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.     

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.     

Uniform strong convergence results for the conditional kaplan-meier estimator and its quantiles

We consider a fixed design model in which the responses are possibly right censored. The aim of this paper is to establish some important almost sure convergence properties of the Kaplan-Meier type estimator for the lifetime distribution at a given covariate value. We also consider the corresponding quantile estimator and obtain a modulus of continuity result. Our rates of uniform strong convergence are obtained via exponential probability bounds.     

Asymptotic Normality for Regression Function Estimate Under Truncation and α-Mixing Conditions

In this article we establish pointwise asymptotic normality of nonparametric kernel estimator of regression function for a left truncation model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. Also, the asymptotic normality of the estimation of the covariable's density is considered. As a by-product, we obtain a uniform weak convergence rate for the product-limit estimator of the lifetime and truncated distributions under dependence, which is interesting independently. Finite sample behavior of the estimator of the regression function is investigated as well.     

Nonparametric regression in a statistical modified Helmholtz equation using the Fourier spectral regularization

In this paper, we deal with the Cauchy problem for the modified Helmholtz equation. We consider two models of data: the bounded variance model and the i.i.d. model. The trigonometric estimators of nonparametric regression is applied to solve the problem. In addition, the general forms of regularization parameter corresponding to the pointwise mean squared error and the mean integrated squared error are discussed in detail. The minimax rate convergence corresponding to the bounded variance model is also presented. In the i.i.d. model, we construct the asymptotic confidence interval for the solution of the problem. Finally, we give some numerical experiments and discuss the obtained results.     

Consistency of the kernel density estimator: a survey

Various consistency proofs for the kernel density estimator have been developed over the last few decades. Important milestones are the pointwise consistency and almost sure uniform convergence with a fixed bandwidth on the one hand and the rate of convergence with a fixed or even a variable bandwidth on the other hand. While considering global properties of the empirical distribution functions is sufficient for strong consistency, proofs of exact convergence rates use deeper information about the underlying empirical processes. A unifying character, however, is that earlier and more recent proofs use bounds on the probability that a sum of random variables deviates from its mean.     

Estimation and simulation of conditional hazard function in the quasi-associated framework when the observations are linked via a functional single-index structure

We consider the estimation of the conditional hazard function of a scalar response variable Y given a Hilbertian random variable X when the observations are linked via a single-index structure in the quasi-associated framework. We establish the pointwise almost complete convergence and the uniform almost complete convergence (with the rate) of the estimate of this model. A simulation is given to illustrate the good behavior in the practice of our methodology.     

Functional data: local linear estimation of the conditional density and its application

In this paper, we introduce a new nonparametric estimation procedure of the conditional density of a scalar response variable given a random variable taking values in a semi-metric space. Under some general conditions, we establish both the pointwise and the uniform almost-complete consistencies with convergence rates of the conditional density estimator related to this estimation procedure. Moreover, we give some particular cases of our results which can also be considered as novel in the finite-dimensional setting. Notice also that the results of this paper are used to derive some asymptotic properties of the local linear estimator of the conditional mode.     

Comparison between the rates of convergence of extremes under linear and under power normalization

By using the theory of the second-order regular variation, we study the rates of the weak convergence of the maximum order statistics under power normalization. The exact rates are obtained in the uniform metric and the total variation metric. The relationship between the rates of convergence under linear and under power normalization is derived. Some illustrative examples are given for comparing the rates of convergence.     

Local linear estimation of a generalized regression function with functional dependent data

This work deals with a local linear non parametric estimation of the generalized regression function in the case of a scalar response variable given a random variable taking values in a semimetric space. The rates of pointwise and uniform almost complete convergence are established for the studied estimator when the sample is an α-mixing sequence. Two real datasets are used to illustrate the performance of a studied estimator with respect to the kernel method.     

On Rates of Convergence for Bayesian Density Estimation

Abstract.  We consider the problem of estimating a compactly supported density taking a Bayesian nonparametric approach. We define a Dirichlet mixture prior that, while selecting piecewise constant densities, has full support on the Hellinger metric space of all commonly dominated probability measures on a known bounded interval. We derive pointwise rates of convergence for the posterior expected density by studying the speed at which the posterior mass accumulates on shrinking Hellinger neighbourhoods of the sampling density. If the data are sampled from a strictly positive, α -Hölderian density, with α  ∈ ( 0,1] , then the optimal convergence rate n^{− α / (2 α +1)} is obtained up to a logarithmic factor. Smoothing histograms by polygons, a continuous piecewise linear estimator is obtained that for twice continuously differentiable, strictly positive densities satisfying boundary conditions attains a rate comparable up to a logarithmic factor to the convergence rate n ^−4/5 for integrated mean squared error of kernel type density estimators.     

Consistency of the GMLE with Mixed Case Interval-Censored Data

In this paper we consider an interval censorship model in which the endpoints of the censoring intervals are determined by a two stage experiment. In the first stage the value k of a random integer is selected; in the second stage the endpoints are determined by a case k interval censorship model. We prove the strong consistency in the L ₁( μ )-topology of the non-parametric maximum likelihood estimate of the underlying survival function for a measure μ which is derived from the distributions of the endpoints. This consistency result yields strong consistency for the topologies of weak convergence, pointwise convergence and uniform convergence under additional assumptions. These results improve and generalize existing ones in the literature.     

A Positive False Discovery Rate Convergence Result

The positive false discovery rate (pFDR) is the average proportion of false rejections given that the overall number of rejections is greater than zero. Assuming that the proportion of true null hypotheses, proportion of false positives, and proportion of true positives all converge pointwise, the pFDR converges to a continuous limit uniformly over all significance levels. We are showing that the uniform convergence still holds given a weaker assumption that the proportion of true positives converges in L ¹.     

Bootstrap in functional linear regression

We have considered the functional linear model with scalar response and functional explanatory variable. One of the most popular methodologies for estimating the model parameter is based on functional principal components analysis (FPCA). In recent literature, weak convergence for a wide class of FPCA-type estimates has been proved, and consequently asymptotic confidence sets can be built. In this paper, we have proposed an alternative approach in order to obtain pointwise confidence intervals by means of a bootstrap procedure, for which we have obtained its asymptotic validity. Besides, a simulation study allows us to compare the practical behaviour of asymptotic and bootstrap confidence intervals in terms of coverage rates for different sample sizes.     

Non-parametric Regression with Dependent Censored Data

Abstract.  Let ( X _i , Y _i ) ( i = 1 ,…, n ) be n replications of a random vector ( X , Y  ), where Y is supposed to be subject to random right censoring. The data ( X _i , Y _i ) are assumed to come from a stationary α -mixing process. We consider the problem of estimating the function m ( x ) = E ( φ ( Y ) |  X = x ), for some known transformation φ . This problem is approached in the following way: first, we introduce a transformed variable     , that is not subject to censoring and satisfies the relation     , and then we estimate m ( x ) by applying local linear regression techniques. As a by-product, we obtain a general result on the uniform rate of convergence of kernel type estimators of functionals of an unknown distribution function, under strong mixing assumptions.     

The conditional cumulative distribution function in single functional index model

ABSTRACT

We consider the estimation of the conditional cumulative distribution function of a scalar response variable Y given a Hilbertian random variable X when the observations are linked via a single-index structure. We establish the pointwise and the uniform almost complete convergence (with the rate) of the kernel estimate of this model. As an application, we show how our result can be applied in the prediction problem via the conditional median estimate. Also, the choice of the functional index via the cross-validation procedure is also discussed but not attacked.     

Density and hazard rate estimation for right-censored data by using wavelet methods

This paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right-censored data. We adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form. The method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval. The number of events and the survival function of the observations are then separately smoothed over time via linear wavelet smoothers, and then the hazard rate function estimators are obtained by taking the ratio. We prove that the estimators have pointwise and global mean-square consistency, obtain the best possible asymptotic mean integrated squared error convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show that these estimators are reasonably reliable in practice. The method is illustrated with two real examples. The first uses survival time data for patients with liver metastases from a colorectal primary tumour without other distant metastases. The second is concerned with times of unemployment for women and the wavelet estimate, through its flexibility, provides a new and interesting interpretation.     

Complete convergence for weighted sums of i.i.d. random variables with applications in regression estimation and EV model

In this paper, we obtain complete convergence results for Stout type weighted sums of i.i.d. random variables. A strong law for weighted sums of i.i.d. random variables is also obtained. As the applications of the strong law, the strong consistency and rate of the nonparametric regression estimations and the rates of the strong consistency of LS estimators for the unknown parameters of the simple linear errors in variables (EV) model are given.     

On histograms for linear processes

Sharp rates of convergence of histogram estimates of the marginal density of a linear process are obtained. Histograms can achieve optimal rates of convergence (n⁻¹ log n)^1·3 under general conditions. The assumptions involved are easily verifiable. Histograms appear to be very good estimators from the point of view of uniform convergence.     

Large Deviations Limit Theorems for the Kernel Density Estimator

We establish pointwise and uniform large deviations limit theorems of Chernoff-type for the non-parametric kernel density estimator based on a sequence of independent and identically distributed random variables. The limits are well-identified and depend upon the underlying kernel and density function. We derive then some implications of our results in the study of asymptotic efficiency of the goodness-of-fit test based on the maximal deviation of the kernel density estimator as well as the inaccuracy rate of this estimate