LIMITING BEHAVIOR OF RECURSIVE M-ESTIMATORS IN MULTIVARIATE LINEAR REGRESSION MODELS AND THEIR ASYMPTOTIC EFFICIENCIES

Recursive algorithms are very useful for computing M-estimators of regression coefficients and scatter parameters. In this article, it is shown that for a nondecreasing ul （t）, under some mild conditions the recursive M-estimators of regression coefficients and scatter parameters are strongly consistent and the recursive M-estimator of the regression coefficients is also asymptotically normal distributed. Furthermore, optimal recursive M-estimators, asymptotic efficiencies of recursive M-estimators and asymptotic relative efficiencies between recursive M-estimators of regression coefficients are studied.

Algorithmic Study of M-Estimators for Multi-Function Sensor Data Reconstruction

This paper describes a data reconstruction technique for a multi-function sensor based on the Mestimator, which uses least squares and weighted least squares method. The algorithm has better robustness than conventional least squares which can amplify the errors of inaccurate data. The M-estimator places particular emphasis on reducing the effects of large data errors, which are further overcome by an iterative regression process which gives small weights to large off-group data errors and large weights to small data errors. Simulation results are consistent with the hypothesis with 81 groups of regression data having an average accuracy of 3.5%, which demonstrates that the M-estimator provides more accurate and reliable data reconstruction.

Moderate Deviations for M-estimators in Linear Models with φ-mixing Errors

In this paper, the moderate deviations for the M-estimators of regression parameter in a linear model are obtained when the errors form a strictly stationary Ф-mixing sequence. The results are applied to study many different types of M-estimators such as Huber＇s estimator, L^P-regression estimator, least squares estimator and least absolute deviation estimator.

Rates of convergence of M-estimators for partly linear models involving time series

The asymptotic behaviour of M-estimalors constructed with B-spline method based on strictly stationary β-mixing observations of a partly linear model is dealt with. Under some regular conditions, it is proved that the M-estimators of the vector of parameters are asymptotically normal and the M-estimators of the nonparametric component achieve the optimal convergence rates for nonparametric regression. Our asymptotic theory includes L1-, L2-, Lp-norm, and Huber estimators as special cases.

ON B-SPLINE M-ESTIMATORS IN A SEMIPARAMETRIC REGRESSION MODEL

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Bahadur Representation of Nonparametric M-Estimators for Spatial Processes

Under some mild conditions, we establish a strong Bahadur representation of a general class of nonparametric local linear M-estimators for mixing processes on a random field. If the socalled optimal bandwidth hn = O（｜n｜^-1/5）, n ∈ Z^d, is chosen, then the remainder rates in the Bahadur representation for the local M-estimators of the regression function and its derivative are of order O（｜n｜^-4/5 log ｜n｜）. Moreover, we derive some asymptotic properties for the nonparametric local linear M-estimators as applications of our result.

Study of M-estimator Variational Retrieval Using Simulated Feng Yun-3A Data

This paper adopts satellite channel brightness temperature simulation to study M-estimator variational retrieval. This approach combines both the advantages of classical variational inversion and robust M-estimators. Classical variational inversion depends on prior quality control to elim- inate outliers, and its errors follow a Gaussian distribution. We coupled the M-estimators to the framework of classical variational inversion to obtain a M-estimator variational inversion. The cost function contains the M-estimator to guarantee the robustness to outliers and improve the retrieval re- sults. The experimental evaluation adopts Feng Yun-3A （FY-3A） simulated data to add to the Gaussian and Non-Gaussian error. The variational in- version is used to obtain the inversion brightness temperature, and temperature and humidity data are used for validation. The preliminary results demonstrate the potential of M-estimator variational retrieval.

The M-estimate of Local Linear Regression with Variable Window Breadth

In this paper, by using the Brouwer fixed point theorem, we consider the existence and uniqueness of the solution for local linear regression with variable window breadth.

基于自适应权值的点云三维物体重建算法研究

基于三维扫描点云数据的三维物体重建是计算机图形学中非常重要的课题,在计算机动画、医学图像处理等多方面都有应用。其中基于最小二乘问题的Levenberg-Marquart算法和基于极大似然估计的M-Estimator算法都是不错的方案。但是当点的数量过多过少或者点云中有噪声时,这些方案产生的结果都会有较大的误差,影响重建的效果。为了解决这两个问题,结合Levenberg-Marquart算法和M-Estimator算法,提出了一种新的算法。该算法结合Levenberg-Marquart算法较快的收敛性和M-Estimator算法的抗噪性,能很好地解决点数量较多和噪声点影响结果的问题。通过在M-Estimator的权重函数上进行改进,提出自适应的权值函数,用灵活变动和自适应的值代替原来的固定值,使算法在噪声等级较高时也能表现良好。最后将算法应用在球体和圆柱上,并和最新的研究成果进行对比,数据说明算法无论是在点云数量较多还是在噪声等级较高的情况下都明显优于其他已知算法。

Likelihood and Quadratic Distance Methods for the Generalized Asymmetric Laplace Distribution for Financial Data

Maximum likelihood (ML) estimation for the generalized asymmetric Laplace (GAL) distribution also known as Variance gamma using simplex direct search algorithms is investigated. In this paper, we use numerical direct search techniques for maximizing the log-likelihood to obtain ML estimators instead of using the traditional EM algorithm. The density function of the GAL is only continuous but not differentiable with respect to the parameters and the appearance of the Bessel function in the density make it difficult to obtain the asymptotic covariance matrix for the entire GAL family. Using M-estimation theory, the properties of the ML estimators are investigated in this paper. The ML estimators are shown to be consistent for the GAL family and their asymptotic normality can only be guaranteed for the asymmetric Laplace (AL) family. The asymptotic covariance matrix is obtained for the AL family and it completes the results obtained previously in the literature. For the general GAL model, alternative methods of inferences based on quadratic distances (QD) are proposed. The QD methods appear to be overall more efficient than likelihood methods infinite samples using sample sizes n ≤5000 and the range of parameters often encountered for financial data. The proposed methods only require that the moment generating function of the parametric model exists and has a closed form expression and can be used for other models.

Unified Asymptotic Results for Maximum Spacing and Generalized Spacing Methods for Continuous Models

Asymptotic results are obtained using an approach based on limit theorem results obtained for α-mixing sequences for the class of general spacings (GSP) methods which include the maximum spacings (MSP) method. The MSP method has been shown to be very useful for estimating parameters for univariate continuous models with a shift at the origin which are often encountered in loss models of actuarial science and extreme models. The MSP estimators have also been shown to be as efficient as maximum likelihood estimators in general and can be used as an alternative method when ML method might have numerical difficulties for some parametric models. Asymptotic properties are presented in a unified way. Robustness results for estimation and parameter testing results which facilitate the applications of the GSP methods are also included and related to quasi-likelihood results.

Robust Inference for Time-Varying Coefficient Models with Longitudinal Data

Time-varying coefficient models are useful in longitudinal data analysis. Various efforts have been invested for the estimation of the coefficient functions, based on the least squares principle. Related work includes smoothing spline and kernel methods among others, but these methods suffer from the shortcoming of non-robustness. In this paper, we introduce a local M-estimation method for estimating the coefficient functions and develop a robustified generalized likelihood ratio (GLR) statistic to test if some of the coefficient functions are constants or of certain parametric forms. The robustified GLR test is robust against outliers and the error distribution. This provides a useful robust inference tool for the models with longitudinal data. The bandwidth selection issue is also addressed to facilitate the implementation in practice. Simulations show that the proposed testing method is more powerful in some situations than its counterpart based on the least squares principle. A real example is also given for illustration.

MetaCost与重采样结合的不平衡分类算法——RS-MetaCost

不平衡分类是当今机器学习中的研究热点与难点。为提高不平衡数据的分类效果,提出MetaCost与重采样结合的不平衡分类算法——RS-MetaCost。首先在MetaCost划分子集前对不平衡数据集进行重采样,即过采样少数类或欠采样多数类,以降低或消除数据不平衡程度;其次在预测概率阶段,利用m-estimation提高少数类预测概率。采用6组模拟数据集与10组实例数据集,将RS-MetaCost与经典算法进行比较实验。结果表明,在大多数数据集上,RS-MetaCost在保证整体分类精度很高的前提下,还能提高少数类的分类精度,且过采样下的RS-MetaCost优于欠采样下的RS-MetaCost。

Robust and Efficient Reliability Estimation for Exponential Distribution

In modeling reliability data,the exponential distribution is commonly used due to its simplicity.For estimating the parameter of the exponential distribution,classical estimators including maximum likelihood estimator represent the most commonly used method and are well known to be efficient.However,the maximum likelihood estimator is highly sensitive in the presence of contamination or outliers.In this study,a robust and efficient estimator of the exponential distribution parameter was proposed based on the probability integral transform statistic.To examine the robustness of this new estimator,asymptotic variance,breakdown point,and gross error sensitivity were derived.This new estimator offers reasonable protection against outliers besides being simple to compute.Furthermore,a simulation study was conducted to compare the performance of this new estimator with the maximum likelihood estimator,weighted likelihood estimator,and M-scale estimator in the presence of outliers.Finally,a statistical analysis of three reliability data sets was conducted to demonstrate the performance of the proposed estimator.

A Yield Mapping Procedure Based on Robust Fitting Paraboloid Cones on Moving Elliptical Neighborhoods and the Determination of Their Size Using a Robust Variogram

The yield map is generated by fitting the yield surface shape of yield monitor data mainly using paraboloid cones on floating neighborhoods. Each yield map value is determined by the fit of such a cone on an elliptical neighborhood that is wider across the harvest tracks than it is along them. The coefficients of regression for modeling the paraboloid cones and the scale parameter are estimated using robust weighted M-estimators where the weights decrease quadratically from 1 in the middle to zero at the border of the selected neighborhood. The robust way of estimating the model parameters supersedes a procedure for detecting outliers. For a given neighborhood shape, this yield mapping method is implemented by the Fortran program paraboloidmapping.exe, which can be downloaded from the web. The size of the selected neighborhood is considered appropriate if the variance of the yield map values equals the variance of the true yields, which is the difference between the variance of the raw yield data and the error variance of the yield monitor. It is estimated using a robust variogram on data that have not had the trend removed.

Nonlocal-Means Image Denoising Technique Using Robust M-Estimator

Edge preserved smoothing techniques have gained importance for the purpose of image processing applications A good edge preserving filter is given by nonlocal-means filter rather than any other linear model based approaches. This paper explores a different approach of nonlocal-means filter by using robust M-estimator function rather than the exponential function for its weight calculation. Here the filter output at each pixel is the weighted average of pixels with surrounding neighborhoods using the chosen robust M-estimator function. The main direction of this paper is to identify the best robust M-estimator function for nonlocal-means denoising algorithm. In order to speed up the computation, a new patch classification method is followed to eliminate the uncorrelated patches from the weighted averaging process. This patch classification approach compares favorably to existing techniques in respect of quality versus computational time. Validations using standard test images and brain atlas images have been analyzed and the results were compared with the other known methods. It is seen that there is reason to believe that the proposed refined technique has some notable points.

ASYMPTOTIC NORMALITY OF M-ESTIMATES IN THE EV MODEL

The M-estimate of parameters in the errors-in-variables (EV) model Y =xτβ0+∈,X =x+u ((∈,uτ)τ is a (p+1)-dimensional spherical error, Coy[(∈, uτ)τ] =σ2Ip+1)being considered. The M-estimate βn,, of β0 under a general ρ(·) function and the estimateof σ2 are given, the strong consistency and asymptotic normality of βn as well as are obtained. The conditions for the ρ(·) function in this paper are similar to that of linearexpression of M-estimates in the linear regression model.

Variable bandwidth and one-step local M-estimator

A robust version of local linear regression smoothers augmented with variable bandwidth is studied. The proposed method inherits the advantages of local polynomial regression and overcomes the shortcoming of lack of robustness of least-squares techniques. The use of variable bandwidth enhances the flexibility of the resulting local M- estimators and makes them possible to cope well with spatially inho-mogeneous curves, heteroscedastic errors and nonuniform design densities. Under appropriate regularity conditions, it is shown that the proposed estimators exist and are asymptotically normal. Based on the robust estimation equation, one-step local M-estimators are introduced to reduce computational burden. It is demonstrated that the one-step local M-estimators share the same asymptotic distributions as the fully iterative M-estimators, as long as the initial estimators are good enough. In other words, the one-step local M-estimators reduce significantly the computation cost of the fully iterative

Strong consistency of M-estimates in linear models

The strong consistency of M-estimates of the regression coefficients in a linear modelunder some mild conditions is established, which is an essential improvement over the relevantresults in the literature on the moment condition. Especially, in some important circumstances, onlyE ψ(ek) q for some q＞1 is needed, where ψ(eκ) is some score function of random error.

A note on constrained M-estimation and its recursive analog in multivariate linear regression models

In this paper,the constrained M-estimation of the regression coeffcients and scatter parameters in a general multivariate linear regression model is considered.Since the constrained M-estimation is not easy to compute,an up-dating recursion procedure is proposed to simplify the com-putation of the estimators when a new observation is obtained.We show that,under mild conditions,the recursion estimates are strongly consistent.In addition,the asymptotic normality of the recursive constrained M-estimators of regression coeffcients is established.A Monte Carlo simulation study of the recursion estimates is also provided.Besides,robustness and asymptotic behavior of constrained M-estimators are briefly discussed.