Finite Sample Breakdown Point Research Articles

Projection-based depth functions and associated medians

order √ n uniform consistency. Depth regions and contours induced from projection-based depth functions are investigated. Structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are obtained. Affine equivariant multivariate medians induced from projection-based depth functions are probed. The limiting distributions as well as the strong and order √ n consistency of the sample projection medians are established. The finite sample performance of projection medians is compared with that of a leading depth-induced median, the Tukey halfspace median (induced from the Tukey halfspace depth function). It turns out that, with appropriate choices of univariate location and scale estimators, the projection medians have a very high finite sample breakdown point and relative efficiency, much higher than those of the halfspace median. Based on the results obtained, it is found that projection depth functions and projection medians behave very well overall compared with their competitors and consequently are good alternatives to statistical depth functions and affine equivariant multivariate location estimators, respectively.

Finite sample tail behavior of multivariate location estimators

A finite sample performance measure of multivariate location estimators is introduced based on “tail behavior”. The tail performance of multivariate “monotone” location estimators and the halfspace depth based “non-monotone” location estimators including the Tukey halfspace median and multivariate L-estimators is investigated. The connections among the finite sample performance measure, the finite sample breakdown point, and the halfspace depth are revealed. It turns out that estimators with high breakdown point or halfspace depth have “appealing” tail performance. The tail performance of the halfspace median is very appealing and also robust against underlying population distributions, while the tail performance of the sample mean is very sensitive to underlying population distributions. These findings provide new insights into the notions of the halfspace depth and breakdown point and identify the important role of tail behavior as a quantitative measure of robustness in the multivariate location setting.

A class of robust and fully efficient regression estimators

This paper introduces a new class of robust estimators for the linear regression model. They are weighted least squares estimators, with weights adaptively computed using the empirical distribution of the residuals of an initial robust estimator. It is shown that under certain general conditions the asymptotic breakdown points of the proposed estimators are not less than that of the initial estimator, and the finite sample breakdown point can be at most $1/n$ less. For the special case of the least median of squares as initial estimator, hard rejection weights and normal errors and carriers, the maximum bias function of the proposed estimators for point-mass contaminations is numerically computed, with the result that there is almost no worsening of bias. Moreover–and this is the original contribution of this paper–if the errors are normally distributed and under fairly general conditions on the design the proposed estimators have full asymptotic efficiency. A Monte Carlo study shows that they have better behavior than the initial estimators for finite sample sizes.

The largest nonidentifiable outlier: a comparison of multivariate simultaneous outlier identification rules

The aim of detecting outliers in a multivariate sample can be pursued in different ways. We investigate here the performance of several simultaneous multivariate outlier identification rules based on robust estimators of location and scale. It has been shown that the use of estimators with high finite-sample breakdown point in such procedures yields a good behaviour with respect to the prevention of breakdown by the masking effect (Becker, Gather, 1999. J. Amer. Statist. Assoc. 94, 947–955). In this article, we investigate by simulation, at which distance from the centre of an underlying model distribution outliers can be placed until certain simultaneous identification rules will detect them as outliers. We consider identification procedures based on the minimum volume ellipsoid, the minimum covariance determinant, and S-estimators.

Some quantitative relationships between two types of finite sample breakdown point

The two versions of finite-sample breakdown points, the addition breakdown point and the replacement breakdown point, have been employed in the derivation of the breakdown points for various estimators. However, for a given estimator, frequently only the version that is convenient in the derivation is selected. The connections between the two versions have not yet been formally explored in the literature. In this paper, quantitative relationships between the two versions of the breakdown point are formally established. The result obtained allows one to obtain the ABP directly from the RBP of estimators and vice versa and thus to avoid some unnecessary, sometimes tedious, derivations of breakdown points.

Tail Behavior and Breakdown Properties of Equivariant Estimators of Location

For translation and scale equivariant estimators of location, inequalities connecting tail behavior and the finite-sample breakdown point are proved, analogous to those established by He et al. (1990, Econometrika, 58, 1195–1214) for monotone and translation equivariant estimators. Some other inequalities are given as well, enabling to establish refined bounds and in some cases exact values for the tail behavior under heavy- and light-tailed distributions. The inequalities cover translation and scale equivariant estimators in great generality, and they involve new breakdown-related quantities, whose relations to the breakdown point are discussed. The worth of tail-behavior considerations in robustness theory is demonstrated on examples, showing the impact of the basic two techniques in robust estimation: trimming and averaging. The mathematical language employs notions from regular variation theory.

Finite sample tail behavior of the multivariate trimmed mean based on Tukey-Donoho halfspace depth

Finite sample tail behavior of the Tukey-Donoho halfspace depth based multivariate trimmed mean is investigated with respect to a tail performance measure. It turns out that the tails of the sampling distribution of the α-depth-trimmed mean approach zero at least ⌊αn⌋ times as fast as the tails of the underlying population distribution and could be n−⌊αn⌋+ 1 times as fast. In addition, there is an intimate relationship among the tail behavior, the halfspace depth, and the finite sample breakdown point of the estimator. It is shown that the lower tail performance bound of the depth based trimmed mean is essentially the same as its halfspace depth and the breakdown point. This finding offers a new insight into the notion of the halfspace depth and extends the important role of the tail behavior as a quantitative assessment of robustness in the regression (He, Jureckova, Koenker and Portnoy (1990)) and the univariate location settings (Jureckova (1981)) to the multivariate location setting.

A note on finite sample breakdown points of projection based multivariate location and scatter statistics

In this note, general results of finite sample breakdown point are obtained for two classes of projection based location and scatter statistics: the Stahel-Donoho statistics and the Maronna-Yohai statistics. It is shown that these projection based location and scatter statistics can achieve the maximum breakdown point of affine equivariant multivariate location and scatter statistics. General relationships between the finite sample breakdown point of these statistics and the uniform finite sample breakdown point of the sample median and a modified sample median absolute deviation are formally established.

Breakdown point of Schuster–Narvarte's location estimator

The Schuster–Narvarte's (Ann. Statist. 1 (1973) 1096–1104) location parameter of a distribution F, the center of a symmetric distribution closest to F, is an appealing definition for the center of distributions. It is noticed that this parameter is a special case of Donoho and Liu's (Ann. Statist. 16 (1988) 552–586) semiparametric minimum distance functional. For this estimator, an algorithm is available in Schuster–Narvarte (1973); asymptotics results were established in Rao et al. (Ann. Statist. 3 (1975) 862–873) and in Arcones and Giné (Ann. Statist. 19 (1991) 1496–1511); validity of bootstraps is proved in Arcones and Giné (1991) while simulation results can be found in Schuster and Barker (Comm. Statist. Simulation Comput. 16 (1987) 69–84). In this article, we compliment the study by establishing the robustness properties for the Schuster–Narvarte's estimator. A general result for the (enlargement) finite sample breakdown point is established. In particular, when the underlying distribution is symmetric, the limiting finite sample breakdown point is almost surely 1 3 .

A note on the robustness of multivariate medians

In this note we investigate the extent to which some of the fundamental properties of univariate median are retained by different multivariate versions of median with special emphasis on robustness and breakdown properties. We show that transformation retransformation medians, which are affine equivariant, n 1/2-consistent and asymptotically normally distributed under standard regularity conditions, can also be very robust with high breakdown points. We prove that with some appropriate adaptive choice of the transformation matrix based on a high breakdown estimate of the multivariate scatter matrix (e.g. S-estimate or minimum covariance determinant estimate), the finite sample breakdown point of a transformation retransformation median will be as high as n −1[( n− d+1)/2], where n= the sample size, d= the dimension of the data, and [ x] denotes the largest integer smaller than or equal to x. This implies that as n→∞, the asymptotic breakdown point of a transformation retransformation median can be made equal to 50% in any dimension just like the univariate median. We present a brief comparative study of the robustness properties of different affine equivariant multivariate medians using an illustrative example.

The Masking Breakdown Point of Multivariate Outlier Identification Rules

Abstract In this article, we consider simultaneous outlier identification rules for multivariate data, generalizing the concept of so-called α outlier identifiers, as presented by Davies and Gather for the case of univariate samples. Such multivariate outlier identifiers are based on estimators of location and covariance. Therefore, it seems reasonable that characteristics of the estimators influence the behavior of outlier identifiers. Several authors mentioned that using estimators with low finite-sample breakdown point is not recommended for identifying outliers. To give a formal explanation, we investigate how the finite-sample breakdown points of estimators used in these identification rules influence the masking behavior of the rules.

The Masking Breakdown Point of Multivariate Outlier Identification Rules

In this article, we consider simultaneous outlier identification rules for multivariate data, generalizing the concept of so-called α outlier identifiers, as presented by Davies and Gather for the case of univariate samples. Such multivariate outlier identifiers are based on estimators of location and covariance. Therefore, it seems reasonable that characteristics of the estimators influence the behavior of outlier identifiers. Several authors mentioned that using estimators with low finite-sample breakdown point is not recommended for identifying outliers. To give a formal explanation, we investigate how the finite-sample breakdown points of estimators used in these identification rules influence the masking behavior of the rules.

The Masking Breakdown Point of Multivariate Outlier Identification Rules

In this paper, we consider one-step outlier identification rules for multivariate data-generalizing the concept of so-called a - outlier identifiers_ as presented in Davies and Gather (1993) for the case of univariate samples. We investigate how the finite sample breakdown points of estimators used in these identification rules influence the masking behaviour of the rules.

Reliabilities of χ2- and F Tests in Gauss-Markov Model

Power and level breakdown points are used in robust statistics to determine the global reliability of a test. A method for calculating the finite-sample versions of these for tests such as χ2 or F is given here when observations have the Gauss-Markov model. Outliers are compared with the generating model using a coordinate transformation and simulated samples. Outliers are considered in two groups, namely, “doubtful” and “clear.” The reliability of a χ2- or F test changes with the magnitude, number, and kind of outliers (i.e., random outliers or influential outliers), and also with the level of significance, α. The χ2- and F tests are generally insensitive to both doubtful and clear outliers. The χ2 test can reliably reject the null hypothesis at α = 0.05, when the observations contain multiple clear random outliers, only if the magnitude of one of them is at least greater than 5σ. Also, the F test can reliably reject the null hypothesis at α = 0.05, when the observations contain multiple clear random outliers, only if the magnitude of one of them is at least greater than 6σ. The greater the magnitude and number of clear outliers that are contained in the observations, the more successful is the χ2- or F test in rejecting the null hypothesis. The reliabilities of the χ2- and F tests decrease as the number of unknowns increases. For this case, the power and level finite-sample breakdown points of the χ2- and F tests are estimated here to be approximately 1/2.

Application of Equiredundancy Design to M-Estimation

A new method of weighting that realizes equiredundancy design is introduced, and its convergence is better than the method introduced by Koch. However, if Koch's method is modified, it gives the best results compared to the others. Furthermore, it is shown that balanced weights depend only on the positions of observations and not on the initial weights. If the equiredundancy design is taken into consideration in the generalized form of maximum likelihood estimation (M-estimation), the finite sample breakdown point (FSBP) (or reliability) increases. This problem has been studied using a coordinate transformation simulation. After obtaining the balanced weights, which provide the equiredundancy, they are used as the pseudoweights of observations in M-estimation. This is called the equiredundancy-designed generalized M-estimation (GM-estimation). In addition, another GM-estimation is developed by balancing the partial redundancy numbers at every step of the reweighted and iterated least-squares solution of M-estimation. This is called the GM-estimation with balanced case. Both methods have the same FSBP. Using both methods of GM-estimation in the robustifying conventional outlier detection procedures, such as the method of Baarda or Pope, the new robustifying outlier detection procedures are obtained with a higher FSBP.

About Regression Estimators with High Breakdown Point

A generalisation of a theorem by Vandev (1993) concerning the finite sample breakdown point is given. Using this result the breakdown point properties of the LMS and LTS estimators of Rousseeuw (1984) and the rank-based regression estimator of Hössjer (1994) are studied. Moreover, the breakdown point properties of the weighted least trimmed estimators of order k in the case of grouped logistic regression are investigated, as well as linear regression with an exponential q-th power distribution.

A note on Tyler's modification of the MAD for the Stahel-Donoho estimator

In 1994 Tyler proposed a modification of the MAD in order to achieve the maximal possible finite sample breakdown point for the Stahel-Donoho estimator based upon the median and the modified MAD. A proof is not given, however, and as far as we know has not been published yet. Moreover there seems to be an error concerning the exact definition of the modified MAD which is needed to achieve this goal. For the sake of theoretical completeness and in order to correct this error we calculate the finite sample breakdown point of modifications of the MAD and point out which of these modifications lead to a Stahel-Donoho estimator with maximal finite sample breakdown point.

Finite Sample Breakdown Points of Outlier Detection Procedures

The conventional iterative outlier detection procedures (CIODP), such as the Baarda-, Pope-, or t-testing procedure, based on the least-squares estimation (LSE) are used to detect the outliers in geodesy. Since the finite sample breakdown point (FSBP) of LSE is about 1/n, the FSBPs of the CIODP are also expected to be the same, about 1/n. In this paper, this problem is studied in view of the robust statistics for coordinate transformation with simulated data. Outliers have been examined in two groups: “random” and “jointly influential.” Random outliers are divided again into two subgroups: “random scattered” and “adjacent.” The single point displacements can be thought of as jointly influential outliers. These are modeled as the shifts along either the x- and y-axis or parallel to any given direction. In addition, each group is divided into two subgroups according to the magnitude of outliers: “small” and “large.” The FSBPs of either the Baarda-, Pope-, or t-testing procedure are the same and about 1/n. I...

Partial breakdown in two-factor models

The usual (global) breakdown point describes the worst effect that a given number of gross errors can have. In a two-way layout, without interaction, one is frustrated by the small number of gross errors such a design can tolerate. However, neither the whole fit nor all parameter estimates need to be affected by such a breakdown. An example from molecular spectroscopy serves to illustrate such partial breakdown in a large, “sparse” two-factor model. Because the global finite sample breakdown point is zero for all usual estimators in this example, this concept does not make sense in such problems. The more appropriate concept of partial breakdown point is discussed in this paper. It also provides a crude quantification of the robustness properties of an estimator, yet for any linear combination of the estimated parameters. The maximum number of gross errors to which the linear combination of the estimated parameters can resist is related to the minimum number of observations that must be omitted to make the linear function a non-estimable function. In the example, we are mainly interested in differences of parameters. Then the maximal partial breakdown point for regression equivariant estimators is one half, and Huber-type regression M-estimators with bounded ψ-function reach this limit.

On the Influence Functions of Certain Bivariate Medians

SUMMARY The asymptotic efficiency, influence function and breakdown properties of the multivariate median proposed by Oja are investigated and comparisons are made with two other generalizations of the median: the spatial median and the vector of marginal medians. It is shown that, for distributions with finite expectations, the Oja median has a bounded influence function although this median has 0% breakdown. For circular distributions, the influence functions of the Oja median and the spatial median coincide. The Oja median is then as efficient as the spatial median in the circular case but strictly better for any other elliptical case. It is noteworthy that the spatial median, however, has the best possible 50% breakdown. This phenomenon is analysed in the paper: it is shown that the finite sample breakdown point of the Oja median depends on the dispersion of the corrupted observations.