimation, the rate of convergence depends on the dimension of the regressor. This fact, known as the curse of the dimensionality, motivated several works. The additive model, introduced by Stone (10), offers an efficient response to this problem. In the setting of continuous time processes, using the marginal integration method, we obtain the quadratic convergence rate and the asymptotic normality of the components of the additive model.
This paper constructs a doubly robust estimator for continuous dose-response estimation. An outcome regression model is augmented with a set of inverse generalized propensity score covariates to correct for potential misspecification bias. From the augmented model we can obtain consistent estimates of mean average potential outcomes for distinct strata of the treatment. A polynomial regression is then fitted to these point estimates to derive a Taylor approximation to the continuous dose-response function. The bootstrap is used for variance estimation. Analytical results and simulations show t…
Associated kernels have been introduced to improve the classical continuous kernels for smoothing any functional on several kinds of supports such as bounded continuous and discrete sets. This work deals with the effects of combined associated kernels on nonparametric multiple regression functions. Using the Nadaraya-Watson estimator with optimal bandwidth matrices selected by cross-validation procedure, different behaviours of multiple regression estimations are pointed out according the type of multivariate associated kernels with correlation or not. Through simulation studies, there are no …
In this article we consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov noises observed in discrete time moments. An adaptive model selection procedure is proposed. A sharp non-asymptotic oracle inequality for the robust risks is obtained. We obtain sufficient conditions on the frequency observations under which the robust efficiency is shown. It turns out that for the semi-Markov models the robust minimax convergence rate may be faster or slower than the classical one.
This paper considers the problem of robust adaptive efficient estimating of a periodic function in a continuous time regression model with the dependent noises given by a general square integrable semimartingale with a conditionally Gaussian distribution. An example of such noise is the non-Gaussian Ornstein-Uhlenbeck-Levy processes. An adaptive model selection procedure, based on the improved weighted least square estimates, is proposed. Under some conditions on the noise distribution, sharp oracle inequality for the robust risk has been proved and the robust efficiency of the model selection…
In the setting of additive regression model for continuous time process, we establish the optimal uniform convergence rates and optimal asymptotic quadratic error of additive regression. To build our estimate, we use the marginal integration method.
A continuous-time nonlinear regression model with Lévy-driven linear noise process is considered. Sufficient conditions of consistency and asymptotic normality of the Whittle estimator for the parameter of the noise spectral density are obtained in the paper.
We show that prediction performance for global-local shrinkage regression can overcome two major difficulties of global shrinkage regression: (i) the amount of relative shrinkage is monotone in the singular values of the design matrix and (ii) the shrinkage is determined by a single tuning parameter. Specifically, we show that the horseshoe regression, with heavy-tailed component-specific local shrinkage parameters, in conjunction with a global parameter providing shrinkage towards zero, alleviates both these difficulties and consequently, results in an improved risk for prediction. Numerical …
The TREX is a recently introduced approach to sparse linear regression. In contrast to most well-known approaches to penalized regression, the TREX can be formulated without the use of tuning parameters. In this paper, we establish the first known prediction error bounds for the TREX. Additionally, we introduce extensions of the TREX to a more general class of penalties, and we provide a bound on the prediction error in this generalized setting. These results deepen the understanding of TREX from a theoretical perspective and provide new insights into penalized regression in general.
The subject of this work is two treatment groups random coefficient regression models, in which observational units receive some group-specific treatments. We provide A- and D-optimality criteria for the estimation of the fixed parameter and the prediction of the random effects. We illustrate the behavior of optimal designs by a simple example.
We tackle estimation and prediction at non-visted sites in a spatial semi-functional linear regression model with derivatives that combines a functional linear model with a nonparametric regression one. The parametric part is estimated by a method of moments and the other one by a local linear estimator. We establish the convergence rate of the resulting estimators and predictor. A simulation study and an application to ozone pollution prediction at non-visted sites are proposed to illustrate our results.
We analyse the prediction error of principal component regression (PCR) and prove non-asymptotic upper bounds for the corresponding squared risk. Under mild assumptions, we show that PCR performs as well as the oracle method obtained by replacing empirical principal components by their population counterparts. Our approach relies on upper bounds for the excess risk of principal component analysis.
We consider a spatial functional linear regression, where a scalar response is related to a square integrable spatial functional process. We use a smoothing spline estimator for the functional slope parameter and establish a finite sample bound for variance of this estimator under mixing spatial dependence. Then, we give a bound of the prediction error. Finally, we illustrate our results by simulations
In this paper we develop the James - Stein improved estimation method for a nonparametric periodic function observed with the Levy noises in continuous time. An adaptive model selection procedure based on the improved weighted least square estimates is constructed. The improvement effect for the nonparametric models is obtained. It turns out that in the nonasymptotic studies the accuracy improvement for nonparametric problems is more significantly than for the parametric one. Moreover, sharp oracle inequalities for the robust risks have been shown and the efficiency property for the improved m…
In this paper the problem of best linear unbiased estimation is investigated for continuous-time regression models. We prove several general statements concerning the explicit form of the best linear unbiased estimator (BLUE), in particular when the error process is a smooth process with one or several derivatives of the response process available for construction of the estimators. We derive the explicit form of the BLUE for many specific models including the cases of continuous autoregressive errors of order two and integrated error processes (such as integrated Brownian motion). The results…
We provide new theoretical results in the field of inverse regression methods for dimension reduction. Our approach is based on the study of some empirical processes that lie close to a certain dimension reduction subspace, called the central subspace. The study of these processes essentially includes weak convergence results and the consistency of some general bootstrap procedures. While such properties are used to obtain new results about sliced inverse regression, they mainly allow to define a natural family of methods for dimension reduction. First the estimation methods are shown to have …
Prediction error is critical to assessing the performance of statistical methods and selecting statistical models. We propose the cross-validation and approximated cross-validation methods for estimating prediction error under a broad q-class of Bregman divergence for error measures which embeds nearly all of the commonly used loss functions in regression, classification procedures and machine learning literature. The approximated cross-validation formulas are analytically derived, which facilitate fast estimation of prediction error under the Bregman divergence. We then study a data-driven op…
In data analysis problems where we are not able to rely on distributional assumptions, what types of inference guarantees can still be obtained? Many popular methods, such as holdout methods, cross-validation methods, and conformal prediction, are able to provide distribution-free guarantees for predictive inference, but the problem of providing inference for the underlying regression function (for example, inference on the conditional mean $\mathbb{E}[Y|X]$) is more challenging. In the setting where the features $X$ are continuously distributed, recent work has established that any confidence…
Most data sets comprise of measurements on continuous and categorical variables. In regression and classification Statistics literature, modeling high-dimensional mixed predictors has received limited attention. In this paper we study the general regression problem of inferring on a variable of interest based on high dimensional mixed continuous and binary predictors. The aim is to find a lower dimensional function of the mixed predictor vector that contains all the modeling information in the mixed predictors for the response, which can be either continuous or categorical. The approach we pro…
We consider Bayesian shrinkage predictions for the Normal regression problem under the frequentist Kullback-Leibler risk function. Firstly, we consider the multivariate Normal model with an unknown mean and a known covariance. While the unknown mean is fixed, the covariance of future samples can be different from training samples. We show that the Bayesian predictive distribution based on the uniform prior is dominated by that based on a class of priors if the prior distributions for the covariance and future covariance matrices are rotation invariant. Then, we consider a class of priors for t…
In the one-parameter regression model with AR(1) and AR(2) errors we find explicit expressions and a continuous approximation of the optimal discrete design for the signed least square estimator. The results are used to derive the optimal variance of the best linear estimator in the continuous time model and to construct efficient estimators and corresponding optimal designs for finite samples. The resulting procedure (estimator and design) provides nearly the same efficiency as the weighted least squares and its variance is close to the optimal variance in the continuous time model. The resul…
In this paper, we are interested in nonparametric kernel estimation of a generalized regression function, including conditional cumulative distribution and conditional quantile functions, based on an incomplete sample $(X_t, Y_t, ζ_t)_{t\in \mathbb{ R}^+}$ copies of a continuous-time stationary ergodic process $(X, Y, ζ)$. The predictor $X$ is valued in some infinite-dimensional space, whereas the real-valued process $Y$ is observed when $ζ= 1$ and missing whenever $ζ= 0$. Pointwise and uniform consistency (with rates) of these estimators as well as a central limit theorem are established. Con…
We tackle the problem of building a prediction interval in heteroscedastic Gaussian regression. We focus on prediction intervals with constrained expected length in order to guarantee interpretability of the output. In this framework, we derive a closed form expression of the optimal prediction interval that allows for the development a data-driven prediction interval based on plug-in. The construction of the proposed algorithm is based on two samples, one labeled and another unlabeled. Under mild conditions, we show that our procedure is asymptotically as good as the optimal prediction interv…
We consider the on-line predictive version of the standard problem of linear regression; the goal is to predict each consecutive response given the corresponding explanatory variables and all the previous observations. We are mainly interested in prediction intervals rather than point predictions. The standard treatment of prediction intervals in linear regression analysis has two drawbacks: (1) the classical prediction intervals guarantee that the probability of error is equal to the nominal significance level epsilon, but this property per se does not imply that the long-run frequency of err…
As with classic statistics, functional regression models are invaluable in the analysis of functional data. While there are now extensive tools with accompanying theory available for linear models, there is still a great deal of work to be done concerning nonlinear models for functional data. In this work we consider the Additive Function-on-Function Regression model, a type of nonlinear model that uses an additive relationship between the functional outcome and functional covariate. We present an estimation methodology built upon Reproducing Kernel Hilbert Spaces, and establish optimal rates …
Predict a new response from a covariate is a challenging task in regression, which raises new question since the era of high-dimensional data. In this paper, we are interested in the inverse regression method from a theoretical viewpoint. Theoretical results have already been derived for the well-known linear model, but recently, the curse of dimensionality has increased the interest of practitioners and theoreticians into generalization of those results for various estimators, calibrated for the high-dimension context. To deal with high-dimensional data, inverse regression is used in this pap…
High-dimensional prediction typically comprises two steps: variable selection and subsequent least-squares refitting on the selected variables. However, the standard variable selection procedures, such as the lasso, hinge on tuning parameters that need to be calibrated. Cross-validation, the most popular calibration scheme, is computationally costly and lacks finite sample guarantees. In this paper, we introduce an alternative scheme, easy to implement and both computationally and theoretically efficient.
We consider the problem of link prediction, based on partial observation of a large network, and on side information associated to its vertices. The generative model is formulated as a matrix logistic regression. The performance of the model is analysed in a high-dimensional regime under a structural assumption. The minimax rate for the Frobenius-norm risk is established and a combinatorial estimator based on the penalised maximum likelihood approach is shown to achieve it. Furthermore, it is shown that this rate cannot be attained by any (randomised) algorithm computable in polynomial time un…
We study online prediction of bounded stationary ergodic processes. To do so, we consider the setting of prediction of individual sequences and build a deterministic regression tree that performs asymptotically as well as the best L-Lipschitz constant predictors. Then, we show why the obtained regret bound entails the asymptotical optimality with respect to the class of bounded stationary ergodic processes.
In this paper we adopt the familiar sparse, high-dimensional linear regression model and focus on the important but often overlooked task of prediction. In particular, we consider a new empirical Bayes framework that incorporates data in the prior in two ways: one is to center the prior for the non-zero regression coefficients and the other is to provide some additional regularization. We show that, in certain settings, the asymptotic concentration of the proposed empirical Bayes posterior predictive distribution is very fast, and we establish a Bernstein--von Mises theorem which ensures that …
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…
An important problem in network analysis is predicting a node attribute using both network covariates, such as graph embedding coordinates or local subgraph counts, and conventional node covariates, such as demographic characteristics. While standard regression methods that make use of both types of covariates may be used for prediction, statistical inference is complicated by the fact that the nodal summary statistics are often dependent in complex ways. We show that under a mild joint exchangeability assumption, a network analog of conformal prediction achieves finite sample validity for a w…
This work is devoted to the finite sample prediction risk analysis of a class of linear predictors of a response $Y\in \mathbb{R}$ from a high-dimensional random vector $X\in \mathbb{R}^p$ when $(X,Y)$ follows a latent factor regression model generated by a unobservable latent vector $Z$ of dimension less than $p$. Our primary contribution is in establishing finite sample risk bounds for prediction with the ubiquitous Principal Component Regression (PCR) method, under the factor regression model, with the number of principal components adaptively selected from the data -- a form of theoretical…
