Search Results (19)

This paper examines the asymptotic behavior of the conditional hazard function using kernel-based methods, with particular emphasis on functional weakly dependent data. In particular, we establish the asymptotic normality of the proposed estimator when the covariate follows a functional quasi-associated process. This contribution extends the scope of nonparametric inference under weak dependence within the framework of functional data analysis. The estimator is constructed through kernel smoothing techniques inspired by the classical Nadaraya–Watson approach, and its theoretical properties are rigorously derived under appropriate regularity conditions. To evaluate its practical performance, we carried out an extensive simulation study, where finite-sample outcomes were compared with their asymptotic counterparts. The results showed the robustness and reliability of the estimator across a range of scenarios, thereby confirming the validity of the proposed limit theorem in empirical settings. Full article

Survival analysis is a fundamental tool for modeling time-to-event data in healthcare, engineering, and finance, where censored observations pose significant challenges. While traditional methods like the Beran estimator offer nonparametric solutions, they often struggle with the complex data structures and heavy censoring. This paper introduces three novel survival models, iSurvM (imprecise Survival model based on Mean likelihood functions), iSurvQ (imprecise Survival model based on Quantiles of likelihood functions), and iSurvJ (imprecise Survival model based on Joint learning), that combine imprecise probability theory with attention mechanisms to handle censored data without parametric assumptions. The first idea behind the models is to represent censored observations by interval-valued probability distributions for each instance over time intervals between event moments. The second idea is to employ the kernel-based Nadaraya–Watson regression with trainable attention weights for computing the imprecise probability distribution over time intervals for the entire dataset. The third idea is to consider three decision strategies for training, which correspond to the proposed three models. Experiments on synthetic and real datasets demonstrate that the proposed models, especially iSurvJ, consistently outperform the Beran estimator from accuracy and computational complexity points of view. Codes implementing the proposed models are publicly available. Full article

A geographically and temporally weighted regression (GTWR) model is an effective tool for dealing with spatial heterogeneity and temporal non-stationarity simultaneously. As an important characteristic of spatiotemporal data, spatiotemporal autocorrelation should be considered when constructing spatiotemporally varying coefficient models. The proposed local autoregressive geographically and temporally weighted regression (GTWRLAR) model can simultaneously handle spatiotemporal autocorrelations among response variables and the spatiotemporal heterogeneity of regression relationships. The two-stage weighted least squares (2SLS) estimation can effectively reduce computational complexity. However, the weighted least squares estimation is essentially a Nadaraya–Watson kernel-smoothing approach for nonparametric regression models, and it suffers from a boundary effect. For spatiotemporally varying coefficient models, the three-dimensional spatiotemporal coefficients (longitude, latitude, and time) inherently exhibit larger boundaries than one-dimensional intervals. Therefore, the boundary effect of the 2SLS estimation of GTWRLAR will be more serious. A local–linear geographically and temporally weighted 2SLS (GTWRLAR-L) estimation is proposed to correct the boundary effect in both the spatial and temporal dimensions of GTWRLAR and simultaneously improve parameter estimation accuracy. The simulation experiment shows that the GTWRLAR-L method reduces the root mean square error (RMSE) of parameter estimates compared to the standard GTWRLAR approach. Empirical analyses of carbon emissions in China’s Yellow River Basin (2017–2021) show that GTWRLAR-L enhances the adjusted $R^2$ from 0.888 to 0.893. Full article

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 $r^{\left(m\right)}(\phi ,\mathbf{t})$, defined as $\mathbb{E}\left[\phi (Y_1,\dots ,Y_m)\right|(X_1,\dots ,X_m)=\mathbf{t}]$ for $\mathbf{t}\in {\mathcal{X}}^m$. 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. Full article

The main aim of this paper is to consider a new risk metric that permits taking into account the spatial interactions of data. The considered risk metric explores the spatial tail-expectation of the data. Indeed, it is obtained by combining the ideas of expected shortfall regression with an expectile risk model. A spatio-functional Nadaraya–Watson estimator of the studied metric risk is constructed. The main asymptotic results of this work are the establishment of almost complete convergence under a mixed spatial structure. The claimed asymptotic result is obtained under standard assumptions covering the double functionality of the model as well as the data. The impact of the spatial interaction of the data in the proposed risk metric is evaluated using simulated data. A real experiment was conducted to measure the feasibility of the Spatio-Functional Expectile Shortfall Regression (SFESR) in practice. Full article

This paper treats the problem of risk management through a new conditional expected shortfall function. The new risk metric is defined by the expectile as the shortfall threshold. A nonparametric estimator based on the Nadaraya–Watson approach is constructed. The asymptotic property of the constructed estimator is established using a functional time-series structure. We adopt some concentration inequalities to fit this complex structure and to precisely determine the convergence rate of the estimator. The easy implantation of the new risk metric is shown through real and simulated data. Specifically, we show the feasibility of the new model as a risk tool by examining its sensitivity to the fluctuation in financial time-series data. Finally, a comparative study between the new shortfall and the standard one is conducted using real data. Full article

The functional partially linear regression model comprises a functional linear part and a non-parametric part. Testing the linear relationship between the response and the functional predictor is of fundamental importance. In cases where functional data cannot be approximated with a few principal components, we develop a second-order U-statistic using a pseudo-estimate for the unknown non-parametric component. Under some regularity conditions, the asymptotic normality of the proposed test statistic is established using the martingale central limit theorem. The proposed test is evaluated for finite sample properties through simulation studies and its application to real data. Full article

In his work published in (Ann. Probab. 19, No. 2 (1991), 812–825), W. Stute introduced the notion of conditional U-statistics, expanding upon the Nadaraya–Watson estimates used for regression functions. Stute illustrated the pointwise consistency and asymptotic normality of these statistics. Our research extends these concepts to a broader scope, establishing, for the first time, an asymptotic framework for single-index conditional U-statistics applicable to locally stationary random fields $\{{\mathbf{X}}_{s,A_n}:sin{R}_n\}$ observed at irregularly spaced locations in $R_n$, a subset of ${\mathbb{R}}^d$. We introduce an estimator for the single-index conditional U-statistics operator that accommodates the nonstationary nature of the data-generating process. Our method employs a stochastic sampling approach that allows for the flexible creation of irregularly spaced sampling sites, covering both pure and mixed increasing domain frameworks. We establish the uniform convergence rate and weak convergence of the single conditional U-processes. Specifically, we examine weak convergence under bounded or unbounded function classes that satisfy specific moment conditions. These findings are established under general structural conditions on the function classes and underlying models. The theoretical advancements outlined in this paper form essential foundations for potential breakthroughs in functional data analysis, laying the groundwork for future research in this field. Moreover, in the same context, we show the uniform consistency for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship, which is of its own interest. Potential applications of our findings encompass, among many others, the set-indexed conditional U-statistics, the Kendall rank correlation coefficient, and the discrimination problems. Full article

A method for estimating the conditional average treatment effect under the condition of censored time-to-event data, called BENK (the Beran Estimator with Neural Kernels), is proposed. The main idea behind the method is to apply the Beran estimator for estimating the survival functions of controls and treatments. Instead of typical kernel functions in the Beran estimator, it is proposed to implement kernels in the form of neural networks of a specific form, called neural kernels. The conditional average treatment effect is estimated by using the survival functions as outcomes of the control and treatment neural networks, which consist of a set of neural kernels with shared parameters. The neural kernels are more flexible and can accurately model a complex location structure of feature vectors. BENK does not require a large dataset for training due to its special way for training networks by means of pairs of examples from the control and treatment groups. The proposed method extends a set of models that estimate the conditional average treatment effect. Various numerical simulation experiments illustrate BENK and compare it with the well-known T-learner, S-learner and X-learner for several types of control and treatment outcome functions based on the Cox models, the random survival forest and the Beran estimator with Gaussian kernels. The code of the proposed algorithms implementing BENK is publicly available. Full article

This paper studies the estimation problem for semi-varying coefficient heteroscedastic instrumental variable models with missing responses. First, we propose the adjusted estimators for unknown parameters and smooth functional coefficients utilizing the ordinary profile least square method and instrumental variable adjustment technique with complete data. Second, we present an adjusted estimator of the stochastic error variance by employing the Nadaraya–Watson kernel estimation technique. Third, we apply the inverse probability-weighted method and instrumental variable adjustment technique to construct the adaptive-weighted adjusted estimators for unknown parameters and smooth functional coefficients. The asymptotic properties of our proposed estimators are established under some regularity conditions. Finally, numerous simulation studies and a real data analysis are conducted to examine the finite sample performance of the proposed estimators. Full article

In this paper, a nonlinear time series model is developed for the case when the underlying time series data are reported by $LR$ fuzzy numbers. To this end, we present a three-stage nonparametric kernel-based estimation procedure for the center as well as the left and right spreads of the unknown nonlinear fuzzy smooth function. In each stage, the nonparametric Nadaraya–Watson estimator is used to evaluate the center and the spreads of the fuzzy smooth function. A hybrid algorithm is proposed to estimate the unknown optimal bandwidths and autoregressive order simultaneously. Various goodness-of-fit measures are utilized for performance assessment of the fuzzy nonlinear kernel-based time series model and for comparative analysis. The practical applicability and superiority of the novel approach in comparison with further fuzzy time series models are demonstrated via a simulation study and some real-life applications. Full article

A new method for estimating the conditional average treatment effect is proposed in this paper. It is called TNW-CATE (the Trainable Nadaraya–Watson regression for CATE) and based on the assumption that the number of controls is rather large and the number of treatments is small. TNW-CATE uses the Nadaraya–Watson regression for predicting outcomes of patients from control and treatment groups. The main idea behind TNW-CATE is to train kernels of the Nadaraya–Watson regression by using a weight sharing neural network of a specific form. The network is trained on controls, and it replaces standard kernels with a set of neural subnetworks with shared parameters such that every subnetwork implements the trainable kernel, but the whole network implements the Nadaraya–Watson estimator. The network memorizes how the feature vectors are located in the feature space. The proposed approach is similar to transfer learning when domains of source and target data are similar, but the tasks are different. Various numerical simulation experiments illustrate TNW-CATE and compare it with the well-known T-learner, S-learner, and X-learner for several types of control and treatment outcome functions. The code of proposed algorithms implementing TNW-CATE is publicly available. Full article

Stute presented the so-called conditional U-statistics generalizing the Nadaraya–Watson estimates of the regression function. Stute demonstrated their pointwise consistency and the asymptotic normality. In this paper, we extend the results to a more abstract setting. We develop an asymptotic theory of conditional U-statistics for locally stationary random fields $\{{\mathit{X}}_{s,A_n}:s\mathrm{in}{R}_n\}$ observed at irregularly spaced locations in $R_n={[0,A_n]}^d$ as a subset of ${\mathbb{R}}^d$. We employ a stochastic sampling scheme that may create irregularly spaced sampling sites in a flexible manner and includes both pure and mixed increasing domain frameworks. We specifically examine the rate of the strong uniform convergence and the weak convergence of conditional U-processes when the explicative variable is functional. We examine the weak convergence where the class of functions is either bounded or unbounded and satisfies specific moment conditions. These results are achieved under somewhat general structural conditions pertaining to the classes of functions and the underlying models. The theoretical results developed in this paper are (or will be) essential building blocks for several future breakthroughs in functional data analysis. Full article

One of the main tasks in the problems of machine learning and curve fitting is to develop suitable models for given data sets. It requires to generate a function to approximate the data arising from some unknown function. The class of kernel regression estimators is one of main types of nonparametric curve estimations. On the other hand, fractal theory provides new technologies for making complicated irregular curves in many practical problems. In this paper, we are going to investigate fractal curve-fitting problems with the help of kernel regression estimators. For a given data set that arises from an unknown function m, one of the well-known kernel regression estimators, the Nadaraya–Watson estimator $\widehat{m}$, is applied. We consider the case that m is Hölder-continuous of exponent $\beta$ with $0<\beta \le 1$, and the graph of m is irregular. An estimation for the expectation of $|\widehat{m}{-m|}^2$ is established. Then a fractal perturbation $f_{\left[\widehat{m}\right]}$ corresponding to $\widehat{m}$ is constructed to fit the given data. The expectations of $|f_{\left[\widehat{m}\right]}-\widehat{m}{|}^2$ and $|f_{\left[\widehat{m}\right]}{-m|}^2$ are also estimated. Full article

This article deals with the problem of designing regression models for evaluating the parameters of the operation of complex technological equipment—hydroturbine units. A promising approach to the construction of regression models based on nonparametric Nadaraya–Watson kernel estimates is considered. A known problem in applying this approach is to determine the effective values of kernel-smoothing coefficients. Kernel-smoothing factors significantly impact the accuracy of the regression model, especially under conditions of variability of noise and parameters of samples in the input space of models. This fully corresponds to the characteristics of the problem of estimating the parameters of hydraulic turbines. We propose to use the evolutionary genetic algorithm with an addition in the form of a local-search stage to adjust the smoothing coefficients. This ensures the local convergence of the tuning procedure, which is important given the high sensitivity of the quality criterion of the nonparametric model. On a set of test problems, the results were obtained showing a reduction in the modeling error by 20% and 28% for the methods of adjusting the coefficients by the standard and hybrid genetic algorithms, respectively, in comparison with the case of an arbitrary choice of the values of such coefficients. For the task of estimating the parameters of the operation of a hydroturbine unit, a number of promising approaches to constructing regression models based on artificial neural networks, multidimensional adaptive splines, and an evolutionary method of genetic programming were included in the research. The proposed nonparametric approach with a hybrid smoothing coefficient tuning scheme was found to be most effective with a reduction in modeling error of about 5% compared with the best of the alternative approaches considered in the study, which, according to the results of numerical experiments, was the method of multivariate adaptive regression splines. Full article