Finite sample breakdown point of Tukey’s halfspace median

Abstract

SummaryTukey's halfspace median (HM), servicing as the multivariate counterpart of the univariate median, has been introduced and extensively studied in the literature. It is supposed and expected to preserve robustness property (the most outstanding property) of the univariate median. One of prevalent quantitative assessments of robustness is finite sample breakdown point (FSBP). Indeed, the FSBP of many multivariate medians have been identified, except for the most prevailing one-the Tukey's halfspace median. This paper presents a precise result on FSBP for Tukey's halfspace median. The result here depicts the complete prospect of the global robustness of HM in the finite sample practical scenario, revealing the dimension effect on the breakdown point robustness and complimenting the existing asymptotic breakdown point result.

Multiple-Try Simulated Annealing Algorithm for Global Optimization

“…Third, we anticipate that a parallel SA algorithm can be used efficiently for variable selection in high dimensional cases [42][43][44][45] because the variable selection problem is a special case of the optimization problem. Finally, data depth [32,33,35,46] is an important tool for multidimensional data analysis, but the computation of data depth in high dimensional cases is challenging. The example of half-space depth computation in Section 4 shows the advantage of the MTMSA algorithm in low dimensional case.…”

“…As a powerful tool for nonparametric multivariate analysis, halfspace depth (HD also known as Tukey depth) has been eliciting increased interest since it was introduced by Tukey [31,32]. HD, which extends univariate order-related statistics to multivariate settings, provides a center-outward ordering of multivariate samples and visualizes data in high dimensional cases [33,34]. However, the computation of HD is challenging, and the exact algorithm is often inefficient, especially when the dimension is high [35].…”

The Limit of Finite Sample Breakdown Point of Tukey’s Halfspace Median for General Data

“…This contradicts with z j * ∈ B z j−1 when j * > j by (o1)-(o2). Then, based on lim k→∞ g(z i k −1 ) = 0 and (F1), we can obtain B z 0 \ {z 0 } = ∅ similar to Lemma 3 of Liu et al (2015b). Hence, we may let x 0 = z 0 .…”

“…(II). By denoting ℓ u = {z ∈ R d : z = A u x 0 + γu, ∀γ ∈ R 1 } and using a similar method to the first proof part of Theorem 1 in Liu et al (2015b), we can obtain that, for an any given y 0 ∈ cov(X n ) \ ℓ u , it holds sup x ∈cov(X n ) D(x , F n+m ) ≤ nλ * u n+m , where F n+m denotes the empirical distribution related to X n ∪ Y m , and Y m contains m repetitions of y 0 .…”

“…We will consider this issue under the combination of halfspace symmetry and a weak smooth condition (see Section 2 for details). Recently, Liu et al (2015b) have derived the exact finite sample breakdown point for fixed n. Their result nevertheless depends on the assumption that X n is in general position and could not be directly utilized under the current setting, because when the underlying F only satisfies the weak smooth condition, the random sample X n generated from F may not be in general position in some scenarios.…”