Density Estimation Using Gumbel Kernel Estimator

In this article, our proposed kernel estimator, named as Gumbel kernel, which broadened the class of non-negative, asymmetric kernel density estimators. Such kernel estimator can be used in nonparametric estimation of the probability density function (pdf). When the density functions have limited bounded support on [0, ∞) and they are liberated of boundary bias, always non-negative and obtain the optimal rate of convergence for the mean integrated squared error (MISE). The bias, variance and the optimal bandwidth of the proposed estimators are investigated on theoretical grounds as well as on simulation basis. Further, the applicability of the proposed estimator is compared to Weibull kernel estimator, where performance of newly proposed kernel is outstanding.

Improved estimator of the continuous-time kernel estimator

There have been many papers presenting kernel density estimators for a strictly stationary continuous time process observed over the time interval [0, T ]. However the estimators do not satisfy the property of mean-square continuity if the process is mean-square continuous. In this paper we present a modified kernel estimator and substantiate that the modified estimator satisfies the property of mean-square continuity. In a simulation study the results show the modified estimator is better than the original estimator in some cases.

Asymptotic Confidence Bands for Copulas Based on the Local Linear Kernel Estimator

In this paper, we establish asymptotically optimal simultaneous confidence bands for the copula function based on the local linear kernel estimator proposed by Chen and Huang [1]. For this, we prove under smoothness conditions on the derivatives of the copula a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this estimator from its expectation. We also show that the bias term converges uniformly to zero with a precise rate. The performance of these bands is illustrated by a simulation study. An application based on pseudo-panel data is also provided for modeling the dependence structure of Senegalese households’ expense data in 2001 and 2006.

MODERATE DEVIATIONS AND LARGEDE VIATIONS FOR A TEST OF SYMMETRY BASED ON KERNEL DENSITY ESTIMATOR

Let fn be a non-parametric kernel density estimator based on a kernel function K. and a sequence of independent and identically distributed random variables taking values in R. The goal of this article is to prove moderate deviations and large deviations for the statistic sup ｜fn（x） - fn（-x） ｜.

A KERNEL-TYPE ESTIMATOR OF A QUANTILE FUNCTION UNDER RANDOMLY TRUNCATED DATA

A kernel-type estimator of the quantile function Q（p） = inf{t：F（t） ≥ p}, 0 ≤ p ≤ 1, is proposed based on the kernel smoother when the data are subjected to random truncation. The Bahadur-type representations of the kernel smooth estimator are established, and from Bahadur representations the authors can show that this estimator is strongly consistent, asymptotically normal, and weakly convergent.

Central limit theorem for integrated square error of kernel estimators of spherical density

LetX 1,…,X n be iid observations of a random variableX with probability density functionf(x) on the q-dimensional unit sphere Ωq in Rq+1,q ? 1. Let $f_n (x) = n^{ - 1} c(h)\sum\nolimits_{i = 1}^n {K[(1 - x'X_i )/h^2 ]} $ be a kernel estimator off(x). In this paper we establish a central limit theorem for integrated square error off n under some mild conditions.

Some uniform convergence results for kernel estimators

This paper derives some uniform convergence rates for kernel regression of some index functions that may depend on infinite dimensional parameter. The rates of convergence are computed for independent, strongly mixing and weakly dependent data respectively. These results extend the existing literature and are useful for the derivation of large sample properties of the estimators in some semiparametric and nonparametric models.

A KERNEL ESTIMATOR OF A DENSITY FUNCTION IN MULTIVARIATE CASE FROM RANDOMLY CENSORED DATA

A kernel density estimator is proposed when tile data are subject to censorship in multivariate case. The asymptotic normality, strong convergence and asymptotic optimal bandwidth which minimize the mean square error of the estimator are studied.

Enhancing microseismic/acoustic emission source localization accuracy with an outlier-robust kernel density estimation approach

Monitoring sensors in complex engineering environments often record abnormal data,leading to significant positioning errors.To reduce the influence of abnormal arrival times,we introduce an innovative,outlier-robust localization method that integrates kernel density estimation(KDE)with damping linear correction to enhance the precision of microseismic/acoustic emission(MS/AE)source positioning.Our approach systematically addresses abnormal arrival times through a three-step process:initial location by 4-arrival combinations,elimination of outliers based on three-dimensional KDE,and refinement using a linear correction with an adaptive damping factor.We validate our method through lead-breaking experiments,demonstrating over a 23%improvement in positioning accuracy with a maximum error of 9.12 mm(relative error of 15.80%)—outperforming 4 existing methods.Simulations under various system errors,outlier scales,and ratios substantiate our method’s superior performance.Field blasting experiments also confirm the practical applicability,with an average positioning error of 11.71 m(relative error of 7.59%),compared to 23.56,66.09,16.95,and 28.52 m for other methods.This research is significant as it enhances the robustness of MS/AE source localization when confronted with data anomalies.It also provides a practical solution for real-world engineering and safety monitoring applications.

Bayesian Classifier Based on Robust Kernel Density Estimation and Harris Hawks Optimisation

In real-world applications, datasets frequently contain outliers, which can hinder the generalization ability of machine learning models. Bayesian classifiers, a popular supervised learning method, rely on accurate probability density estimation for classifying continuous datasets. However, achieving precise density estimation with datasets containing outliers poses a significant challenge. This paper introduces a Bayesian classifier that utilizes optimized robust kernel density estimation to address this issue. Our proposed method enhances the accuracy of probability density distribution estimation by mitigating the impact of outliers on the training sample’s estimated distribution. Unlike the conventional kernel density estimator, our robust estimator can be seen as a weighted kernel mapping summary for each sample. This kernel mapping performs the inner product in the Hilbert space, allowing the kernel density estimation to be considered the average of the samples’ mapping in the Hilbert space using a reproducing kernel. M-estimation techniques are used to obtain accurate mean values and solve the weights. Meanwhile, complete cross-validation is used as the objective function to search for the optimal bandwidth, which impacts the estimator. The Harris Hawks Optimisation optimizes the objective function to improve the estimation accuracy. The experimental results show that it outperforms other optimization algorithms regarding convergence speed and objective function value during the bandwidth search. The optimal robust kernel density estimator achieves better fitness performance than the traditional kernel density estimator when the training data contains outliers. The Naïve Bayesian with optimal robust kernel density estimation improves the generalization in the classification with outliers.

ESTIMATORS AND SOME BEHAVIORS FORA PARTIALLY LINEAR MODEL WITH CENSORED DATA

This paper considers the local linear regression estimators for partially linear model with censored data. Which have some nice large-sample behaviors and are easy to implement. By many simulation runs, the author also found that the estimators show remarkable in the small sample case yet.

Functional Kernel Estimation of the Conditional Extreme Quantile under Random Right Censoring

The study of estimation of conditional extreme quantile in incomplete data frameworks is of growing interest. Specially, the estimation of the extreme value index in a censorship framework has been the purpose of many investigations when finite dimension covariate information has been considered. In this paper, the estimation of the conditional extreme quantile of a heavy-tailed distribution is discussed when some functional random covariate (i.e. valued in some infinite-dimensional space) information is available and the scalar response variable is right-censored. A Weissman-type estimator of conditional extreme quantiles is proposed and its asymptotic normality is established under mild assumptions. A simulation study is conducted to assess the finite-sample behavior of the proposed estimator and a comparison with two simple estimations strategies is provided.

Spatiotemporal Patterns of Road Network and Road Development Pri-ority in Three Parallel Rivers Region in Yunnan,China:An Evaluation Based on Modified Kernel Distance Estimate

Road network is a critical component of public infrastructure,and the supporting system of social and economic development.Based on a modified kernel density estimate(KDE)algorithm,this study evaluated the road service capacity provided by a road network composed of multi-level roads(i.e.national,provincial,county and rural roads),by taking account of the differences of effect extent and intensity for roads of different levels.Summarized at town scale,the population burden and the annual rural economic income of unit road service capacity were used as the surrogates of social and economic demands for road service.This method was applied to the road network of the Three Parallel River Region,the northwestern Yunnan Province,China to evaluate the development of road network in this region.In results,the total road length of this region in 2005 was 3.70×104km,and the length ratio between national,provincial,county and rural roads was 1∶2∶8∶47.From 1989 to 2005,the regional road service capacity increased by 13.1%,of which the contributions from the national,provincial,county and rural roads were 11.1%,19.4%,22.6%,and 67.8%,respectively,revealing the effect of′All Village Accessible′policy of road development in the mountainous regions in the last decade.The spatial patterns of population burden and economic requirement of unit road service suggested that the areas farther away from the national and provincial roads have higher road development priority(RDP).Based on the modified KDE model and the framework of RDP evaluation,this study provided a useful approach for developing an optimal plan of road development at regional scale.

APPROXIMATION RATES OF ERROR DISTRIBUTION OF DOUBLE KERNEL ESTIMATES OF CONDITIONAL DENSITY

In this paper, the normal approximation rate and the random weighting approximation rate of error distribution of the kernel estimator of conditional density function f(y|x) are studied. The results may be used to construct the confidence interval of f(y|x) .

Probability distribution of wind power volatility based on the moving average method and improved nonparametric kernel density estimation

In the process of large-scale,grid-connected wind power operations,it is important to establish an accurate probability distribution model for wind farm fluctuations.In this study,a wind power fluctuation modeling method is proposed based on the method of moving average and adaptive nonparametric kernel density estimation(NPKDE)method.Firstly,the method of moving average is used to reduce the fluctuation of the sampling wind power component,and the probability characteristics of the modeling are then determined based on the NPKDE.Secondly,the model is improved adaptively,and is then solved by using constraint-order optimization.The simulation results show that this method has a better accuracy and applicability compared with the modeling method based on traditional parameter estimation,and solves the local adaptation problem of traditional NPKDE.

Strong Convergence Rates of Double Kernel Estimates of Conditional Desity Under Stationary Sequences

In the paper,we study the strong convergence rates of double kernel estimates of conditional density under stationary sequences.

Kernel density estimation and marginalized-particle based probability hypothesis density filter for multi-target tracking

In order to improve the performance of the probability hypothesis density(PHD) algorithm based particle filter(PF) in terms of number estimation and states extraction of multiple targets, a new probability hypothesis density filter algorithm based on marginalized particle and kernel density estimation is proposed, which utilizes the idea of marginalized particle filter to enhance the estimating performance of the PHD. The state variables are decomposed into linear and non-linear parts. The particle filter is adopted to predict and estimate the nonlinear states of multi-target after dimensionality reduction, while the Kalman filter is applied to estimate the linear parts under linear Gaussian condition. Embedding the information of the linear states into the estimated nonlinear states helps to reduce the estimating variance and improve the accuracy of target number estimation. The meanshift kernel density estimation, being of the inherent nature of searching peak value via an adaptive gradient ascent iteration, is introduced to cluster particles and extract target states, which is independent of the target number and can converge to the local peak position of the PHD distribution while avoiding the errors due to the inaccuracy in modeling and parameters estimation. Experiments show that the proposed algorithm can obtain higher tracking accuracy when using fewer sampling particles and is of lower computational complexity compared with the PF-PHD.

AN EFFECTIVE IMAGE RETRIEVAL METHOD BASED ON KERNEL DENSITY ESTIMATION OF COLLAGE ERROR AND MOMENT INVARIANTS

In this paper, we propose a new method that combines collage error in fractal domain and Hu moment invariants for image retrieval with a statistical method - variable bandwidth Kernel Density Estimation (KDE). The proposed method is called CHK (KDE of Collage error and Hu moment) and it is tested on the Vistex texture database with 640 natural images. Experimental results show that the Average Retrieval Rate (ARR) can reach into 78.18%, which demonstrates that the proposed method performs better than the one with parameters respectively as well as the commonly used histogram method both on retrieval rate and retrieval time.

Bandwidth adaption for kernel particle filter

A novel particle filter bandwidth adaption for kernel particle filter （BAKPF） is proposed. Selection of the kernel bandwidth is a critical issue in kernel density estimation （KDE）. The plug-in method is adopted to get the global fixed bandwidth by optimizing the asymptotic mean integrated squared error （AMISE） firstly. Then, particle-driven bandwidth selection is invoked in the KDE. To get a more effective allocation of the particles, the KDE with adap- tive bandwidth in the BAKPF is used to approximate the posterior probability density function （PDF） by moving particles toward the posterior. A closed-form expression of the true distribution is given. The simulation results show that the proposed BAKPF performs better than the standard particle filter （PF）, unscented particle filter （UPF） and the kernel particle filter （KPF） both in efficiency and estimation precision.

SAR images classification method based on Dempster-Shafer theory and kernel estimate

To study the scene classification in the Synthetic Aperture Radar （SAR） image, a novel method based on kernel estimate, with the Maxkov context and Dempster-Shafer evidence theory is proposed. Initially, a nonpaxametric Probability Density Function （PDF） estimate method is introduced, to describe the scene of SAR images. And then under the Maxkov context, both the determinate PDF and the kernel estimate method axe adopted respectively, to form a primary classification. Next, the primary classification results are fused using the evidence theory in an unsupervised way to get the scene classification. Finally, a regularization step is used, in which an iterated maximum selecting approach is introduced to control the fragments and modify the errors of the classification. Use of the kernel estimate and evidence theory can describe the complicated scenes with little prior knowledge and eliminate the ambiguities of the primary classification results. Experimental results on real SAR images illustrate a rather impressive performance.