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.     

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.     

Local linear estimate of the point at high risk: Spatial functional data case

Abstract

The purpose of the present paper is to investigate by the local linear method a nonparametric estimator of the point at high risk of scalar response variable given a functional variable when the observations are spatially dependent. The main goal is to establish the almost complete convergence with rate of this estimator under some general conditions. A practical example on the climatological data shows the usefulness of our theoretical study.     

On the rate of convergence of the Robbins–Monro's algorithm in a linear stochastic ill-posed problem with α-mixing data

In this paper we consider a recursive method of Robbins–Monro type to estimate the solution of the linear problem Ax = u, in which the second member is measured with α-mixing errors. We also show the almost complete convergence (a.co) of this algorithm specifying its convergence rate.     

Lp convergence and complete convergence for weighted sums of AANA random variables

Let {X_n, n ? 1} be a sequence of asymptotically almost negatively associated (AANA, for short) random variables which is stochastically dominated by a random variable X, and {d_ni, 1 ? i ? n, n ? 1} be a sequence of real function, which is defined on a compact set E. Under some suitable conditions, we investigate some convergence properties for weighted sums of AANA random variables, especially the L^p convergence and the complete convergence. As an application, the Marcinkiewicz–Zygmund-type strong law of large numbers for AANA random variables is obtained.     

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.     

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.     

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.     

Complete convergence and complete moment convergence for widely orthant-dependent random variables

In this paper, we first establish the complete convergence for weighted sums of widely orthant-dependent (WOD, in short) random variables by using the Rosenthal type maximal inequality. Based on the complete convergence, we further study the complete moment convergence for weighted sums of arrays of rowwise WOD random variables which is stochastically dominated by a random variable X. The results obtained in the paper generalize the corresponding ones for some dependent random variables.     

Complete moment convergence and Lr convergence for asymptotically almost negatively associated random variables

In this paper, we investigate the complete moment convergence and L_r convergence for maximal partial sums of asymptotically almost negatively associated random variables under some general conditions. The results obtained in the paper generalize some corresponding ones for negatively associated random variables.     

UPPER BOUNDS FOR THE HARMONIC MEAN, WITH AN APPLICATION TO EXPERIMENTAL DESIGN

For positive-valued random variables, the paper provides a sequence of upper bounds for the harmonic mean, the ith of these bounds being exact if and only if the random variable is essentially i-valued. Sufficient conditions for the convergence of the bounds to the harmonic mean are given. The bounds have a number of applications, particularly in experimental design where they may be used to check how close a given design is to A-optimality     

Conditional Density Estimation in the Single Functional Index Model for α-Mixing Functional Data

In this article, we investigate the nonparametric estimation of the conditional density of a scalar response variable Y, given the explanatory variable X taking value in a Hilbert space when the observations are linked with a single index structure. The goal of this article is to present the asymptotic results such as pointwise almost complete consistency and the uniform almost complete convergence of the kernel estimation with rate for the conditional density in the setting of the α-mixing functional data, which extend the i.i.d case in Attaoui et al. (2011 Attaoui , S. , Laksaci , A. , Ould-Said , E. ( 2011 ). A note on the conditional density estimate in the single functional index model . Statist. Probab. Lett. 81 ( 1 ): 45 – 53 .[Crossref], [Web of Science ®] , [Google Scholar]) to the dependence setting. As an application, the convergence rate of the kernel estimation for the conditional mode is also obtained.     

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.     

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.     

Estimation suroptimale de la densité par projection

Superefficiency of a projection density estimator The author constructs a projection density estimator with a data‐driven truncation index. This estimator reaches the superoptimal rates 1/n in mean integrated square error and {In ln(n/n}^1/2 in uniform almost sure convergence over a given subspace which is dense in the class of all possible densities; the rate of the estimator is quasi‐optimal everywhere else. The subspace in question may be chosen a priori by the statistician.     

A modified kernel quantile estimator under censoring

Based on right-censored data from a lifetime distribution F₀, a modification of the kernel quantile estimator is proposed. The advantage of this estimator is that the data play a role in the degree of smoothing of the estimator while retaining the desirable features of the kernel estimator. Convergence in probability and almost sure convergence of the estimator are discussed. Also, asymptotic normality and confidence bands are presented and some examples are given.     

Patched approximations and their convergence

Abstract

Patched approximations of copulas unify ordinal sums, shuffles of Min, checkerboard, and checkmin approximations. We give a characterization of patched approximations and an error bound of the approximations in Sobolev norm. Patched approximations with uniform marginal conditional distributions are shown to arise naturally. We prove that these uniform patched approximations converge uniformly and in the Sobolev norm. The latter convergence is settled by showing the convergence almost everywhere of the first partial derivatives. We also show that the independence copula can be approximated by conditional mutual complete copulas in the Sobolev norm.     

Asymptotic normality of a conditional hazard function estimate in the single index for quasi-associated data

Abstract

The main goal of this paper is to study the estimation of the conditional hazard function of a scalar response variable Y given a hilbertian random variable X in functional single-index model. We construct an estimator of this nonparametric function and we study its asymptotic properties, under quasi-associated structure. Precisely, we establish the asymptotic normality of the constructed estimator. We carried out simulation experiments to examine the behavior of this asymptotic property over finite sample data.     

Complete and complete moment convergence with applications to the EV regression models

In this paper, the complete convergence and complete moment convergence for arrays of rowwise m-extended negatively dependent (m-END, for short) random variables are established. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for m-END random variables is also achieved. By using the results that we established, we further investigate the strong consistency of the least square estimator in the simple linear errors-in-variables models, and provide some simulations to verify the validity of our theoretical results.     

On the mean L1-error in the heteroscedastic deconvolution problem with compactly supported noises

We study the heteroscedastic deconvolution problem when random noises have compactly supported densities. In this context, the Fourier transforms of the densities can vanish on the real line. We propose a truncated type of estimator for target density and derive the convergence rate of the mean L¹-error uniformly over a class of target densities. A lower bound for the mean L¹-error is also established. Some simulations will be given to illustrate the performance of the proposed estimator.