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Change-of-variance sensitivities in regression analysis
Abstract
The change-of-variance function is defined for estimators of regression coefficients. Both an unstandardized and a standardized form of the change-of-variance sensitivity are introduced, and their relation with the corresponding gross-error-sensitivities is investigated. The problems of optimal robustness lead to the Hampel-Krasker and the Krasker-Welsch estimators. At the same time, also the scale parameter has to be estimated robustly. By means of the change-of-variance sensitivity, optimal robust redescending scale estimators are constructed.
... Section 1.5 will briefly describe these concepts. Their extensions to estimators of multiple linear regression parameters have been developed (Hampel, 1973;Huber, 1973 ;Ronchetti and Rousseeuw, 1985). However, there are still many concepts that have not been explored for GLMs and GLMMs, for example the Change-of-Variance Function (CVF). ...
... to linear regression problems (Ronchetti and Rousseeuw, 1985). We summarize the latter below. ...
... The CVF in Linear Models Ronchetti and Rousseeuw (1985) extended the notion of Change-of-Variance Function to regression problems. Working in the framework of GM -estimators, presented in Definition 1.5.3, ...
... 7 If one is to have stable inference, then it is desirable that no small subset of the data have a very large effect on the (asymptotic) variances of the parameter estimates. We would like, therefore, to be able to assess (and control) the influence of a single observation (x.,y.) on the standard -J J errors of Bas well as on B. One way of describing this influence is by means of the change-of-variance function, first introduced in the singleparameter case by Hampel, Rousseeuw and Ronchetti (1981) and later extended to the multivariate case by Ronchetti and Rousseeuw (1983). ...
... We have the following result, relating K 2 and Y 2 : Proof: This follows directly from Theorem l' and Corollary l' of Ronchetti and Rousseeuw (1983). ...
... The proof combines the results of Ronchetti and Rousseeuw (1983) for the regression case and the case of estimating a I-dimensional scale parameter. We first show that Y~S K 2q. ...
Robust regression methods require much more computing than least squares, and also require some assumptions to be made about the down-weighting procedure to be employed. This chapter opens with a discussion on least absolute deviations regression and M-Estimators. It talks about robust analyses that could be made, using criteria other than Huber's. The chapter explains the standard errors of estimated coefficients. Values for standard errors of the robust regression estimates can be obtained from the square roots of the diagonal entries. The chapter further focuses on least median of squares (LMS) regression and robust regression with ranked residuals (rreg).
In letzter Zeit werden Vergleichsstudien beliebt, bei denen die Prognosen verschiedener Verfahren an empirischem Datenmaterial zueinander in Beziehung gesetzt werden. Nicht selten schneiden dabei relativ einfache Verfahren mit Trendextrapolationen oder Exponential Smoothing besser ab als ausgefeilte und flexible Ansätze wie etwa ARIMA-Modelle. Man fragt sich dann natürlich, woran das wohl liegen könnte: Sicher nicht daran, daß die zugrunde liegenden Modelle ein besseres Abbild der Wirklichkeit liefern. Daß die Entwicklung eines wirtschaftlichen Phänomens etwa einer einfachen analytischen Funktion oder einer Linearkombination solcher Funktionen folgt, ist wenig glaubhaft. die Gründe müssen woanders liegen.
We are delighted to have the opportunity to introduce Ray's papers on robustness in statistics. These papers include important breakthrough developments such as the first rigorous asymptotic analysis of trimmed least squares, the first robust heteroscedastic regression estimators, the first efficient bounded-influence estimators for generalized linear regression, and the first bounded-influence regression estimators with high breakdown points. Like so many of Ray's publications, each of these widely cited papers is a marvel of clarity, innovation, and deep analysis. © 2014 Springer International Publishing Switzerland. All rights are reserved.
This paper provides a summary of the influence function approach to robust estimation of parametric models. Hampel’s optimality results for M-estimators with a bounded influence function is generalized to allow for arbitrary choices of the asymptotic efficiency criterion and the norm of the influence function. Further extensions to other cases of practical interest are also considered.
This article presents robust methods for the random calibration problem. Many calibration techniques are based on regression models or measurement-error models. Prediction from such models is known to be highly nonrobust, and robust techniques should prove quite valuable. Robust-calibration procedures are procedures that work well even if there is some contamination in the data or if the model assumptions used in deriving the procedure are not quite true for the given data. Several approaches to robustifying calibration are compared theoretically, by Monte Carlo simulation, and on real data.
Research directions in robust statistics are listed and partly commented. The intention is to help stimulating research in topics which are important according to the author’s perception. The reference list includes more than 500 items, mainly of the years 1985–1990.
We consider two methods of defining a regression analog to a trimmed mean. The first was suggested by Koenker and Bassett and uses their concept of regression quantiles. Its asymptotic behavior is completely analogous to that of a trimmed mean. The second method uses residuals from a preliminary estimator. Its asymptotic behavior depends heavily on the preliminary estimate; it behaves, in general, quite differently than the estimator proposed by Koenker and Bassett, and it can be inefficient at the normal model even if the percentage of trimming is small. However, if the preliminary estimator is the average of the two regression quantiles used with Koenker and Bassett's estimator, then the first and second methods are asymptotically equivalent for symmetric error distributions.
Thesis (Ph. D.)--Harvard University, 1977.
We introduce general tests for the linear model and we compute their influence function and their asymptotic distribution. We also solve an optimality problem based on a criterion for infinitesimal robustness and we derive optimally bounded-influence tests.
Simple “one-step” versions of Huber’s (M) estimates for the linear model are introduced. Some relevant Monte Carlo results obtained in the Princeton project [1] are singled out and discussed. The large sample behavior of these procedures is examined under very mild regularity conditions.
The least squares estimator for β in the classical linear regression model is strongly efficient under certain conditions. However, in the presence of heavy-tailed errors and/or anomalous data, the least squares efficiency can be markedly reduced. In this article we propose an estimator that limits the influence of any small subset of the data and show that it satisfies a first-order condition for strong efficiency subject to the constraint. We then show that the estimator is asymptotically normal.The article concludes with an outline of an algorithm for computing a bounded-influence regression estimator and with an example comparing least squares, robust regression as developed by Huber, and the estimator proposed in this article.
Influence functions for testing are defined by applying Hampel's influence function to transformed P values. These influence functions describe the effect of an observation and an underlying distribution on the behavior of a test. Because these influence functions are not based on the test statistic alone, they are applicable to both unconditional and conditional (e.g., randomization) tests. The influence function of a transformed P value at a null hypothesis distribution is related to an influence function based on limiting power. The influence functions of several one- and two-sample tests for location are discussed to illustrate the utility of this approach.