Influence functions of two families of robust estimators under proportional scatter matrices

In this paper, under a proportional model, two families of robust estimates for the proportionality constants, the common principal axes and their size are discussed. The first approach is obtained by plugging robust scatter matrices on the maximum likelihood equations for normal data. A projection- pursuit and a modified projection-pursuit approach, adapted to the proportional setting, are also considered. For all families of estimates, partial influence functions are obtained and asymptotic variances are derived from them. The performance of the estimates is compared through a Monte Carlo study.     

Robust analysis of variance for a randomized block design

In this paper we study the asymptotic theory of M-estimates and their associated tests for a one-factor experiment in a randomized block design. In this case one natural asymptotic theory corresponds to leaving the number of treatments fixed and letting the number of blocks tend to infinity. The classic asymptotic theory of M-estimates does not apply here, because the number of parameters and the number of observations are of the same order. In this paper we prove the consistency and asymptotic normality of the estimators of the treatment effects. It turns out that the asymptotic covariance matrix of the treatment effects estimators differs from the one derived from the classic theory of M-estimates for the linear model with a fixed number of parameters. We also study a test for treatment effects derived from M-estimates and we compare by Monte Carlo simulation the efficiency of this test with respect to the F-test, the Friedman test and the test based on aligned ranks.     

Robust tests for the common principal components model

When dealing with several populations, the common principal components (CPC) model assumes equal principal axes but different variances along them. In this paper, a robust log-likelihood ratio statistic allowing to test the null hypothesis of a CPC model versus no restrictions on the scatter matrices is introduced. The proposal plugs into the classical log-likelihood ratio statistic robust scatter estimators. Using the same idea, a robust log-likelihood ratio and a robust Wald-type statistic for testing proportionality against a CPC model are considered. Their asymptotic distributions under the null hypothesis and their partial influence functions are derived. A small simulation study allows to compare the behavior of the classical and robust tests, under normal and contaminated data.     

Robust M-estimators of scale: Minimax bias versus maximal variance

In the location-scale estimation problem, we study robustness properties of M-estimators of the scale parameter under unknown ?-contamination of a fixed symmetric unimodal error distribution F₀. Within a general class of M-estimators, the estimator with minimax asymptotic bias is shown to lie within the subclass of α-interquantile ranges of the empirical distribution symmetrized about the sample median. Our main result is that as ? → 0, the limiting minimax asymptotic bias estimator is sometimes (e.g., when F_o is Cauchy), but not always, the median absolute deviation about the median. It is also shown that contamination in the neighbourhood of a discontinuity of the influence function of a minimax bias estimator can sometimes inflate the asymptotic variance beyond that achieved by placing all the ?-contamination at infinity. This effect is quantified by a new notion of asymptotic efficiency that takes into account the effect of infinitesimal contamination of the parametric model for the error distribution.     

Robust M- and L-Estimators of Scale Parameter

We consider first the class of M-estimators of scale that are location-scale equivariant and Fisher consistent at the error distribution of the shrinking contamination neighborhood and derive an expression for the maximal asymptotic mean-squared-error, for a suitably regular score function, followed by a lower bound on it. We next show that the minimax asymptotic mean-squzred-error is attained at an M-estimator of scale with the truncated MLE score function which, when specialized to the Standard Normal error distribution has the form of Huber's Proposal 2. The latter minimax property is also shown to hold for α-trimmed variance as an L-estimator of scale.     

Consistency of completely outlier-adjusted simultaneous redescending M-estimators of location and scale

This paper gives conditions for the consistency of simultaneous redescending M-estimators for location and scale. The consistency postulates the uniqueness of the parameters μ and σ, which are defined analogously to the estimations by using the population distribution function instead of the empirical one. The uniqueness of these parameters is no matter of course, because redescending ψ- and χ-functions, which define the parameters, cannot be chosen in a way that the parameters can be considered as the result of a common minimizing problem where the sum of ρ-functions of standardized residuals is to be minimized. The parameters arise from two minimizing problems where the result of one problem is a parameter of the other one. This can give different solutions. Proceeding from a symmetrical unimodal distribution and the usual symmetry assumptions for ψ and χ leads, in most but not in all cases, to the uniqueness of the parameters. Under this and some other assumptions, we can also prove the consistency of the according M-estimators, although these estimators are usually not unique even when the parameters are. The present article also serves as a basis for a forthcoming paper, which is concerned with a completely outlier-adjusted confidence interval for μ. So we introduce a ñ where data points far away from the bulk of the data are not counted at all.     

Controlling the size of multivariate outlier tests with the MCD estimator of scatter

Multivariate outlier detection requires computation of robust distances to be compared with appropriate cut-off points. In this paper we propose a new calibration method for obtaining reliable cut-off points of distances derived from the MCD estimator of scatter. These cut-off points are based on a more accurate estimate of the extreme tail of the distribution of robust distances. We show that our procedure gives reliable tests of outlyingness in almost all situations of practical interest, provided that the sample size is not much smaller than 50. Therefore, it is a considerable improvement over all the available MCD procedures, which are unable to provide good control over the size of multiple outlier tests for the data structures considered in this paper.     

A kalman filter in the presence of outliers

A Kalman Filtering algorithm which is robust to observational outliers is developed by assuming that the measurement error may come from either one of two Normal distributions, and that the transition between these distributions is governed by a Markov Chain. The resulting algorithm is very simple, and consists of two parallel Kalman Filters having different gains. The state estimate is obtained as a weighted average of the estimates from the two parallel filters, where the weights are the posterior probabilities that the current observation comes from either of the two distributions. The large improvements obtained by this Robust Kalman Filter in the presence of outliers is demonstrated with examples.     

Robust fitting of claim severity distributions and the method of trimmed moments

Many quantities arising in non-life insurance depend on claim severity distributions, which are usually modeled assuming a parametric form. Obtaining good estimates of the quantities, therefore, reduces to having good estimates of the model parameters. However, the notion of ‘good estimate’ depends on the problem at hand. For example, the maximum likelihood estimators (MLEs) are efficient, but they generally lack robustness. Since outliers are common in insurance loss data, it is therefore important to have a method that allows one to balance between efficiency and robustness. Guided by this philosophy, in the present paper we suggest a general estimation method that we call the method of trimmed moments (MTM). This method is appropriate for various model-fitting situations including those for which a close fit in one or both tails of the distribution is not required. The MTM estimators can achieve various degrees of robustness, and they also allow the decision maker to easily see the actions of the estimators on the data, which makes them particularly appealing. We illustrate these features with detailed theoretical analyses and simulation studies of the MTM estimators in the case of location–scale families and several loss distributions such as lognormal and Pareto. As a further illustration, we analyze a real data set concerning hurricane damages in the United States from 1925 to 1995.     

Minimax-variance L- and R-estimators of location

We consider the problem of minimax-variance, robust estimation of a location parameter, through the use of L- and R-estimators. We derive an easily checked necessary condition for L-estimation to be minimax, and a related sufficient condition for R-estimation to be minimax. Those cases in the literature in which L-estimation is known not to be minimax, and those in which R-estimation is minimax, are derived as consequences of these conditions. New classes of examples are given in each case. As well, we answer a question of Scholz (1974), who showed essentially that the asymptotic variance of an R-estimator never exceeds that of an L-estimator, if both are efficient at the same strongly unimodal distribution. Scholz raised the question of whether or not the assumption of strong unimodality could be dropped. We answer this question in the negative, theoretically and by examples. In the examples, the minimax property fails both for L-estimation and for R-estimation, but the variance of the L-estimator, as the distribution of the observation varies over the given neighbourhood, remains unbounded. That of the R-estimator is unbounded.     

Robust estimates of ordered means in normal models

In this paper we consider the problem of estimating the locations of several normal populations when an order relation between them is known to be true. We compare the maximum likelihood estimator, the M-estimators based on Huber’s ψ function, a robust weighted likelihood estimator, the Gastworth estimator and the trimmed mean estimator. A Monte-Carlo study illustrates the performance of the methods considered.     

Robust estimation in the multivariate normal model with variance components

Fisher consistent and Fréchet differentiable statistical functionals have been already used by Bednarski and Zontek [Robust estimation of parameters in a mixed unbalanced model. Ann Statist. 1996;24(4):1493–1510] to get a robust estimator of parameters in a two-way crossed classification mixed model. This way of robust estimation appears also in the variance components model with a commutative covariance matrix [Zmy?lony, Zontek. Robust M-estimator of parameters in variance components model. Discuss Math Probab Stat. 2002;22:61–71]. In this paper it is shown that a modification of this method does not involve any assumptions about commutation of covariance matrix. The theoretical results have been completed with computer simulation studies. Robustness of considered estimator and possibility of approximation of the estimator's distribution with some multivariate normal distribution for both model and contaminated data have been confirmed there.     

a bayesian analysis of the multivariate mixed-linear model

The Bayesian analysis of the multivariate mixed linear model is considered. The exact posterior distribution for the fixed effects matrix and the error covariance matrix are obtained. The exact posterior means and variances of the Bayesian estimators for the covariance matrices of random effects are also derived. These posterior moments are computed without constrained optimization and numerical integration. The calculations are feasible for arbitrary models. Reasonable approximations for the posterior distributions for the covariance matrices associated with the random effects are obtained also. Results are illustrated with a numerical example.     

Second order asymptotic relations of M-estimators and R-estimators in two-sample location model

Let T_M be an M-estimator (maximum likelihood type estimator) and T_R be an R-estimator (Hodges-Lehmann's estimator) of the shift parameter Δ in the two-sample location model. The asymptotic representation of √N(T_M-T_R) up to a term of the order O_p(N^{-$\text{1}\text{4}$}) is derived which is valid if the functions Ψ and ? generating T_M and T_R, respectively, decompose into an absolutely continuous and a step-function components; the order O_p(N^{-$\text{1}\text{4}$}) cannot be improved unless the discontinuous components vanish. As a consequence, the conditions under which √N(T_M-T_R)=O_p(N^{-$\text{1}\text{4}$}) are obtained. The main tool for obtaining the results is the second order asymptotic linearity of the pertaining linear rank statistics which is proved here under the assumption that the score-generating function ? has some jump-discontinuities.     

Robust slippage rank tests for k location parameters in the presence of gross errors

A robust slippage test problem of k location parameters in the presence of gross errors is formulated from the point of view of Huber's robust test theory. Under an asymptotic model of the robust slippage test problem an asymptotic level α slippage rank test based on k linear rank statistics is constructed by applying majorization methods and its asymptotic minimum power is evaluated by applying weak majorization methods. It is also shown that the slippage rank test is asymptotically unbiased.     

Performance Evaluation Based on the Robust Mahalanobis Distance and Multilevel Modeling Using Two New Strategies

In this article, we propose a general framework for performance evaluation of organizations and individuals over time using routinely collected performance variables or indicators. Such variables or indicators are often correlated over time, with missing observations, and often come from heavy-tailed distributions shaped by outliers. Two new double robust and model-free strategies are used for evaluation (ranking) of sampling units. Strategy 1 can handle missing data using residual maximum likelihood (RML) at stage two, while strategy two handles missing data at stage one. Strategy 2 has the advantage that overcomes the problem of multicollinearity. Strategy one requires independent indicators for the construction of the distances, where strategy two does not. Two different domain examples are used to illustrate the application of the two strategies. Example one considers performance monitoring of gynecologists and example two considers the performance of industrial firms.     

A characterization of the independence - distribution - preserving covariance structure for the multivariate maximum squared - radii statistic

Necessary and sufficient conditions on the observation covariance structure and on the set of linear transformations are given for which the distribution of the multivariate maximum squared - radii statistic for detecting a single multivariate outlier is invariant from the distribution assuming the usual independence covariance structure. Thus, we extend the work of Baksalary and Puntanen (1990), who have given necessary and sufficient conditions for an independence-distribution-preserving covariance structure for Grubbs' statistic for detecting a univariate outlier. We also extend the work of Marco, Young, and Turner (1987) and Pavur and Young (1991), who have given sufficient conditions for an independence-distribution-preserving dependency structure for the multivariate squared - radii statistic.     

稳健统计以及几种统计量的稳健性比较分析

稳健统计作为统计学的一个较为活跃的研究领域,是对传统统计方法的完善和补充。其中最为简单和通俗易懂的是统计量的稳健性。本文对几种常用的统计量进行深入地剖析,进而揭示其稳健性的不足。同时给出几种稳健统计量,并与传统的统计量进行比较。通过比较来展现稳健统计量的优势及其应用价值。     

Robust Estimation of Multivariate Linear Model Based on Depth Weighted Mean and Scatter

Based on the projection depth weighted mean and scatter estimation of the joint distribution of (x, y), we introduce a robust estimator of the regression coefficients for the multivariate linear model. The new estimator possesses desirable properties including affine invariance, Fisher consistency, and asymptotic normality. Also, we study the robustness of the estimator in terms of breakdown point and influence function. Extensive simulation studies are performed to investigate the finite sample behavior of robustness and efficiency. The methodology is illustrated with a real data example.