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
An adaptive nonparametric estimation procedure is constructed for the estimation problem of heteroscedastic regression when the noise variance depends on the unknown regression. A non-asymptotic upper bound for a quadratic risk (an oracle inequality) is constructed.
Functional linear regression analysis aims to model regression relations which include a functional predictor. The analog of the regression parameter vector or matrix in conventional multivariate or multiple-response linear regression models is a regression parameter function in one or two arguments. If, in addition, one has scalar predictors, as is often the case in applications to longitudinal studies, the question arises how to incorporate these into a functional regression model. We study a varying-coefficient approach where the scalar covariates are modeled as additional arguments of the …
We establish the asymptotic normality of the kernel type estimator for the regression function constructed from quasi-associated data when the explanatory variable takes its values in a separable Hilbert space.
This paper deals with the estimation of hidden periodicities in a non-linear regression model with stationary noise displaying cyclical dependence. Consistency and asymptotic normality are established for the least-squares estimates.
This manuscript is a supplemental document providing the omitted material for our paper entitled "Nonparametric kernel estimation of the probability density function of regression errors using estimated residuals" [arXiv:1010.0439]. The paper is submitted to Journal of Nonparametric Statistics.
In this study, we develop an asymptotic theory of nonparametric regression for a locally stationary functional time series. First, we introduce the notion of a locally stationary functional time series (LSFTS) that takes values in a semi-metric space. Then, we propose a nonparametric model for LSFTS with a regression function that changes smoothly over time. We establish the uniform convergence rates of a class of kernel estimators, the Nadaraya-Watson (NW) estimator of the regression function, and a central limit theorem of the NW estimator.
We study regression models for the situation where both dependent and independent variables are square-integrable stochastic processes. Questions concerning the definition and existence of the corresponding functional linear regression models and some basic properties are explored for this situation. We derive a representation of the regression parameter function in terms of the canonical components of the processes involved. This representation establishes a connection between functional regression and functional canonical analysis and suggests alternative approaches for the implementation of…
The paper deals with asymptotic properties of the adaptive procedure proposed in the author paper (2007) for estimation of unknown nonparametric regression. We prove that this procedure is asymptotically efficient for a quadratic risk. It means that the asymptotic quadratic risk for this procedure coincides with a sharp lower bound.
Estimation for the log-logistic and Weibull distributions can be performed by using the equations used for probability plotting. The equations leads to highly heteroscedastic regression. Exact expressions for the variances of the residuals are derived which can be used to perform weighted regression. In large samples maximum likelihood performs best, but it is shown that in smaller samples the weighted regression outperforms maximum likelihood estimation with respect to bias and mean square error.
This paper is devoted to the nonparametric estimation of the derivative of the regression function in a nonparametric regression model. We implement a very efficient and easy to handle statistical procedure based on the derivative of the recursive Nadaraya-Watson estimator. We establish the almost sure convergence as well as the asymptotic normality for our estimates. We also illustrate our nonparametric estimation procedure on simulated and real life data associated with sea shores water quality and valvometry.
We consider the related problems of estimating the $l_2$-norm and the squared $l_2$-norm in sparse linear regression with unknown variance, as well as the problem of testing the hypothesis that the regression parameter is null under sparse alternatives with $l_2$ separation. We establish the minimax optimal rates of estimation (respectively, testing) in these three problems.
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 introduce and study a local linear nonparametric regression estimator for censorship model. The main goal of this paper is, to establish the uniform almost sure consistency result with rate over a compact set for the new estimate. To support our theoretical result, a simulation study has been done to make comparison with the classical regression estimator.
A nonparametric procedure for robust regression estimation and for quantile regression is proposed which is completely data-driven and adapts locally to the regularity of the regression function. This is achieved by considering in each point M-estimators over different local neighbourhoods and by a local model selection procedure based on sequential testing. Non-asymptotic risk bounds are obtained, which yield rate-optimality for large sample asymptotics under weak conditions. Simulations for different univariate median regression models show good finite sample properties, also in comparison t…
We introduce a new model of linear regression for random functional inputs taking into account the first order derivative of the data. We propose an estimation method which comes down to solving a special linear inverse problem. Our procedure tackles the problem through a double and synchronized penalization. An asymptotic expansion of the mean square prevision error is given. The model and the method are applied to a benchmark dataset of spectrometric curves and compared with other functional models.
We propose a novel test statistic for testing exogeneity in the functional linear regression model. In contrast to Hausman-type tests in finite dimensional linear regression setups, a direct extension to the functional linear regression model is not possible. Instead, we propose a test statistic based on the sum of the squared difference of projections of the two estimators for testing the null hypothesis of exogeneity in the functional linear regression model. We derive asymptotic normality under the null and consistency under general alternatives. Moreover, we prove bootstrap consistency res…
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.
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.
This paper is devoted to the estimation of the shift parameter in a semiparametric regression model when the distribution of the observation times is unknown. Hence, we propose to use a stochastic algorithm which takes into account the estimation of the distribution of the observation times. We establish the almost sure convergence of our estimator and the asymptotic normality. The main result of the paper is that, with little assumptions on the regularity of the regression function, the asymptotic variance obtained is the same as when the distribution is known. In that sense, we improve the r…
This paper studies density estimation and regression analysis with contaminated data observed on the unit hypersphere S^d. Our methodology and theory are based on harmonic analysis on general S^d. We establish novel nonparametric density and regression estimators, and study their asymptotic properties including the rates of convergence and asymptotic distributions. We also provide asymptotic confidence intervals based on the asymptotic distributions of the estimators and on the empirical likelihood technique. We present practical details on implementation as well as the results of numerical st…
We propose in this work to derive a CLT in the functional linear regression model to get confidence sets for prediction based on functional linear regression. The main difficulty is due to the fact that estimation of the functional parameter leads to a kind of ill-posed inverse problem. We consider estimators that belong to a large class of regularizing methods and we first show that, contrary to the multivariate case, it is not possible to state a CLT in the topology of the considered functional space. However, we show that we can get a CLT for the weak topology under mild hypotheses and in p…
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 study a non linear regression model with functional data as inputs and scalar response. We propose a pointwise estimate of the regression function that maps a Hilbert space onto the real line by a local linear method. We provide the asymptotic mean square error. Computations involve a linear inverse problem as well as a representation of the small ball probability of the data and are based on recent advances in this area. The rate of convergence of our estimate outperforms those already obtained in the literature on this model.
We consider a re-sampling scheme for estimation of the population parameters in the mixed effects nonlinear regression models of the type use for example in clinical pharmacokinetics, say. We provide an estimation procedure which {\it recycles}, via random weighting, the relevant two-stage parameters estimates to construct consistent estimates of the sampling distribution of the various estimates. We establish the asymptotic consistency and asymptotic normality of the resampled estimates and demonstrate the applicability of the {\it recycling} approach in a small simulation study and via examp…
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.
A general theory for Gaussian mean estimation that automatically adapts to unknown sparsity under arbitrary norms is proposed. The theory is applied to produce adaptively minimax rate-optimal estimators in high dimensional regression and matrix estimation that involve no tuning parameters.
We consider the robust adaptive nonparametric estimation problem for a periodic function observed in the framework of a continuous time regression model with semimartingale noises.
Motivated by finance and technical applications, the objective of this paper is to consider adaptive estimation of regression and density distribution based on Fourier-Legendre expansion, and construction of confidence intervals - also adaptive. The estimators are asymptotically optimal and adaptive in the sense that they can adapt to unknown smoothness.
In this paper, we study statistical inference in functional quantile regression for scalar response and a functional covariate. Specifically, we consider a functional linear quantile regression model where the effect of the covariate on the quantile of the response is modeled through the inner product between the functional covariate and an unknown smooth regression parameter function that varies with the level of quantile. The objective is to test that the regression parameter is constant across several quantile levels of interest. The parameter function is estimated by combining ideas from f…
Regressing a scalar response on a random function is nowadays a common situation. In the nonparametric setting, this paper paves the way for making the local linear regression based on a projection approach a prominent method for solving this regression problem. Our asymptotic results demonstrate that the functional local linear regression outperforms its functional local constant counterpart. Beyond the estimation of the regression operator itself, the local linear regression is also a useful tool for predicting the functional derivative of the regression operator, a promising mathematical ob…
This paper is devoted to the parametric estimation of a shift together with the nonparametric estimation of a regression function in a semiparametric regression model. We implement a very efficient and easy to handle Robbins-Monro procedure. On the one hand, we propose a stochastic algorithm similar to that of Robbins-Monro in order to estimate the shift parameter. A preliminary evaluation of the regression function is not necessary to estimate the shift parameter. On the other hand, we make use of a recursive Nadaraya-Watson estimator for the estimation of the regression function. This kernel…
We investigate the nonparametric estimation for regression in a fixed-design setting when the errors are given by a field of dependent random variables. Sufficient conditions for kernel estimators to converge uniformly are obtained. These estimators can attain the optimal rates of uniform convergence and the results apply to a large class of random fields which contains martingale-difference random fields and mixing random fields.
We consider nonparametric regression in the context of functional data, that is, when a random sample of functions is observed on a fine grid. We obtain a functional asymptotic normality result allowing to build simultaneous confidence bands (SCB) for various estimation and inference tasks. Two applications to a SCB procedure for the regression function and to a goodness-of-fit test for curvilinear regression models are proposed. The first one has improved accuracy upon the other available methods while the second can detect local departures from a parametric shape, as opposed to the usual goo…
We study a functional linear regression model that deals with functional responses and allows for both functional covariates and high-dimensional vector covariates. The proposed model is flexible and nests several functional regression models in the literature as special cases. Based on the theory of reproducing kernel Hilbert spaces (RKHS), we propose a penalized least squares estimator that can accommodate functional variables observed on discrete sample points. Besides a conventional smoothness penalty, a group Lasso-type penalty is further imposed to induce sparsity in the high-dimensional…
Shape restrictions on functional regression coefficients such as non-negativity, monotonicity, convexity or concavity are often available in the form of a prior knowledge or required to maintain a structural consistency in functional regression models. A new estimation method is developed in shape-constrained functional regression models using Bernstein polynomials. Specifically, estimation approaches from nonparametric regression are extended to functional data, properly accounting for shape-constraints in a large class of functional regression models such as scalar-on-function regression (SO…
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 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.
Finite mixtures of regression models provide a flexible modeling framework for many phenomena. Using moment-based estimation of the regression parameters, we develop unbiased estimators with a minimum of assumptions on the mixture components. In particular, only the average regression model for one of the components in the mixture model is needed and no requirements on the distributions. The consistency and asymptotic distribution of the estimators is derived and the proposed method is validated through a series of simulation studies and is shown to be highly accurate. We illustrate the use of…
We develop uniformly valid confidence regions for regression coefficients in a high-dimensional sparse median regression model with homoscedastic errors. Our methods are based on a moment equation that is immunized against non-regular estimation of the nuisance part of the median regression function by using Neyman's orthogonalization. We establish that the resulting instrumental median regression estimator of a target regression coefficient is asymptotically normally distributed uniformly with respect to the underlying sparse model and is semi-parametrically efficient. We also generalize our …
We want to analyse EEG recordings in order to investigate the phonemic categorization at a very early stage of auditory processing. This problem can be modelled by a supervised classification of functional data. Discrimination is explored via a logistic functional linear model, using a wavelet representation of the data. Different procedures are investigated, based on penalized likelihood and principal component reduction or partial least squares reduction.
In this paper, we consider an unknown functional estimation problem in a general nonparametric regression model with the feature of having both multiplicative and additive noise.We propose two new wavelet estimators in this general context. We prove that they achieve fast convergence rates under the mean integrated square error over Besov spaces. The obtained rates have the particularity of being established under weak conditions on the model. A numerical study in a context comparable to stochastic frontier estimation (with the difference that the boundary is not necessarily a production funct…
This paper studies the non-asymptotic merits of the double $\ell_1$-regularized for heterogeneous overdispersed count data via negative binomial regressions. Under the restricted eigenvalue conditions, we prove the oracle inequalities for Lasso estimators of two partial regression coefficients for the first time, using concentration inequalities of empirical processes. Furthermore, derived from the oracle inequalities, the consistency and convergence rate for the estimators are the theoretical guarantees for further statistical inference. Finally, both simulations and a real data analysis demo…
Functional linear regression is a useful extension of simple linear regression and has been investigated by many researchers. However, functional variable selection problems when multiple functional observations exist, which is the counterpart in the functional context of multiple linear regression, is seldom studied. Here we propose a method using group smoothly clipped absolute deviation penalty (gSCAD) which can perform regression estimation and variable selection simultaneously. We show the method can identify the true model consistently and discuss construction of pointwise confidence int…
A general method of minimization using correlation coefficients and order statistics is evaluated relative to least squares procedures in the estimation of parameters for normal data in simple linear regression.
The R add-on package FDboost is a flexible toolbox for the estimation of functional regression models by model-based boosting. It provides the possibility to fit regression models for scalar and functional response with effects of scalar as well as functional covariates, i.e., scalar-on-function, function-on-scalar and function-on-function regression models. In addition to mean regression, quantile regression models as well as generalized additive models for location scale and shape can be fitted with FDboost. Furthermore, boosting can be used in high-dimensional data settings with more covari…
In this paper, we apply empirical likelihood method to inference for the regression parameters in the partial functional linear regression models based on B spline. We prove that the empirical log likelihood ratio for the regression parameters converges in law to a weighted sum of independent chi square distributions and run simulations to assess the finite sample performance of our method.
For a class of parametric modal regression models with measurement error, a simulation extrapolation estimation procedure is proposed in this paper for estimating the modal regression coefficients. Large sample properties of the proposed estimation procedure, including the consistency and asymptotic normality, are thoroughly investigated. Simulation studies are conducted to evaluate its robustness to potential outliers and the effectiveness in reducing the bias caused by the measurement error.
We investigate nonparametric regression methods based on spatial depth and quantiles when the response and the covariate are both functions. As in classical quantile regression for finite dimensional data, regression techniques developed here provide insight into the influence of the functional covariate on different parts, like the center as well as the tails, of the conditional distribution of the functional response. Depth and quantile based nonparametric regressions are useful to detect heteroscedasticity in functional regression. We derive the asymptotic behaviour of nonparametric depth a…
We 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…
A characteristic feature of functional data is the presence of phase variability in addition to amplitude variability. Existing functional regression methods do not handle time variability in an explicit and efficient way. In this paper we introduce a functional regression method that incorporates time warping as an intrinsic part of the model. The method achieves good predictive power in a parsimonious way and allows unified statistical inference about phase and amplitude components. The asymptotic distribution of the estimators is derived and the finite-sample properties are studied by simul…
In the functional linear regression model, many methods have been proposed and studied to estimate the slope function while the functional predictor was observed in the entire domain. However, works on functional linear regression models with partially observed trajectories have received less attention. In this paper, to fill the literature gap we consider the scenario where individual functional predictor may be observed only on part of the domain. Depending on whether measurement error is presented in functional predictors, two methods are developed, one is based on linear functionals of the…
We propose a prediction procedure for the functional linear quantile regression model by using partial quantile covariance techniques and develop a simple partial quantile regression (SIMPQR) algorithm to efficiently extract partial quantile regression (PQR) basis for estimating functional coefficients. We further extend our partial quantile covariance techniques to functional composite quantile regression (CQR) defining partial composite quantile covariance. There are three major contributions. (1) We define partial quantile covariance between two scalar variables through linear quantile regr…
We place ourselves in a functional regression setting and propose a novel methodology for regressing a real output on vector-valued functional covariates. This methodology is based on the notion of signature, which is a representation of a function as an infinite series of its iterated integrals. The signature depends crucially on a truncation parameter for which an estimator is provided, together with theoretical guarantees. An empirical study on both simulated and real-world datasets shows that the resulting methodology is competitive with traditional functional linear models, in particular …
