Complete Convergence Rates Research Articles

Uniform in Number of Neighbor Consistency and Weak Convergence of k-Nearest Neighbor Single Index Conditional Processes and k-Nearest Neighbor Single Index Conditional U-Processes Involving Functional Mixing Data

U-statistics are fundamental in modeling statistical measures that involve responses from multiple subjects. They generalize the concept of the empirical mean of a random variable X to include summations over each m-tuple of distinct observations of X. W. Stute introduced conditional U-statistics, extending the Nadaraya–Watson estimates for regression functions. Stute demonstrated their strong pointwise consistency with the conditional expectation r(m)(φ,t), defined as E[φ(Y1,…,Ym)|(X1,…,Xm)=t] for t∈Xm. This paper focuses on estimating functional single index (FSI) conditional U-processes for regular time series data. We propose a novel, automatic, and location-adaptive procedure for estimating these processes based on k-Nearest Neighbor (kNN) principles. Our asymptotic analysis includes data-driven neighbor selection, making the method highly practical. The local nature of the kNN approach improves predictive power compared to traditional kernel estimates. Additionally, we establish new uniform results in bandwidth selection for kernel estimates in FSI conditional U-processes, including almost complete convergence rates and weak convergence under general conditions. These results apply to both bounded and unbounded function classes, satisfying certain moment conditions, and are proven under standard Vapnik–Chervonenkis structural conditions and mild model assumptions. Furthermore, we demonstrate uniform consistency for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship. This result is independently valuable and has potential applications in areas such as set-indexed conditional U-statistics, the Kendall rank correlation coefficient, and discrimination problems.

Two highly accurate and efficient numerical methods for solving the fractional Liénard’s equation arising in oscillating circuits

In this paper, we investigate the fractional model of the Liénard and Duffing equations with the Liouville-Caputo fractional derivative. These equations grow with the evolution of radio and vacuum tube technology, which describe oscillating circuits and generalize the spring–mass device equation. We compare two numerical approaches, namely Jacobi and Haar wavelet collocation methods. The given approaches are used to discretize and transform the equation into a system of algebraic equations, and the Broyden-Quasi Newton algorithm is applied to solve the resulting nonlinear system of equations. A complete error analysis and convergence rates for different grid sizes are derived for both methods, which are used to compare the accuracy and efficiency of the two approaches. While both approaches produce correct solutions, according to the numerical findings, the Jacobi collocation method is more efficient and accurate than the Haar wavelet collocation method.

On the solution of MHD Jeffery–Hamel problem involving flow between two nonparallel plates with a blood flow application

AbstractThe Jeffery–Hamel flow phenomenon appears in a variety of real‐world applications involving the flow of two nonparallel plates. BY using a similarity transformation derived from the equation of continuity, partial differential equations determining flow characteristics are translated into nonlinear ordinary differential equations. The problem involves the flow of a specific type of fluid, namely, an incompressible and electrically conducting fluid, between two nonparallel plates. The flow is assumed to be steady, two‐dimensional, and subject to certain boundary conditions. Specifically, the plates are impermeable, and the fluid adheres to a no‐slip condition, resulting in zero fluid velocity at the plates' surfaces. Moreover, the problem incorporates the effects of magnetic fields and pressure fluctuations, making it highly applicable to scenarios, such as blood flow through arteries in the human body, which can be modeled as a special case of the magnetohydrodynamic (MHD) Jeffery–Hamel problem referred to as the (MHD) blood pressure equation. This work compares two numerical approaches for solving the MHDs Jeffery–Hamel problem: B‐spline and Bernstein polynomial collocation. The given approaches are used to discretize and transform the equation into a system of algebraic equations. Matrix algebra techniques are then used to solve the resultant system. A complete error analysis and convergence rates for different grid sizes are derived for both methods and are used to compare the accuracy and efficiency of the two approaches. Both approaches produce correct solutions, according to the numerical findings, although the Bernstein polynomial collocation method is more efficient and accurate than the B‐spline collocation.

Asymptotic Normality of Single Functional Index Quantile Regression for Functional Data with Missing Data at Random

This work addresses the problem of the nonparametric estimation of the regression function, namely the conditional distribution and the conditional quantile in the single functional index model (SFIM) under the independent and identically distributed condition with randomly missing data. The main result of this study was the establishment of the asymptotic properties of the estimator, such as the almost complete convergence rates. Moreover, the asymptotic normality of the constructs was obtained under certain mild conditions. Lastly, the authors discussed how to apply the result to construct confidence intervals.

Estimation in nonparametric functional-on-functional models with surrogate responses

We construct an estimator for the regression operator of a functional response variable using surrogate data, given a functional random variable. The almost complete uniform convergence rate of the estimator is then established. Finally, to demonstrate the predictive utility and superiority of the estimator when dealing with incomplete data, we apply the methodology to both simulated data and meteorological data.

Limit behaviors of the estimator of non parametric regression model based on extended negatively dependent errors

In this article, we study the complete consistency for the estimator of non parametric regression model based on extended negatively dependent errors, and obtain the convergence rates of the complete consistency by using the inequalities for extended negatively dependent random variables. As an application, the complete consistency and convergence rate for the nearest neighbor estimator are obtained. Finally, some simulations are illustrated.

K-Nearest Neighbor Estimation of Functional Nonparametric Regression Model under NA Samples

Functional data, which provides information about curves, surfaces or anything else varying over a continuum, has become a commonly encountered type of data. The k-nearest neighbor (kNN) method, as a nonparametric method, has become one of the most popular supervised machine learning algorithms used to solve both classification and regression problems. This paper is devoted to the k-nearest neighbor (kNN) estimators of the nonparametric functional regression model when the observed variables take values from negatively associated (NA) sequences. The consistent and complete convergence rate for the proposed kNN estimator is first provided. Then, numerical assessments, including simulation study and real data analysis, are conducted to evaluate the performance of the proposed method and compare it with the standard nonparametric kernel approach.

Complete consistency and convergence rate of the nearest neighbor estimator of the density function based on WOD samples

Complete consistency and convergence rate of the nearest neighbor estimator of the density function based on WOD samples

Strong convergence of the functional nonparametric relative error regression estimator under right censoring

Abstract In this paper, we investigate the asymptotic properties of a nonparametric estimator of the relative error regression given a functional explanatory variable, in the case of a scalar censored response, we use the mean squared relative error as a loss function to construct a nonparametric estimator of the regression operator of these functional censored data. We establish the strong almost complete convergence rate and asymptotic normality of these estimators. A simulation study is performed to illustrate and compare the higher predictive performances of our proposed method to those obtained with standard estimators.

On Complete Consistency for the Estimator of Nonparametric Regression Model Based on Asymptotically Almost Negatively Associated Errors

In this paper, we mainly study the consistency for the estimator of nonparametric regression model based on asymptotically almost negatively associated (AANA, in short) errors. Firstly, the Bernstein type inequality for AANA random variables is established. By using the Bernstein type inequality and moment inequalities, we investigate the complete consistency and convergence rate for the estimator of nonparametric regression model based on AANA errors. As applications, the complete consistency and convergence rate for the nearest neighbor estimator are obtained.

Complete consistency for recursive probability density estimator of widely orthant dependent samples

In this paper, we will study the recursive density estimators of the probability density function for widely orthant dependent WOD random variables. The complete consistency and complete convergence rate are established under some general conditions.

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

ABSTRACTIn 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.

M-estimation of the regression function under random left truncation and functional time series model

In this paper we study the M-estimation of the functional nonparametric regression when the response variable is subject to left-truncation by an other random variable. Under standard assumptions, we get the almost complete convergence rate of this robust estimate when the sample is an $$\alpha $$-mixing sequence. This approach can be applied in time series analysis to the prediction problem. Our asymptotic results are confronted by some simulations study.

Complete Convergence of Weighted Sums for Asymptotically Almost Negatively Associated Sequences

For weighted sums of asymptotically almost negatively associated (AANA) random variables sequences, we use the Rosenthal type moment inequalities and prove the Marcinkiewicz-Zygmund type complete convergence and obtain the complete convergence rates. Our results extend some known ones.

Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications

In this paper, an exponential inequality for the maximal partial sums of negatively superadditive-dependent (NSD, in short) random variables is established. By using the exponential inequality, we present some general results on the complete convergence for arrays of rowwise NSD random variables, which improve or generalize the corresponding ones of Wang et al. [28] and Chen et al. [2]. In addition, some sufficient conditions to prove the complete convergence are provided. As an application of the complete convergence that we established, we further investigate the complete consistency and convergence rate of the estimator in a nonparametric regression model based on NSD errors.

The large deviation results for the nonlinear regression model with dependent errors

In this paper, we investigate the least squares (LS) estimator of the nonlinear regression model based on the extended negatively dependent errors which are widely dependent structures. Under the general conditions, we establish some large deviation results for the LS estimator of the nonlinear regression parameter, which can be applied to obtain a weak uniform consistency and a complete convergence rate for this estimator. In addition, some examples and simulations are presented for illustration.

A recursive kernel estimate of the functional modal regression under ergodic dependence condition

In this article, we consider an alternative estimator of the conditional mode when the explanatory variable takes values in a semimetric space. This alternative estimate is based in a recursive kernel method. Under the ergodicity hypothesis, we quantify the asymptotic properties of this estimate, by giving the almost complete convergence rate. The asymptotic normality of this estimate is also given. Our approach is illustrated by a real data application.

Complete consistency of estimators for regression models based on extended negatively dependent errors

In this paper, we investigate the consistency of the estimators of nonparametric regression model and multiple linear regression model based on extended negatively dependent errors. The complete convergence rates of the estimators of nonparametric regression model are presented. In addition, the rth-mean consistency and complete consistency of the least squares estimator of the multiple linear regression model are obtained too. Finally, some examples and some simulations are illustrated.

Asymptotic results of a nonparametric conditional cumulative distribution estimator in the single functional index modeling for time series data with applications

In this paper, we treat nonparametric estimation of the conditional cumulative distribution with a scalar response variable conditioned by a functional Hilbertian regressor. We establish asymptotic normality and uniform almost complete convergence rates of the conditional cumulative distribution estimator for dependent variables, linked semiparametrically by the single index structure. Furthermore, we provide some applications and simulations to illustrate our methodology.

Non‐repeatedly marking traceback model for wireless sensor networks

When networks are under attack, traceback schemes can be used to locate the compromised node and remove the compromised node from the network. To resolve the problem that marked information has been easy to be covered by the information of down-stream node in the traceback schemes based on probabilistic packet marking (PPM), ‘a non-repeatedly marking traceback scheme for wireless sensor networks (NMtrace) is proposed to locate the compromised nodes in this study’. In the NMtrace, when a packet is forwarding to the base station, if the edge information of the packet has been marked by forwarding nodes, the packet will not be marked again by the other nodes. The security shows that the NMtrace can resilience against various attacks which are launched by the compromised forwarding nodes collude with compromised source node. The performance evaluation shows that with the length of the attack path increasing, ‘the NMtrace needs fewer’ packets to reconstruct the attack path than the PPM-based schemes, the complete convergence rate less affected by the marking probability p and the energy ‘consumption of the NMtrace is also’ less than PPM-based schemes.