Nonparametric Kernel Estimation Research Articles

Medical Data for the Nadaraya-Watson Kernel Method in Nonparametric Regression Models Based on Hyperbolic Secant Function

Nonparametric kernel estimates used in this work aim to compare different treatment options by examining the recorded medical data. The following use of the suggested strategy depends on a kernel function and a parameter called bandwidth. The Nadaraya-Watson kernel (NWK) estimation is a necessary nonparametric kernel estimator used in regression models. A new Nadaraya-Watson regression estimate depends on the hyperbolic secant kernel (HSK) with fixed bandwidth (FNW) and Variable Bandwidth (VNW) is proposed.. We calculated some properties of the unknown regression function estimator, including bias, variance, optimal bandwidth, and a global measure of error criterion mean square error. Finally, simulation and three real data sets are used to evaluate its performance. Results from simulation and real data showed that the VNW using HSK is more effective than the FNW based on Average Mean Square Error Criterion. Also, Nadaraya-Watson using HSK function is more effective than Nadaraya-Watson using the Gaussian kernel density function.

Probabilistic estimation of the source component of seismic hazard in North-Eastern Brazil

Stable continental regions pose unique challenges for conducting Probabilistic Seismic Hazard Analysis because the earthquake activity driving mechanisms are poorly understood. For instance, the lower seismicity (hence the paucity of data) and the absence of well-defined active fault systems complicate accurately determining seismic source parameters. Northeastern Brazil is a stable continental region exhibiting moderate-size events recorded with significant seismic intensities and provoking the collapse of poorly constructed buildings in the last century. Thus, assessing the seismic hazard is critical for seismic risk mitigation. The seismic hazard depends on three components: source, path, and site, and here, we present the probabilistic seismic hazard analysis of the source component for NE Brazil. Spatial aggregation of earthquake sources outlined four areal seismic zones. A goodness-of-fit test rejected the Gutenberg-Richter model of magnitude frequency distribution in one of the studied seismic zones. For this reason, we estimated the magnitude probability distribution function in that zone using a nonparametric adaptive kernel estimator. In other zones the Gutenberg-Richter magnitude frequency model was applied. In either way of the magnitude probability distribution modelling we considered the upper bound for magnitude equal to 6.6 mR, based on the upper bound of a 95 % confidence interval for the standard normal distribution of palaeoearthquake sizes. Our findings suggests that potentially damaging events are likely to occur, and we cannot neglect chances for the occurrence of earthquakes exceeding 5.2 mR. The calculated mean return periods indicate significantly shorter intervals between consecutive large events than palaeoseismic records.

Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample ( X t , Y t , ζ t ) t ∈ [ 0 , T ] copies of a continuous-time stationary and ergodic process ( X , Y , ζ ) . The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process ζ = 1 and missing whenever ζ = 0 . Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.

On the dynamic residual measure of inaccuracy based on extropy in order statistics

AbstractIn this paper, we introduce a novel way to quantify the remaining inaccuracy of order statistics by utilizing the concept of extropy. We explore various properties and characteristics of this new measure. Additionally, we expand the notion of inaccuracy for ordered random variables to a dynamic version and demonstrate that this dynamic information measure provides a unique determination of the distribution function. Moreover, we investigate specific lifetime distributions by analyzing the residual inaccuracy of the first-order statistics. Nonparametric kernel estimation of the proposed measure is suggested. Simulation results show that the kernel estimator with bandwidth selection using the cross-validation method has the best performance. Finally, an application of the proposed measure on the model selection is provided.

An Approach to Constructing Explicit Estimators in Nonlinear Regression

We consider the problem of constructing explicit consistent estimators of finite-dimensionalparameters of nonlinear regression models using various nonparametric kernel estimators.

Study on Regional Differentiation of Quantitative and Qualitative Dividends of Population under Economic Growth Objectives Based on Panel Models and Nonparametric Kernel Estimation

In this paper, on the basis of combing through the relevant studies and theories in the field of China's demographic dividend, provincial panel data of 31 provinces, autonomous regions, and municipalities directly under the central government (excluding Hong Kong, Macao, and Taiwan) in China for the period of 2010-2020 are selected. Firstly, China's demographic dividend and the current situation of economic development are analyzed, and then panel regression models and non-parametric kernel regression models are constructed to explore the relationship between the demographic dividend's impact on economic growth in terms of both population size and population quality, while long-term equilibrium and short-term volatility are taken into account to realize the analysis of large-scale panel data. The empirical results show that China's population quantity dividend is disappearing, while the population quality dividend is increasing. The population quality dividend has a significant positive impact on economic growth, while the population quantity dividend has a convergence in relatively less economically developed regions. However, both the population quantity dividend and the population quality dividend have significant differences among different provinces. Based on the above theoretical analysis and empirical research, this paper puts forward some suggestions, including optimizing the population structure, improving the population quality, and rationally allocating regional resources. It is hoped that the relevant government departments can seize the current opportunities and advantages, strive to realize the transformation of population size and the improvement of population quality, and actively develop the demographic dividend.

Kernel method for estimating overlapping coefficient using numerical integration methods

In this paper, we proposed three nonparametric kernel estimators for the overlapping Weitzman measure Δ. Due to the difficulty of finding a general expression for Δ when the nonparametric kernel method is adopted, we suggest using the numerical integration method as the first stage of our estimation process. In particular, three numerical integration rules are considered, which are known as, trapezoidal and Simpson rules. The statistical properties of the resulting estimators are studied and investigated by using the simulation technique and their performances are also compared with some existing nonparametric kernel estimators developed by Eidous and Al-Talafhah (2020). The numerical results demonstrated the superiority and usefulness of the proposed technique over that suggested by Eidous and Al-Talafhah in estimating the overlapping measure Δ.

Determining the optimal time to report mortality after lobectomy for lung cancer: An analysis of the time-varying risk of death

ObjectiveSurgical mortality has traditionally been assessed at arbitrary intervals out to 1 year, without an agreed optimum time point. The aim of our study was to investigate the time-varying risk of death after lobectomy to determine the optimum period to evaluate surgical mortality rate after lobectomy for lung cancer. MethodsWe performed a retrospective study of patients undergoing lobectomy for lung cancer at our institution from 2015 to 2022. Parametric survival models were assessed and compared with a nonparametric kernel estimate. The hazard function was plotted over time according to the best-fit statistical distribution. The time points at which instantaneous hazard rate peaked and stabilized in the 1-year period after surgery were then determined. ResultsDuring the study period, 2284 patients underwent lobectomy for lung cancer. Cumulative mortality at 30, 90, and 180 days was 1.3%, 2.9%, and 4.9%, respectively. Log-logistic distribution showed the best fit compared with other statistical distribution, indicated by the lowest Akaike information criteria value. The instantaneous hazard rate was greatest during the immediate postoperative period (0.129; 95% confidence interval, 0.087-0.183) and diminishes rapidly within the first 30 days after surgery. Instantaneous hazard rate continued to decrease past 90 days and stabilized only at approximately 180 days. ConclusionsIn-hospital mortality is the optimal follow-up period that captures the early-phase hazard during the immediate postoperative period after lobectomy. Thirty-day mortality is not synonymous to “early mortality,” as instantaneous hazard rate remains elevated well past the 90-day time point and only stabilizes at approximately 180 days after lobectomy.

Boundary-adaptive kernel density estimation: the case of (near) uniform density

We consider nonparametric kernel estimation of density functions in the bounded-support setting having known support [ a , b ] using a boundary-adaptive kernel function and data-driven bandwidth selection, where a and b are finite and known prior to estimation. We observe, theoretically and in finite sample settings, that when bounds are known a priori this kernel approach is capable of outperforming even correctly specified parametric models, in the case of the uniform distribution. We demonstrate that this result has implications for modelling a range of densities other than the uniform case. Furthermore, when bounds [ a , b ] are unknown and the empirical support (i.e. [ min ( x i ) , max ( x i ) ] ) is used in their place, similar behaviour surfaces.

A study on optimal share pledge rate based on bilateral risk-CVaR of pledgee

This paper proposes the optimal pledge rate model based on the pledgee's bilateral risk-CVaR and dual-objective planning. First, a bilateral risk-CVaR model is constructed, for which a nonparametric kernel estimation method is presented, with a comparative analysis of the efficient frontier for “mean-variance”, “mean-CVaR”, and “mean-bilateral risk CVaR”. Second, it establishes a dual-objective planning model by the objectives of bilateral risk-CVaR and expected return of the pledgee, and accordingly develops an optimal pledge rate model combined with objective deviation, priority factor, and entropy method. Finally, the relationships between the pledge rate, the number of pledged shares, and the expected return are explored through simulation. The results illustrate that: (i) There are sequential inclusion relationships between the “mean-bilateral risk CVaR”, the “mean-CVaR” considering only the downside risk, and the “mean-variance” efficient sets of share pledge rate. (ii) As the number of shares increases, the pledgee's expected return also rises, and its sensitivity to the pledge rate grows. (iii) When the pledgee's expected return fixes, the number of pledged shares and the pledge rate have a U-shaped relationship. As it increases, the number of pledged shares rises, but the variation range of the pledge rate gradually decreases, and the pledgor's default risk declines.

Estimating Smoothness and Optimal Bandwidth for Probability Density Functions

The properties of non-parametric kernel estimators for probability density function from two special classes are investigated. Each class is parametrized with distribution smoothness parameter. One of the classes was introduced by Rosenblatt, another one is introduced in this paper. For the case of the known smoothness parameter, the rates of mean square convergence of optimal (on the bandwidth) density estimators are found. For the case of unknown smoothness parameter, the estimation procedure of the parameter is developed and almost surely convergency is proved. The convergence rates in the almost sure sense of these estimators are obtained. Adaptive estimators of densities from the given class on the basis of the constructed smoothness parameter estimators are presented. It is shown in examples how parameters of the adaptive density estimation procedures can be chosen. Non-asymptotic and asymptotic properties of these estimators are investigated. Specifically, the upper bounds for the mean square error of the adaptive density estimators for a fixed sample size are found and their strong consistency is proved. The convergence of these estimators in the almost sure sense is established. Simulation results illustrate the realization of the asymptotic behavior when the sample size grows large.

Extended Glivenko–Cantelli theorem and L 1 strong consistency of innovation density estimator for time-varying semiparametric ARCH model

ABSTRACT This paper extends the classical Glivenko–Cantelli theorem for the empirical cumulative distribution function based on the innovations in the ARCH model with a slowly time-varying trend. In this semiparametric time-varying model, strong consistency for the innovation density estimator via kernel smoothing method is established, given that the trend and ARCH parameter estimators meet some mild conditions. Besides, the strong consistency for the Gaussian quasi maximum likelihood estimator (QMLE) in the time-varying ARCH parameter is established as well. Moreover, in terms of the existence of the trend in the data, two major nonparametric trend estimators, B-spline and kernel estimators, are shown to be appropriate for the strong consistency results.

Strong consistency of nonparametric kernel estimators for integrated diffusion process

The asymptotic properties of nonparametric kernel estimators of diffusion process and integrated diffusion process were studied by scholars through using the theories of local time, giving the properties of consistency and asymptotic normality for nonparametric kernel estimators under appropriate conditions, but not property of strong consistency for integrated diffusion process. Instead of using the local time method, the paper applies the moment inequality of the ρ-mixing sequence to prove the strong consistency of the nonparametric kernel estimators in the integrated diffusion process. Our theorem conditions are mild and canonical, and some of them improve on the existing corresponding conditions. In numerical simulations and analysis of data from real applications, the nonparametric kernel estimators can capture well the variation characteristics of drift coefficient and diffusion coefficient, and that it is possible to fit parametric models with such characteristics, so that the economic interpretation of the models can be obtained.

Asymptotically Minimax Goodness-of-fit Testing for Single-index Models

In the context of non parametric multivariate regression model, we are interested in goodness-of-fit testing for the single-index models. These models are dimension reduction models and are therefore useful in multidimensional nonparametric statistics because of the well-known phenomenon called the curse of dimensionality. Fan and Li [5] have proposed the first consistent test for goodness-of-fit testing of the single-index by using nonparametric kernel estimation method and a central limit theorem for degenerate <img src=image/13427297_05.gif>-statistics of order higher than two. Since then, the minimax properties of this test have not been investigated. Following this work, we use here the asymptotic minimax approach. We are interested in finding the asymptotic minimax rate of testing <img src=image/13427297_01.gif> which gives the minimal distance between the null and alternative hypotheses such that a successful testing is possible. We propose a test procedure of level <img src=image/13427297_02.gif> which can tend to zero when the sample size tends to infinity. We have established the minimax asymptotic properties of our test procedure by showing that it reaches the asymptotic minimax rate <img src=image/13427297_01.gif> for the dimension <img src=image/13427297_03.gif> and there is no test of level <img src=image/13427297_02.gif> reaching this rate for <img src=image/13427297_04.gif>. Because of its minimax asymptotic properties, our test is able to distinguish the null hypothesis of the closest possible alternative. The results obtained were possible thanks to a large deviation result that we established for a degenerate U-statistic of order two appearing in our decision variable.

Nonparametric kernel estimation of the probability of cure in a mixture cure model when the cure status is partially observed.

Cure models are a class of time-to-event models where a proportion of individuals will never experience the event of interest. The lifetimes of these so-called cured individuals are always censored. It is usually assumed that one never knows which censored observation is cured and which is uncured, so the cure status is unknown for censored times. In this paper, we develop a method to estimate the probability of cure in the mixture cure model when some censored individuals are known to be cured. A cure probability estimator that incorporates the cure status information is introduced. This estimator is shown to be strongly consistent and asymptotically normally distributed. Two alternative estimators are also presented. The first one considers a competing risks approach with two types of competing events, the event of interest and the cure. The second alternative estimator is based on the fact that the probability of cure can be written as the conditional mean of the cure status. Hence, nonparametric regression methods can be applied to estimate this conditional mean. However, the cure status remains unknown for some censored individuals. Consequently, the application of regression methods in this context requires handling missing data in the response variable (cure status). Simulations are performed to evaluate the finite sample performance of the estimators, and we apply them to the analysis of two datasets related to survival of breast cancer patients and length of hospital stay of COVID-19 patients requiring intensive care.

Estimating multivariate density and its derivatives for mixed measurement error data

In this paper, we propose a nonparametric mixed kernel estimator for a multivariate density function and its derivatives when the data are contaminated with different sources of measurement errors. The proposed estimator is a mixture of the classical and the deconvolution kernels, accounting for the error-free and error-prone variables, respectively. Large sample properties of the proposed nonparametric estimator, including the order of the mean squares error, the consistency, and the asymptotic normality, are thoroughly investigated. The optimal convergence rates among all nonparametric estimators for different measurement error structures are derived, and it is shown that the proposed mixed kernel estimators achieve the optimal convergence rate. A simulation study is conducted to evaluate the finite sample performance of the proposed estimators.

Composite laminates reliability assessment using diffusion process backed up by perspective forms of non-parametric kernel estimators

Numerous technical systems are modeled to identify dependability and safety measures. The modeling approach here is a real data driven approach. We work with the results obtained by assessing a progressive material—natural fibers reinforced laminate composites. The knowledge of certain properties, namely reliability performance, has not been sufficiently studied. Our approaches include the Wiener diffusion process, parametric and non-parametric estimations and mainly a hazard function which is estimated parametrically and non-parametrically by kernel estimators. We also demonstrate the significance between parametric and non-parametric estimations, the way they influence the selection of a resulting model and prospective practical applicability.

Sequential Bayesian bandwidth selection for multivariate kernel regression with applications

A new Bayesian bandwidth selection procedure is proposed for nonparametric kernel estimates based on the sequential Monte Carlo method. Compared with the existing Bayesian bandwidth selector of Zhang et al. (2009), this new method can enhance the convergence to the global optimum with a substantially faster computation speed. In particular, the method offers an improved out-of-sample performance as shown by simulations. The bandwidth selector is applied to the option state price density, production function, and nonparametric relationship between oil and stock index returns; results indicate that our proposed method outperforms other methods in terms of the mean square error and log-likelihood in all applications.

Probability Function Estimation for the Maximum Precipitation Model Using Kernel Estimators

The distribution of maximum rainfall level is not a homogeneous phenomenon and is often characterized by multimodality, and often the phenomenon of heavy right-hand tail. Modelling this phenomenon using classic probability distributions leads to ignoring multimodality, and thus underestimating or overestimating the predicted values in the tail tails – the most important from the point of view of safe dimen-sioning of drainage systems. To avoid the abovementioned difficulties, a non-parametric kernel estimator method of maximum precipitation density function was used (on the example of rainfall data from a selected station in Poland). The meth-odology proposed in the paper (for use on any rainfall data from other meteorologi-cal stations) will allow the development of more reliable local models of maximum precipitation.

Sensitivity study of ring-shaped electrostatic sensor based on nonparametric estimation

In this paper, aiming at the problem of the electrostatic sensor signal satisfying the gaussian distribution, the non-parametric kernel estimation method is introduced, and the electrode induction model of the electrostatic sensor is finally fitted by combining the goodness of fit and the simulation data samples. This model satisfies the gaussian distribution and the electrostatic signal satisfying the gaussian distribution is given in the theory. Maxwell simulation software was used to simulate the theoretical sensitivity of the electrostatic sensor and the axial and radial spatial sensitivity characteristics of different sensor parameters were obtained. Within a certain range, the relative permittivity of the insulating tube is also discussed. Finally, an insulating tube with a relative permittivity of 3 is selected as the material of the insulating tube. Finally, the experiment is carried out on the experimental equipment and the conclusions obtained in the article are confirmed.