Asymptotic robustness of least median of squares for autoregressions with additive outliers

This paper considers the robustness properties in the time series context of the least median of squares (LMS) estimator. The influence function of the LMS estimator is derived under additive outlier contamination. This influence function is redescending and bounded for fixed values of the AR parameters. The gross-error sensitivity, however, is an unbounded function of the AR parameters. In order to asses the global robustness behavior of the LMS estimator, we consider several notions of breakdown. The breakdown points of the LMS estimator depend on the value of the underlying AR parameter. Generally, the breakdown point is below one half for high values of the AR parameter. The bias curves of the LMS estimator reveal, however, that the magnitude of outliers has to be considerable in order to cause breakdown.     

Robust estimation for ordinal regression

Ordinal regression is used for modelling an ordinal response variable as a function of some explanatory variables. The classical technique for estimating the unknown parameters of this model is Maximum Likelihood (ML). The lack of robustness of this estimator is formally shown by deriving its breakdown point and its influence function. To robustify the procedure, a weighting step is added to the Maximum Likelihood estimator, yielding an estimator with bounded influence function. We also show that the loss in efficiency due to the weighting step remains limited. A diagnostic plot based on the Weighted Maximum Likelihood estimator allows to detect outliers of different types in a single plot.     

THE MAXIMUM RESISTANCE OF TESTS

The breakdown point of an estimator is the smallest fraction of contamination that can force the value of the estimator beyond the boundary of the parameter space. It is well known that the highest possible breakdown point, under equivariance restrictions, is 50% of the sample. However, this upper bound is not always attainable. We give an example of an estimation problem in which the highest possible attainable breakdown point is much less than 50% of the sample. For hypothesis testing, we discuss the resistance of a test and propose new definitions of resistance. The maximum resistance to rejection (acceptance) is the smallest fraction of contamination necessary to force a test to reject (fail to reject) regardless of the original sample. We derive the maximum resistances of the t-test and sign test in the one-sample problem and of the t-test and Mood test in the two-sample problem. We briefly discuss another measure known as the expected resistance.     

A note on robustness of D-optimal block designs for two-colour microarray experiments

Two-colour microarray experiments form an important tool in gene expression analysis. Due to the high risk of missing observations in microarray experiments, it is fundamental to concentrate not only on optimal designs but also on designs which are robust against missing observations. As an extension of Latif et al. (2009), we define the optimal breakdown number for a collection of designs to describe the robustness, and we calculate the breakdown number for various D-optimal block designs. We show that, for certain values of the numbers of treatments and arrays, the designs which are D-optimal have the highest breakdown number. Our calculations use methods from graph theory.     

制造业项目结构分解过程模型研究

针对项目分解中工作分解结构方法的不足,以制造业项目为例,考虑项目产品本身的构造及特点的基础上,综合项目产品的工艺装配过程、产品数据管理技术以及项目组织结构,构建了项目结构分解的过程模型,并对该模型进行了详细分析,应用该模型能够更加科学、合理地进行项目分解,确定项目范围。     

Robust Sparse Regression with High-Breakdown Value

Penalized least squares estimators are sensitive to the influence of outliers like the ordinary least squares estimator. We propose a sparse regression estimator for robust variable selection and estimation based on a robust initial estimator. It is proven that our estimator has at least the same breakdown value as the initial estimator. Numerical examples are presented to illustrate our method.     

Multivariate trimmed means based on the Tukey depth

In univariate statistics, the trimmed mean has long been regarded as a robust and efficient alternative to the sample mean. A multivariate analogue calls for a notion of trimmed region around the center of the sample. Using Tukey's depth to achieve this goal, this paper investigates two types of multivariate trimmed means obtained by averaging over the trimmed region in two different ways. For both trimmed means, conditions ensuring asymptotic normality are obtained; in this respect, one of the main features of the paper is the systematic use of Hadamard derivatives and empirical processes methods to derive the central limit theorems. Asymptotic efficiency relative to the sample mean as well as breakdown point are also studied. The results provide convincing evidence that these location estimators have nice asymptotic behavior and possess highly desirable finite-sample robustness properties; furthermore, relative to the sample mean, both of them can in some situations be highly efficient for dimensions between 2 and 10.     

The generalized kullback-leibler divergence and robust inference

This paper examines robust techniques for estimation and tests of hypotheses using the family of generalized Kullback-Leibler (GKL) divergences. The GKL family is a new group of density based divergences which forms a subclass of disparities defined by Lindsay (1994). We show that the corresponding minimum divergence estimators have a breakdown point of 50% under the model. The performance of the proposed estimators and tests are investigated through an extensive numerical study involving real-data examples and simulation results. The results show that the proposed methods are attractive choices for highly efficient and robust methods.