The breakdown point in its different variants is one of the central notions to quantify the global robustness of a procedure. We propose a simple supplementary variant which is useful in situations where we have no obvious or only partial equivariance: Extending the Donoho and Huber(1983) Finite Sample Breakdown Point, we propose the Expected Finite Sample Breakdown Point to produce less configuration-dependent values while still preserving the finite sample aspect of the former definition. We apply this notion for joint estimation of scale and shape (with only scale-equivariance available), exemplified for generalized Pareto, generalized extreme value, Weibull, and Gamma distributions. In these settings, we are interested in highly-robust, easy-to-compute initial estimators; to this end we study Pickands-type and Location-Dispersion-type estimators and compute their respective breakdown points.
Deep Dive into Yet another breakdown point notion: EFSBP - illustrated at scale-shape models.
The breakdown point in its different variants is one of the central notions to quantify the global robustness of a procedure. We propose a simple supplementary variant which is useful in situations where we have no obvious or only partial equivariance: Extending the Donoho and Huber(1983) Finite Sample Breakdown Point, we propose the Expected Finite Sample Breakdown Point to produce less configuration-dependent values while still preserving the finite sample aspect of the former definition. We apply this notion for joint estimation of scale and shape (with only scale-equivariance available), exemplified for generalized Pareto, generalized extreme value, Weibull, and Gamma distributions. In these settings, we are interested in highly-robust, easy-to-compute initial estimators; to this end we study Pickands-type and Location-Dispersion-type estimators and compute their respective breakdown points.
In an industrial project to compute robust variants of OpVar, i.e.; the regulatory capital as required in Basel II (2006) for a bank to cover its operational risk, we came across the problem of determining the (finite sample) breakdown point of certain considered procedures. Here operational risk is by definition "the risk of direct or indirect loss resulting from inadequate or failed internal processes, people and systems or from external events."
These extremal events, as motivated by the Pickands-Balkema-de Haan Extreme Value Theorem (see Balkema and de Haan (1974), Pickands (1975)) suggest the use of the generalized Pareto distribution (GPD) for modeling in this context. In an intermediate step this modeling involves estimation of the scale and shape parameters of this distribution. To this end, several robust procedures have been proposed in the literature, see Ruckdeschel and Horbenko (2010) for a more detailed discussion.
One of the quantities to judge robustness of a procedure is the breakdown point (see Definition 3.1). In particular, we are interested in the finite sample version FSBP of this notion to be able to quantify the degree of protection a procedure provides in the estimation at an actual (finite) set of observations. It turns out that for our purposes the original definition has some drawbacks, as it depends strongly on the configuration of the actual sample. To get rid of the dependence on possibly highly improbable sample configurations while still preserving the aspect of a finite sample, we propose an expected FSBP, EFSBP, i.e.; to integrate out the FSBP with respect to the ideal distribution.
We illustrate the usefulness of this new concept for scale-shape models by means of two types of robust estimators, quantile-type estimators (Pickands Estimator PE) and robust Location-Dispersion (LD) estimators as introduced by Marazzi and Ruffieux (1999); for the latter type we study estimators based on the median for the location part and several robust scale estimators for the dispersion part: a (new) asymmetric version of the median of absolute deviations kMAD, as well as Qn and Sn from Rousseeuw and Croux (1993)combined to MedkMAD, MedQn, and MedSn, respectively. These estimators are meant to be used as initial estimators with acceptable to good global robustness properties for (more efficient) robust estimators afterwards. In particular, they can be computed without the need of additional (robust, consistent) initial estimators, which precludes otherwise promising alternatives like Minimum Distance estimators, for which we could have read off asymptotic breakdown point values as high as half the optimal value from Donoho and Liu (1988). We have also excluded the method-of-median approach of Peng and Welsh (2001), because in contrast to PE and MedkMAD, MedQn, and MedSn, for this estimator in the GPD and GEVD case, no explicit calculations are possible. We have studied this approach in another paper, though (Ruckdeschel and Horbenko (2010)), and empirically found that in the GPD case its breakdown behavior is worse than the one of MedkMAD and MedQn.
Our paper is organized as follows: In Section 2, we list our reference examples for scaleshape models, i.e.; the generalized Pareto, the generalized extreme value, the Weibull, and the Gamma distribution, as well as the Gross Error model which we use to capture deviations from the ideal model. In Section 3, we recall the standard definitions of the asymptotic and finite sample breakdown points ABP and FSBP and introduce the new concept of EFSBP in Definition 3.2. Section 4 then defines the considered estimators, i.e.; quantile-type estimators PE, and LD estimators MedkMAD, MedQn, MedSn. At these estimators, we demonstrate our new breakdown point notion in Section 5, giving analytic formulae for FSBP, ABP, and EFSBP in Propositions 5.1, 5.2, and 5.3, together with some numerical evaluations of EFSBP at some reference situation and with simulation-based evaluations. Proofs for our results are gathered in Appendix A.
Remark 1.1 This paper is a part of the PhD thesis of the second author; a preliminary version of it may be found in Ruckdeschel and Horbenko (2010).
For notions of invariance of statistical models and equivariance of estimators we refer to Eaton (1989): Given a measurable space (Ω , B), a family of probability measures P defined on B is a statistical model.
Notationally, we use the same symbol for the cumulative distribution function (c.d.f.) and the probability measure; we write F(x -0) to denote left and, correspondingly, +0 for right limits, and F -to denote the right continuous quantile function given by F
Definition 1 Suppose a group G acts measurably on Ω . Model P is called G-invariant iff for each P ∈ P, the image probability gP of P under group action g stays in P.
For simplicity, we assume that g(P 1 ) = g(P 2 ) implies P 1 = P 2 for any two elements of P. In a G-invariant parametric model P = {P θ |θ ∈ Θ }, where Θ
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