Robust and accurate inference for generalized linear models

In the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem, and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on first order asymptotic theory, and their accuracy in moderate to small samples is still an open question. In this paper, we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti [E. Cantoni, E. Ronchetti, Robust inference for generalized linear models, Journal of the American Statistical Association 96 (2001) 1022–1030] and Robinson, Ronchetti and Young [J. Robinson, E. Ronchetti, G.A. Young, Saddlepoint approximations and tests based on multivariate M-estimators, The Annals of Statistics 31 (2003) 1154–1169] to obtain a robust test statistic for hypothesis testing and variable selection, which is asymptotically χ²-distributed as the three classical tests but with a relative error of order O(n⁻¹). This leads to reliable inference in the presence of small deviations from the assumed model distribution, and to accurate testing and variable selection, even in moderate to small samples.     

The bias of k-step M-estimators

It is well-known that k-step M-estimators can yield a high efficiency without losing the breakdown point of the initial estimator. In this note we derive their bias curves. In the location framework the bias increases only slightly with k, but in the scale case the bias curves change considerably.     

部分线性回归模型变窗宽一步局部M-估计

讨论了部分线性回归模型的变窗宽一步局部M-估计.用一步局部M-估计给出未知函数的估计,用平均方法给出参数估计.进一步通过两个引理证明一步M-估计的渐近正态性.所提出的方法继承了局部多项式的优点并且克服了最小二乘法缺乏稳健性的缺点.     

Robust regression credibility: The influence function approach

In classical credibility theory we assume that the vector of claims conditionally on has independent components with identical means. However, this assumption is sometimes unrealistic. To relax this condition Hachemeister (Hachemeister, C.A., 1975. Credibility for regression models with application to trend. In: Kahn, P. (Ed.), Credibility, Theory and Applications. Academic Press, New York) introduced regressors. The presence of large claims can perturb the credibility premium estimation. The lack of robustness of regression credibility estimators, as well as the fairness of tariff evaluation, led to the development of this paper. Our proposal is to apply robust statistics to the regression credibility estimation by using the robust influence function approach of M-estimators.     

Information Criteria in Model Selection for Mixing Processes

We present information criteria for statistical model evaluation problems for stochastic processes. The emphasis is put on the use of the asymptotic expansion of the distribution of an estimator based on the conditional Kullback–Leibler divergence for stochastic processes. Asymptotic properties of information criteria and their improvement are discussed. An application to a diffusion process is presented.     

非线性回归M-估计的信赖域算法

１引言 考虑具有随机载荷的非线性回归模型其中（Ｘｉ,ｙｉ）　　是ｉ．ｉ．ｄ．随机序列,具有公共分布　　　　　　是已知的回归函数,β　Ｒｍ是未知参数向量,误差ｉ与兄独立　　　．为简单计,本文假设　　　＝１且ｆ（ｘ,·）是连续可微函数． 如何根据观察值（Ｘ１,ｙ１）,…,（Ｘｎ,ｙｎ）对回归参数向量β进行有效地估计,这是一个重要的统计问题．由于传统的最小二乘法关于残差异常值不具有稳健性,因此,迫使人们考虑新的估计方法．Ｈｕｂｅｒ最早提出了线性回归的稳健Ｈｕｂｅｒ估计,后又推广成一般形式的Ｍ－估计、ＧＭ…     

Robust estimation of the cerebral blood flow in arterial spin labelling

The introduction of arterial spin labelling (ASL) techniques in magnetic resonance imaging (MRI) has made feasible a non-invasive measurement of the cerebral blood flow (CBF). However, to date, the low signal-to-noise ratio of ASL gives us no option but to repeat the acquisition to accumulate enough data in order to get a reliable signal. The perfusion signal is then usually extracted by averaging across the repetitions. But the sample mean is very sensitive to outliers. A single incorrect observation can therefore be the source of strong detrimental effects on the perfusion-weighted image estimated with the sample mean.     

Large deviations for M-estimators

We study the large deviation principle for M-estimators (and maximum likelihood estimators in particular). We obtain the rate function of the large deviation principle for M-estimators. For exponential families, this rate function agrees with the Kullback–Leibler information number. However, for location or scale families this rate function is smaller than the Kullback–Leibler information number. We apply our results to obtain confidence regions of minimum size whose coverage probability converges to one exponentially. In the case of full exponential families, the constructed confidence regions agree with the ones obtained by inverting the likelihood ratio test with a simple null hypothesis.     

Information Criteria for Small Diffusions via the Theory of Malliavin–Watanabe

Information criteria based on the expected Kullback–Leibler information are presented by means of the asymptotic expansions derived with the Malliavin calculus. We consider the evaluation problem of statistical models for diffusion processes with small noise. The correction terms are essentially different from the ones for ergodic diffusion models presented in Uchida and Yoshida [34, 35].     

Strong convergence of estimators as ε n -minimisers of optimisation problemsof optimisation problems

In the paper we prove strong consistency of estimators as solution of optimisation problems. The approach of the paper covers non-identifiable models, and models for dependent samples. We provide statements about consistency of M-estimators in regression models with random and with non-random design. The research was partially supported by the Deutsche Forschungsgemeinschaft (project number 436TSE113/40) and by the Grant Agency of the Czech Republic under Grant 201/03/1027.