Robust estimation: A condensed partial survey

Hampel, Frank R.

doi:10.1007/bf00536619

Cited by 199 publications

(62 citation statements)

References 39 publications

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“…To see this, note that if 6(6km) = 6km, then the kth observation will have full unit weight, while if ~(3km) < 6km for larger values of 3km, then the corresponding observation will be given less than unit weight in determining Pk. This is, of course, the same approach as that used in robust procedures, where observations which are unduly discrepant are given reduced weight in the calculations (see, e.g., Hampel, 1973;Hogg, 1977), while observations which are reasonable are given full unit weight. The function ~(~ern) can be thought of more generally as an influence function (Hampel, 1974), the appropriate choice of which determines the contribution of an observation to the statistic(s) of interest.…”

Section: Incorporating a Measure Of Typicalitymentioning

confidence: 99%

“…Some would argue that very discrepant observations should have zero influence on the statistic(s) of interest; this is achieved by introducing a redescending influence function (Hampel, 1973), so-called because the influence of an observation descends to zero as the discrepancy of the observation increases. Several choices are discussed by Holland and Welsch (1977), including the tanh function of Collins (1976).…”

Section: Wkr N = W( Akrn ) = T~( Akm)/ Akmmentioning

confidence: 99%

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Mixture models and atypical values

Campbell

1984

Mathematical Geology

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The resolution of a mixture of two or more populations into its component distr~butions may be markedly influenced by one or a few atypical values. The maximum likelihood solution effectively assigns (part of) each observation to one or another of the components via the posterior probabilities, even though the observation may be widely discrepant from all components. This paper presents a modification of the mixture problem, in which the typicality of each observation is considered, as well as the posterior probabilities, with the contribution of atypical observations being downweighted. The extension to the multivariate ease is discussed.

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Section: Incorporating a Measure Of Typicalitymentioning

confidence: 99%

Section: Wkr N = W( Akrn ) = T~( Akm)/ Akmmentioning

confidence: 99%

Mixture models and atypical values

Campbell

1984

Mathematical Geology

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“…Affine equivariant M-estimators for dispersion matrices were first suggested by Hampel (1973), and studied by Maronna (1976) and Huber (1977Huber ( , 1981. Unfortunately, their breakdown point is at most 1Â( p+1).…”

Section: Dispersion Matrix Estimatorsmentioning

confidence: 99%

Highly Robust Estimation of Dispersion Matrices

Genton

2001

Journal of Multivariate Analysis

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In this paper, we propose a new componentwise estimator of a dispersion matrix, based on a highly robust estimator of scale. The key idea is the elimination of a location estimator in the dispersion estimation procedure. The robustness properties are studied by means of the influence function and the breakdown point. Further characteristics such as asymptotic variance and efficiency are also analyzed. It is shown in the componentwise approach, for multivariate Gaussian distributions, that covariance matrix estimation is more difficult than correlation matrix estimation. The reason is that the asymptotic variance of the covariance estimator increases with increasing dependence, whereas it decreases with increasing dependence for correlation estimators. We also prove that the asymptotic variance of dispersion estimators for multivariate Gaussian distributions is proportional to the asymptotic variance of the underlying scale estimator. The proportionality value depends only on the underlying dependence. Therefore, the highly robust dispersion estimator is among the best robust choice at the present time in the componentwise approach, because it is location-free and combines small variability and robustness properties such as high breakdown point and bounded influence function. A simulation study is carried out in order to assess the behavior of the new estimator. First, a comparison with another robust componentwise estimator based on the median absolute deviation scale estimator is performed. The highly robust properties of the new estimator are confirmed. A second comparison with global estimators such as the method of moment estimator, the minimum volume ellipsoid, and the minimum covariance determinant estimator is also performed, with two types of outliers. In this case, the highly robust dispersion matrix estimator turns out to be an interesting compromise between the high efficiency of the method of moment estimator in noncontaminated situations and the highly robust properties of the minimum volume ellipsoid and minimum covariance determinant estimators in contaminated situations. Academic Press

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“…It allows best parameters estimation, without requiring any identification of specific observations as outliers or excluding them. Robust methods are basically developed by Huber's works (1964Huber's works ( , 1981 and Hampel (1973). Use of them in geodesy was initiated by Carosio (1979) and Mäkinen (1981), Becker (1989) especially for gravimetric data.…”

Section: Introductionmentioning

confidence: 99%

Robust and Efficient Weighted Least Squares Adjustment of Relative Gravity Data

Touati

Kahlouche

Idres³

2010

Gravity, Geoid and Earth Observation

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When gravimetric data observations have outliers, using standard least squares (LS) estimation will likely give poor accuracies and unreliable parameter estimates. One of the typical approaches to overcome this problem consists of using the robust estimation techniques. In this paper, we modified the robust estimator of Gervini and Yohai (2002) called REWLSE (Robust and Efficient Weighted Least Squares Estimator), which combines simultaneously high statistical efficiency and high breakdown point by replacing the weight function by a new weight function. This method allows reducing the outlier impacts and makes more use of the information provided by the data. In order to adapt this technique to the relative gravity data, weights are computed using the empirical distribution of the residuals obtained initially by the LTS (Least Trimmed Squares) estimator and by minimizing the mean distances relatively to the LS estimator without outliers. The robustness of the initial estimator is maintained by an adapted cut-off values as suggested by the REWLSE method which allows also a reasonable statistical efficiency. Hereafter we give the advantage and the pertinence of REWLSE procedure on real and semi-simulated gravity data by comparing it with conventional LS and other robust approaches like M and MM estimators.

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Robust estimation: A condensed partial survey

Cited by 199 publications

References 39 publications

Mixture models and atypical values

Mixture models and atypical values

Highly Robust Estimation of Dispersion Matrices

Robust and Efficient Weighted Least Squares Adjustment of Relative Gravity Data

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