Optimal robust influence functions in semiparametric regression

Robust statistics allows the distribution of the observations to be any member of a suitable neighborhood about an ideal model distribution. In this paper, the ideal models are semiparametric with finite-dimensional parameter of interest and a possibly infinite-dimensional nuisance parameter.In the asymptotic setup of shrinking neighborhoods, we derive and study the Hampel-type problem and the minmax MSE-problem. We show that, for all common types of neighborhood systems, the optimal influence function can be approximated by the optimal influence functions for certain parametric models.For general semiparametric regression models, we determine in case of error-in-variables and in case of error-free-variables.Finally, the results are applied to Cox regression where we compare our approach to that of Bednarski [1993. Robust estimation in Cox's regression model. Scand. J. Statist. 20, 213–225] in a small simulation study and on a real data set.     

Comparative Study of Robust Estimators Based on a Sensitivity Coefficient in Principal Component Analysis

This work is devoted to robust principal component analysis (PCA). We give a comparison between some multivariate estimators of location and scatter by computing the influence functions of the sensitivity coefficient ρ corresponding to these estimators, and the mean squared error (MSE) of estimators of ρ. The coefficient ρ measures the closeness between the subspaces spanned by the initial eigenvectors and their corresponding version derived from an infinitesimal perturbation of the data distribution.     

Predictive Influence of Unavailable Values of Future Explanatory Variables in a Linear Model

We consider an approach to prediction in linear model when values of the future explanatory variables are unavailable, we predict a future response y ^f at a future sample point x ^f when some components of x ^f are unavailable. We consider both the cases where x ^f are dependent and independent but normally distributed. A Taylor expansion is used to derive an approximation to the predictive density, and the influence of missing future explanatory variables (the loss or discrepancy) is assessed using the Kullback–Leibler measure of divergence. This discrepancy is compared in different scenarios including the situation where the missing variables are dropped entirely.     

Robustness of the Affine Equivariant Scatter Estimator Based on the Spatial Rank Covariance Matrix

Visuri et al. (2000 Visuri, S., Koivunen, V., Oja, H. (2000). Sign and rank covariance matrices. J. Stat. Plann. Inference 91:557–575.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a technique for robust covariance matrix estimation based on different notions of multivariate sign and rank. Among them, the spatial rank based covariance matrix estimator that utilizes a robust scale estimator is especially appealing due to its high robustness, computational ease, and good efficiency. Also, it is orthogonally equivariant under any distribution and affinely equivariant under elliptically symmetric distributions. In this paper, we study robustness properties of the estimator with respective to two measures: breakdown point and influence function. More specifically, the upper bound of the finite sample breakdown point can be achieved by a proper choice of univariate robust scale estimator. The influence functions for eigenvalues and eigenvectors of the estimator are derived. They are found to be bounded under some assumptions. Moreover, finite sample efficiency comparisons to popular robust MCD, M, and S estimators are reported.     

Impact of Contamination on Training and Test Error Rates in Statistical Clustering

The k-means algorithm is one of the most common non hierarchical methods of clustering. It aims to construct clusters in order to minimize the within cluster sum of squared distances. However, as most estimators defined in terms of objective functions depending on global sums of squares, the k-means procedure is not robust with respect to atypical observations in the data. Alternative techniques have thus been introduced in the literature, e.g., the k-medoids method. The k-means and k-medoids methodologies are particular cases of the generalized k-means procedure. In this article, focus is on the error rate these clustering procedures achieve when one expects the data to be distributed according to a mixture distribution. Two different definitions of the error rate are under consideration, depending on the data at hand. It is shown that contamination may make one of these two error rates decrease even under optimal models. The consequence of this will be emphasized with the comparison of influence functions and breakdown points of these error rates.     

The influence of observations on misclassification probability in multiple discriminant analysis

The influence of observations in estimating the misclassification probability in multiple discriminant analysis is studied using the common omission approach. An empirical influence function for the misclassification probability is also derived, It can give a very good approximation to the omission approach, but the computational load is much reduced, Various extensions of the measures are suggested. The proposed measures are applied to the famous Iris data set. The same three observations are identified as having the most influence under different measures.     

INFLUENCE FUNCTION OF THE LIKELIHOOD RATIO TEST STATISTIC FOR MULTIVARIATE NORMAL SAMPLE

ABSTRACT

We derive the influence function of the likelihood ratio test statistic for multivariate normal sample. The derived influence function does not depend on the influence functions of the parameters under the null hypothesis. So we can obtain directly the empirical influence function with only the maximum likelihood estimators under the null hypothesis. Since the derived formula is a general form, it can be applied to influence analysis on many statistical testing problems.     

字母词与现代汉语书写系统

随着中国社会与世界结合得越来越紧密,在我们的日常生活中,渐渐兴起了一种与汉字形式词语截然不同的“字母词”,比如,早期的“CT”、“B超”以及最近出现的“MP3”、“PK”等等。字母词的大量涌现必然对现代汉语的词汇书写系统即文字体系产生深远的影响,就如同标点符号、左起横书的引入而使汉语的书写进入现代一样,字母词的出现必将打破汉字一元的格局,引领汉语的文字体系走向二元化、国际化。     

清代中朝之间的使行贸易及影响

清代,朝鲜使团利用到北京朝贡的机会同中国进行商品贸易,其形式主要有三种,即政府间的贸易、同中国商人的贸易和同普通市民的贸易。使行贸易对两国经济发展产生了巨大影响。使行贸易带动了中国辽东运输业的兴起,促进了商路经济的繁荣;也刺激了朝鲜手工业、畜牧业的发展,满足了朝鲜方面对部分商品的需求。