The kernel regression estimation for randomly censored functional stationary ergodic data

In this paper, we investigate the asymptotic properties of the kernel estimator for non parametric regression operator when the functional stationary ergodic data with randomly censorship are considered. More precisely, we introduce the kernel-type estimator of the non parametric regression operator with the responses randomly censored and obtain the almost surely convergence with rate as well as the asymptotic normality of the estimator. As an application, the asymptotic (1 ? ζ) confidence interval of the regression operator is also presented (0 < ζ < 1). Finally, the simulation study is carried out to show the finite-sample performances of the estimator.     

Rates of strong consistencies of the regression function estimator for functional stationary ergodic data

The aim of this paper is to study both the pointwise and uniform consistencies of the kernel regression estimate and to derive also rates of convergence whenever functional stationary ergodic data are considered. More precisely, in the ergodic data setting, we consider the regression of a real random variable Y over an explanatory random variable X taking values in some semi-metric separable abstract space. While estimating the regression function using the well-known Nadaraya-Watson estimator, we establish the strong pointwise and uniform consistencies with rates. Depending on the Vapnik-Chervonenkis size of the class over which uniformity is considered, the pointwise rate of convergence may be reached in the uniform case. Notice, finally, that the ergodic data framework extends the dependence setting to cases that are not covered by the usual mixing structures.     

Conditional mode estimation for functional stationary ergodic data with responses missing at random

In this paper, we investigate the asymptotic properties of a non-parametric conditional mode estimation given a functional explanatory variable, when functional stationary ergodic data and missing at random responses are observed. First of all, we establish asymptotic properties for a conditional density estimator from which we derive almost sure convergence (with rate) and asymptotic normality of a conditional mode estimator. This new estimate take into account missing data, and a simulation study is performed to illustrate how this fact allows to get higher predictive performances than those obtained with standard estimates.     

Rate of uniform consistency for nonparametric estimates with functional variables

In this paper we investigate nonparametric estimation of some functionals of the conditional distribution of a scalar response variable Y given a random variable X taking values in a semi-metric space. These functionals include the regression function, the conditional cumulative distribution, the conditional density and some other ones. The literature on nonparametric functional statistics is only concerning pointwise consistency results, and our main aim is to prove the uniform almost complete convergence (with rate) of the kernel estimators of these nonparametric models. Unlike in standard multivariate cases, the gap between pointwise and uniform results is not immediate. So, suitable topological considerations are needed, implying changes in the rates of convergence which are quantified by entropy considerations. These theoretical uniform consistency results are (or will be) key tools for many further developments in functional data analysis.     

Additive regression model for stationary and ergodic continuous time processes

The main purpose of the present work is to introduce and investigate a simple kernel procedure based on marginal integration that estimates the regression function for stationary and ergodic continuous time processes in the setting of the additive model introduced by Stone (1985 Stone, C.J. (1985). Additive regression and other nonparametric models. Ann. Stat. 13(2):689–705.[Crossref], [Web of Science ®] , [Google Scholar]). We obtain the uniform almost sure consistency with exact rate and the asymptotic normality of the kernel-type estimators of the components of the additive model. Asymptotic properties of these estimators are obtained, under mild conditions, by means of martingale approaches. Finally, a general notion of the bootstrapped additive components, constructed by exchangeably weighting sample, is presented.     

Uniform consistency rate of kNN regression estimation for functional time series data

In this paper, we investigate the k-nearest neighbours (kNN) estimation of nonparametric regression model for strong mixing functional time series data. More precisely, we establish the uniform almost complete convergence rate of the kNN estimator under some mild conditions. Furthermore, a simulation study and an empirical application to the real data analysis of sea surface temperature (SST) are carried out to illustrate the finite sample performances and the usefulness of the kNN approach.     

Weak and strong uniform consistency of a kernel error density estimator in nonparametric regression

This paper considers the problem of estimating the kernel error density function in nonparametric regression models. Sufficient conditions are given under which the kernel error density estimator based on nonparametric residuals is uniformly weakly and strongly consistent.     

Function-on-function regression for two-dimensional functional data

ABSTRACT

We present methods for modeling and estimation of a concurrent functional regression when the predictors and responses are two-dimensional functional datasets. The implementations use spline basis functions and model fitting is based on smoothing penalties and mixed model estimation. The proposed methods are implemented in available statistical software, allow the construction of confidence intervals for the bivariate model parameters, and can be applied to completely or sparsely sampled responses. Methods are tested to data in simulations and they show favorable results in practice. The usefulness of the methods is illustrated in an application to environmental data.     

Recursive kernel estimate of the conditional quantile for functional ergodic data

ABSTRACT

In this article, we study the recursive kernel estimator of the conditional quantile of a scalar response variable Y given a random variable (rv) X taking values in a semi-metric space. Two estimators are considered. While the first one is given by inverting the double-kernel estimate of the conditional distribution function, the second estimator is obtained by using the robust approach. We establish the almost complete consistency of these estimates when the observations are sampled from a functional ergodic process. Finally, a simulation study is carried out to illustrate the finite sample performance of these estimators.     

Spatio-temporal functional regression on paleoecological data

There is much interest in predicting the impact of global warming on the genetic diversity of natural populations and the influence of climate on biodiversity is an important ecological question. Since Holocene, we face many climate perturbations and the geographical ranges of plant taxa have changed substantially. Actual genetic diversity of plant is a result of these processes and a first step to study the impact of future climate change is to understand the important features of reconstructed climate variables such as temperature or precipitation for the last 15,000 years on actual genetic diversity of forest. We model the relationship between genetic diversity in the European beech (Fagus sylvatica) forests and curves of temperature and precipitation reconstructed from pollen databases. Our model links the genetic measure to the climate curves. We adapt classical functional linear model to take into account interactions between climate variables as a bilinear form. Since the data are georeferenced, our extensions also account for the spatial dependence among the observations. The practical issues of these methodological extensions are discussed.     

Strong uniform consistency rates and asymptotic normality of conditional density estimator in the single functional index modeling for time series data

In this paper, we investigate a nonparametric estimation of the conditional density of a scalar response variable given a random variable taking values in separable Hilbert space. We establish under general conditions the uniform almost complete convergence rates and the asymptotic normality of the conditional density kernel estimator, when the variables satisfy the strong mixing dependency, based on the single-index structure. The asymptotic \((1-\zeta )\) confidence intervals of conditional density function are given, for \(0 < \zeta < 1\) . We further demonstrate the impact of this functional parameter to the conditional mode estimate. Simulation study is also presented. Finally, the estimation of the functional index via the pseudo-maximum likelihood method is discussed, but not tackled.     

Semiparametric partially linear regression models for functional data

In the context of longitudinal data analysis, a random function typically represents a subject that is often observed at a small number of time point. For discarding this restricted condition of observation number of each subject, we consider the semiparametric partially linear regression models with mean function x^?β$x^{?}\beta$ + g(z), where x and z   are functional data. The estimations of β$\beta$ and g(z) are presented and some asymptotic results are given. It is shown that the estimator of the parametric component is asymptotically normal. The convergence rate of the estimator of the nonparametric component is also obtained. Here, the observation number of each subject is completely flexible. Some simulation study is conducted to investigate the finite sample performance of the proposed estimators.     

Weak and strong uniform consistency rates of kernel density estimates for randomly censored data

Let X₁,., X_n, be i.i.d. random variables with distribution function F, and let Y₁,.,.,Y_n be i.i.d. with distribution function G. For i = 1, 2,.,., n set δ_i, = 1 if X_i ≤ Y_i, and 0 otherwise, and X_i, = min{X_i, K_i}. A kernel-type density estimate of f, the density function of F w.r.t. Lebesgue measure on the Borel o-field, based on the censored data (δ_i, X_i), i = 1,.,.,n, is considered. Weak and strong uniform consistency properties over the whole real line are studied. Rates of convergence results are established under higher-order differentiability assumption on f. A procedure for relaxing such assumptions is also proposed.     

On the local linear estimate for functional regression: Uniform in bandwidth consistency

We consider the problem of local linear estimation of the regression function when the regressor is functional. The main result of this paper is to prove the strong convergence (with rates), uniformly in bandwidth parameters (UIB), of the considered estimator. The main interest of this result is the possibility to derive the asymptotic properties of our estimate even if the bandwidth parameter is a random variable.     

Strong consistency result of a non parametric conditional mode estimator under random censorship for functional regressors

Abstract

Let (T, C, X) be a vector of random variables (rvs) where T, C, and X are the interest variable, a right censoring rv, and a covariate, respectively. In this paper, we study the kernel conditional mode estimation when the covariate takes values in an infinite dimensional space and is α-mixing. Under some regularity conditions, the almost complete convergence of the estimate with rates is established.     

Bayesian consistency for regression models under a supremum distance

This paper studies the consistency of Bayesian nonparametric regression models. We concentrate on the use of the sup metric and dealing with non-stochastic, i.e. designed, covariate values. We illustrate our results on a normal mean regression function and demonstrate the usefulness of a model based on piecewise constant functions.     

Uniform in bandwidth consistency for various kernel estimators involving functional data

The paper investigates various nonparametric models including regression, conditional distribution, conditional density and conditional hazard function, when the covariates are infinite dimensional. The main contribution is to prove uniform in bandwidth asymptotic results for kernel estimators of these functional operators. Then, the application issues, involving data-driven bandwidth selection, are discussed.     

Asymptotic normality of locally modelled regression estimator for functional data

We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As an alternative to the well-known Nadaraya-Watson estimator for regression function in this framework, the locally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu, P. (2010), ‘Locally Modelled Regression and Functional Data’, Journal of Nonparametric Statistics, 22, 617–632]. In this paper, the asymptotic properties of locally modelled regression estimator for functional data are considered. The mean-squared convergence as well as asymptotic normality for the estimator are established. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function and derive the Wilk's phenomenon for the empirical likelihood inference. Furthermore, a simulation study is presented to illustrate our theoretical results.     

Strong uniform consistency results of the weighted average of conditional artificial data points

In this paper, we study strong uniform consistency of a weighted average of artificial data points. This is especially useful when information is incomplete (censored data, missing data …). In this case, reconstruction of the information is often achieved nonparametrically by using a local preservation of mean criterion for which the corresponding mean is estimated by a weighted average of new data points. The present approach enlarges the possible scope for applications beyond just the incomplete data context and can also be useful to treat the estimation of the conditional mean of specific functions of complete data points. As a consequence, we establish the strong uniform consistency of the Nadaraya–Watson [Nadaraya, E.A., 1964. On estimating regression. Theory Probab. Appl. 9, 141–142; Watson, G.S., 1964. Smooth regression analysis. Sankhyā Ser. A 26, 359–372] estimator for general transformations of the data points. This result generalizes the one of Härdle et al. [Strong uniform consistency rates for estimators of conditional functionals. Ann. Statist. 16, 1428–1449]. In addition, the strong uniform consistency of a modulus of continuity will be obtained for this estimator. Applications of those two results are detailed for some popular estimators.