Double k-Class Estimators in Regression Models with Non-spherical Disturbances

In this paper, we consider a family of feasible generalised double k-class estimators in a linear regression model with non-spherical disturbances. We derive the large sample asymptotic distribution of the proposed family of estimators and compare its performance with the feasible generalized least squares and Stein-rule estimators using the mean squared error matrix and risk under quadratic loss criteria. A Monte-Carlo experiment investigates the finite sample behaviour of the proposed family of estimators.     

Risk and Pitman closeness properties of feasible generalized double k-class estimators in linear regression models with non-spherical disturbances under balanced loss function

In this article, a family of feasible generalized double k-class estimator in a linear regression model with non-spherical disturbances is considered. The performance of this estimator is judged with feasible generalized least-squares and feasible generalized Stein-rule estimators under balanced loss function using the criteria of quadratic risk and general Pitman closeness. A Monte-Carlo study investigates the finite sample properties of several estimators arising from the family of feasible double k-class estimators.     

Minimax lower bound for kink location estimators in a nonparametric regression model with long-range dependence

In this paper, a lower bound is determined in the minimax sense for change point estimators of the first derivative of a regression function in the fractional white noise model. Similar minimax results presented previously in the area focus on change points in the derivatives of a regression function in the white noise model or consider estimation of the regression function in the presence of correlated errors.     

Robust nonparametric estimators of monotone boundaries

This paper revisits some asymptotic properties of the robust nonparametric estimators of order-m and order-α quantile frontiers and proposes isotonized version of these estimators. Previous convergence properties of the order-m frontier are extended (from weak uniform convergence to complete uniform convergence). Complete uniform convergence of the order-m (and of the quantile order-α) nonparametric estimators to the boundary is also established, for an appropriate choice of m (and of α, respectively) as a function of the sample size. The new isotonized estimators share the asymptotic properties of the original ones and a simulated example shows, as expected, that these new versions are even more robust than the original estimators. The procedure is also illustrated through a real data set.     

线性模型参数的稳健化有偏估计

本文讨论复共线性和粗差同时存在时线性模型的参数估计问题,基于等价权原理提出了一个稳健有偏估计类（稳健压缩估计）,并且建立了稳健压缩估计的计算方法,为了满足实际问题的需要,构造了许多很有意义的稳健有偏估计,例如稳健岭估计、稳健主成分估计,稳健组合主成估计、稳健单参数主成分估计、稳健根方估计等等,最后通过一个算例表明,本文提出的稳健有偏估计具有既可克服复共线性影响又可抵抗粗差干扰的良好性质。     

1D quintic nonlinear Schrödinger equation with white noise dispersion

In this article, we improve the Strichartz estimates obtained in A. de Bouard, A. Debussche (2010) [12] for the Schrödinger equation with white noise dispersion in one dimension. This allows us to prove global well posedness when a quintic critical nonlinearity is added to the equation. We finally show that the white noise dispersion is the limit of smooth random dispersion.     

Influence curves of general statistics

The influence curve (JC) of a Fisher-consistent functional was introduced by F. Hampel and plays a central role in the search for robust estimators. An extension of this notion to non-Fisher-consistent functionals is proposed in order to investigate the infinitesimal robustness of more general statistics, e.g. those used in hypothesis testing. This new definition inherits many useful properties, including some on asymptotic efficiency. Functionals in two variables, arising from two-sample statistics, are treated too. Connections with Hodges-Lehmann shift estimators are discovered. One- and two-sample rank statistics illustrate the theory.     

Robust goodness-of-fit tests for AR(p) models based onL 1-norm fitting

A robustified residual autocorrelation is defined based onL ₁-regression. Under very general conditions, the asymptotic distribution of the robust residual autocorrelation is obtained. A robustified portmanteau statistic is then constructed which can be used in checking the goodness-of-fit of AR(p) models when usingL ₁-norm fitting. Empirical results show thatL ₁-norm estimators and the proposed portmanteau statistic are robust against outliers, error distributions, and accuracy for a given finite sample. Project supported by the Foundation of State Educational Commission and a research grant from the Doctoral Program Foundation of China (#97000139).     

Simple robust technique using time delay estimation for the control and synchronization of Lorenz systems

This work presents two simple and robust techniques based on time delay estimation for the respective control and synchronization of chaos systems. First, one of these techniques is applied to the control of a chaotic Lorenz system with both matched and mismatched uncertainties. The nonlinearities in the Lorenz system is cancelled by time delay estimation and desired error dynamics is inserted. Second, the other technique is applied to the synchronization of the Lü system and the Lorenz system with uncertainties. The synchronization input consists of three elements that have transparent and clear meanings.Since time delay estimation enables a very effective and efficient cancellation of disturbances and nonlinearities, the techniques turn out to be simple and robust. Numerical simulation results show fast, accurate and robust performance of the proposed techniques, thereby demonstrating their effectiveness for the control and synchronization of Lorenz systems.     

Robust estimation in single-index models when the errors have a unimodal density with unknown nuisance parameter

This paper develops a robust profile estimation method for the parametric and nonparametric components of a single-index model when the errors have a strongly unimodal density with unknown nuisance parameter. We derive consistency results for the link function estimators as well as consistency and asymptotic distribution results for the single-index parameter estimators. Under a log-Gamma model, the sensitivity to anomalous observations is studied using the empirical influence curve. We also discuss a robust K-fold cross-validation procedure to select the smoothing parameters. A numerical study carried on with errors following a log-Gamma model and for contaminated schemes shows the good robustness properties of the proposed estimators and the advantages of considering a robust approach instead of the classical one. A real data set illustrates the use of our proposal.

     

A robust test of specification based on order statistics

The test of misspecification presented compares parametric and nonparametric regressions. The latter is estimated using order statistics, which provide robust and distribution free estimators. The former is estimated using, in turn, least squares and least absolute deviation (LAD). When implementing LAD, robust and distribution free estimators are considered in both parametric and nonparametric regressions. This defines a very homogeneous test which can discriminate misspecification from the impact of outliers and/or skewness. These two effects are instead mixed together in the tests comparing OLS with nonparametric estimators, potentially driving to erroneous conclusions. An example and a Monte Carlo experiment analyze the behavior of the proposed test.     

Numerical experiments with various optimal estimators

Numerical experiments are performed to measure the relative accuracy and computational efficiency of various estimators for the parameters and state of a linear dynamic system with forcing using a finite sequence of measurements containing noise. This nonlinear estimation problem is treated by estimators based on least-squares and maximumlikelihood criteria and, for the linearized problem, two mechanizations of the Kalman-Bucy estimator are applied. A digital computer simulation of an example problem is performed, and a Monte Carlo technique is used to generate statistics of the errors in the estimates empirically. This process is carried out for a range ofa priori error statistics.This work was supported by the National Science Foundation, Systems Grant No. GU-1153.     

Estimation with Sequential Order Statistics from Exponential Distributions

The lifetime of an ordinary k-out-of-n system is described by the (n–k+1)-st order statistic from an iid sample. This set-up is based on the assumption that the failure of any component does not affect the remaining ones. Since this is possibly not fulfilled in technical systems, sequential order statistics have been proposed to model a change of the residual lifetime distribution after the breakdown of some component. We investigate such sequential k-out-of-n systems where the corresponding sequential order statistics, which describe the lifetimes of these systems, are based on one- and two-parameter exponential distributions. Given differently structured systems, we focus on three estimation concepts for the distribution parameters. MLEs, UMVUEs and BLUEs of the location and scale parameters are presented. Several properties of these estimators, such as distributions and consistency, are established. Moreover, we illustrate how two sequential k-out-of-n systems based on exponential distributions can be compared by means of the probability P(X < Y). Since other models of ordered random variables, such as ordinary order statistics, record values and progressive type II censored order statistics can be viewed as sequential order statistics, all the results can be applied to these situations as well.     

A solution to anisotropic suboptimal filtering problem by convex optimization

This paper considers a robust filtering problem for a linear discrete time invariant system with measured and estimated outputs. The system is exposed to random disturbances with imprecisely known distributions generated by an unknown stable shaping filter from the Gaussian white noise. The stochastic uncertainty of the input disturbance is measured by the mean anisotropy functional. The estimation error is quantified by the anisotropic norm which is a stochastic analogue of the H _∞ norm. A sufficient condition for an estimator to exist and ensure that the error is less than a given threshold value is derived in form of a convex inequality on the determinant of a positive definite matrix and two linear matrix inequalities. The suboptimal problem setting results to a set of the estimators ensuring the anisotropic norm of the error to be strictly bounded thereby providing some additional degree of freedom to impose some additional constraints on the estimator performance specification.     

A local breakdown property of robust tests in linear regression

The breakdown slope, as a useful summary measure of local stability for estimators and test statistics, has been studied recently by He, Simpson, and Protnoy (1990, J. Amer. Statist. Assoc., 85). It is shown here that all regression estimates based on residuals alone in linear models have zero breakdown slopes in contamination neighborhoods, even though they can have breakdown points as high as one-half. The breakdown functions of tests based on the S-estimation are investigated. It is also shown that the Generalized M-estimators can have better local breakdown robustness. One way to obtain regression estimators with desirable local and global breakdown properties is discussed.     

Testing model assumptions in functional regression models

In the functional regression model where the responses are curves, new tests for the functional form of the regression and the variance function are proposed, which are based on a stochastic process estimating L²-distances. Our approach avoids the explicit estimation of the functional regression and it is shown that normalized versions of the proposed test statistics converge weakly. The finite sample properties of the tests are illustrated by means of a small simulation study. It is also demonstrated that for small samples, bootstrap versions of the tests improve the quality of the approximation of the nominal level.     

Fractional generalized L関y random fields as white noise functionals

In this paper, we construct the fractional generalized Lévy random fields (FGLRF) as tempered white noise functionals. We find that this white noise approach is very effective in investigating the properties of these fields. Under some conditions, the fractional Lévy fields in the usual sense are obtained. In addition, we also present a method to construct the anisotropic fractional generalized Lévy random fields (AFGLRF).     

Robust autoregressive estimates using quadratic programming

The robust estimation of the autoregressive parameters is formulated in terms of the quadratic programming problem. This article's main contribution is to present an estimator that down weights both types of outliers in time series and improves the forecasting results. New robust estimates are yielded, by combining optimally two weight functions suitable for Innovation and Additive outliers in time series. The technique which is developed here is based on an approach of mathematical programming applications to I_p-approximation. The behavior of the estimators are illustrated numerically, under the additive outlier generating model. Monte Carlo results show that the proposed estimators compared favorably with respect to M-estimators and bounded influence estimators. Based on these results we conclude that one can improve the robust properties of AR(p) estimators using quadratic programming.     

Fractional generalized Lévy random fields as white noise functionals

In this paper, we construct the fractional generalized Lévy random fields (FGLRF) as tempered white noise functionals. We find that this white noise approach is very effective in investigating the properties of these fields. Under some conditions, the fractional Lévy fields in the usual sense are obtained. In addition, we also present a method to construct the anisotropic fractional generalized Lévy random fields (AFGLRF).