Pith. sign in

REVIEW 4 major objections 6 minor 53 references

Discriminating Tail Behavior Using Halfspace Depths: Population and Empirical Perspectives

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that the empirical (sample) halfspace depth tracks the population halfspace depth at growing points almost surely, making the depth-decay rate a reliable signal of light versus heavy multivariate tails.

desk verdict A genuinely useful unification of empirical halfspace depth rates via Alexander's weighted empirical processes, but the per-coordinate tail-discrimination algorithm claims more than the theorems prove. read the letter →

arxiv 2506.01126 v1 pith:VWFK6H4G submitted 2025-06-01 math.ST stat.MEstat.TH

Sibsankar Singha , Marie Kratz , Sreekar Vadlamani This is my paper
classification math.STstat.MEstat.TH MSC 60F1562G2062G3262H05
keywords halfspacedepthTukeytailbehaviourmultivariateregularvariationempiricalprocessesweightedextremequantilescontours
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's goal is to use halfspace depth, the smallest probability mass in any half-space containing a point, as a probe for how a multivariate distribution's tails behave. The authors establish that for growing points $t_n x$, the sample depth $\mathrm{HD}(t_n x, P_n)$ converges almost surely to the population depth $\mathrm{HD}(t_n x, P)$, under conditions on a threshold sequence $\gamma_n$, with no assumption on the tail type. This transfers the population decay rate to the empirical depth: exponential-type decay signals a light tail, and polynomial decay $t^{-\alpha}$ signals a heavy tail under multivariate regular variation. On this basis they propose an exploratory algorithm that plots $\frac{1}{t_n}\log(1/\mathrm{HD}(t_n e_k, P_n))$ along coordinate directions to flag directions as light- or heavy-tailed, and they illustrate the method on simulated and real climate data.

What carries the argument

The central object is halfspace depth, $\mathrm{HD}(x,P)=\inf_{H\ni x}P(H)$, equivalently written as the minimum over unit vectors $h$ of the projected tail probabilities $1-F_h(\langle h,x\rangle)$ and $F_h(\langle h,x\rangle)$. The argument is carried by a weighted empirical process bound for ratio empirical processes indexed by halfspaces: Lemma 3.3 controls the depth ratio by $\sup_{H\in\mathcal{H}_{t_n x}}|P_n(H)/P(H)-1|$, and condition (C2) restricts that supremum to halfspaces with $P(H)\ge\gamma_n$, where the convergence is uniform by a theorem on weighted empirical processes indexed by a VC class of sets. The capacity function $g_c$ of that theorem enters only through condition (C1a), and the authors show that it plays a negligible role compared with the tail-driven choice of $\gamma_n$.

What would settle it

Take the paper's rotated example: independent Gaussian, Laplace, and $t_3$ marginals rotated so every coordinate $X_i$ has a nonzero $t_3$ coefficient, making all three marginal distributions heavy-tailed. Proposition 2.3 and Theorem 2.5 imply that along $e_2$, $\mathrm{HD}(t e_2,P)$ decays at least exponentially because the Gaussian direction has nonzero inner product with $e_2$. Plotting $\frac{1}{t}\log(1/\mathrm{HD}(t e_2,P_n))$ on a large sample should show the light-tail signature even though the $e_2$ marginal is heavy; observing that would settle that directional flags are not per-coordinate marginal tail statements.

Watch

Extended reading notes

Core claim

The anchor result, Theorem 3.1, says that if the depth at the growing points stays above a threshold $\gamma_n$ that is not too small compared with the sample size, then $\sup_{\|x\|\le\varepsilon}\big|\mathrm{HD}(t_n x, P_n)/\mathrm{HD}(t_n x, P)-1\big|\to 0$ almost surely. Combined with the population results of Section 2, this yields explicit rates: under multivariate regular variation with index $\alpha$, $\mathrm{HD}(t_n x, P_n)/(1-P(t_n B^d))$ converges almost surely to the limiting depth $\mathrm{HD}(x,\nu)$, giving polynomial decay; if the moment generating function is finite in some direction, the sample depth obeys an exponential-type lower bound. The comparison of these rates gives a way to tell light from heavy tails from a single sample, within the data hull and without extrapolation.

Load-bearing premise

The algorithm reads each coordinate direction's depth decay as that coordinate's own tail statement, but the upper bound in Proposition 2.3 lets the lightest marginal contributing to a direction control the decay, so a heavy marginal can be masked by a lighter one in the same rotated direction.

Editorial extensions

If this is right

  • For heavy-tailed multivariate regularly varying distributions, the empirical halfspace depth satisfies $\mathrm{HD}(t_n x, P_n)/(1-P(t_n B^d))\to\mathrm{HD}(x,\nu)$ almost surely, so the sample depth inherits the $t_n^{-\alpha}$ decay of the population depth.
  • For light-tailed distributions with a finite moment generating function in some direction, $\liminf_n \frac{1}{t_n}\log(1/\mathrm{HD}(t_n x, P_n))>\langle x,h^*\rangle$ almost surely, so the empirical depth cannot decay faster than exponential in that direction.
  • In the Gaussian example, choosing $\gamma_n=n^{-\beta}$ with $0<\beta<1$ gives $t_n\le\sqrt{2\beta\log n}$, and both the lower and the marginals-based upper bounds are of order $n^{-\beta}$, showing the transfer from population to sample rate is sharp up to constants.
  • The ratio bound of Lemma 3.3, together with the weighted empirical process theorem, transfers population rates to sample rates without requiring Dvoretzky-Kiefer-Wolfowitz inequalities or tail-specific assumptions.
  • The proposed algorithm flags a direction as light-tailed when $\frac{1}{t_n}\log(1/\mathrm{HD}(t_n e_k,P_n))$ grows or stabilizes at a positive constant, and heavy-tailed when the log-log version stabilizes at a finite positive slope; if every coordinate is heavy, the conclusion is that the distribution is heavy-tailed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The per-coordinate reading of directional flags is the fragile part of the method: because Proposition 2.3 bounds the depth along any direction by the lightest marginal contributing to that direction, a rotated heavy marginal can be masked by a lighter component, as the paper's own rotated simulation illustrates. A more robust tool would scan many random projections and report the lightest decay f
  • The heavy-tail case could be pushed from qualitative discrimination to quantitative estimation: in the multivariate regular variation regime, regressing $\log\mathrm{HD}(t_n x,P_n)$ on $\log t_n$ over a range where the theorem's conditions hold would yield an estimator of the tail index $\alpha$, a step the paper leaves implicit.
  • The choice of $\gamma_n$ and hence the usable range of $t_n$ is governed by conditions (C1a)-(C2); this suggests a principled stopping rule for the plots, based on when the sample depth falls below the computable threshold, instead of the visual inspection currently used.
  • Because halfspace depth does not characterize the full joint distribution, the tool's verdict concerns which halfspace probabilities decay fastest; pairing it with a test for multivariate regular variation, as the authors themselves caution, is needed before interpreting a 'heavy' flag as a full tail statement.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper studies the asymptotic decay of Tukey's halfspace depth for both the population and the empirical measure. It proves a general almost-sure uniform convergence result for the ratio of empirical to population halfspace depth at growing points (Theorem 3.1), derives population decay rates for light and heavy tails (Theorems 2.4, 2.5 and Proposition 2.6), and proposes a coordinate-wise algorithm (Algorithm 1) that flags directions as light- or heavy-tailed on the basis of the scaled log-depth plots. The methodology is illustrated on simulated rotated Gaussian/Laplace/Student data and on an outgoing longwave radiation dataset from two Indian cities across two seasons. The proofs rely on Alexander's weighted empirical process results and on the MRV framework of He and Einmahl (2017).

Significance. If Theorem 3.1 is correct, the paper provides a valuable complement to the heavy-tail depth results of He and Einmahl (2017) and the light-tail depth results of Burr and Fabrizio (2017), and the Alexander-based proof is elegant and reasonably general. The population bounds (Theorem 2.4, Theorem 2.5, Proposition 2.6) give useful comparison rates and the explicit example computations (Example 3.10) are instructive. However, the methodological contribution that motivates the title is not supported: Algorithm 1 interprets depth decay along a coordinate direction as a statement about that coordinate's marginal tail, which does not follow from the paper's theorems and is false in general. Since the real-data conclusions in Section 4.3 rest on this interpretation, the applied claims need substantial reworking.

major comments (4)
  1. [Section 4.1, Algorithm 1; Section 4.3; Proposition 2.3] The per-coordinate tail flags produced by Algorithm 1 are not consequences of the paper's theoretical results. Proposition 2.3 gives only an upper bound on HD(t e_k,P) in terms of the coordinate marginals; it does not identify the halfspace attaining (or nearly attaining) the infimum in (2). That halfspace may be tilted with respect to the coordinate axes. Concretely, in R^2 take X_1 Pareto with P(X_1 >= s) = s^{-alpha}, Z standard normal independent, and X_2 = -X_1 + Z. The halfspace H = {y_1 + y_2 >= t} contains t e_1 and has P(H) = P(Z >= t) ~ (1/t) e^{-t^2/2}; hence HD(t e_1,P) <= P(Z >= t), so (1/t) log(1/HD(t e_1,P)) -> infinity and Algorithm 1 flags e_1 as light-tailed, although the first coordinate marginal is Pareto heavy. This is not a pathological edge case: the simulation discussion in Section 4.2 explicitly observes that the depth in e_2 and e_3 appears exponential/Gaussian even though all marginals are heavy after rotation, and attributes the phenomenon to coefficient sizes. The real-data conclusions in Section 4.3 (e.g., 'Prayagraj summer has the heaviest left tail') rely on the same unjustified per-coordinate reading. The authors should either prove conditions under which the minimizing halfspace is aligned with the coordinate halfspace (e.g., for independent marginals with suitably ordered tails), or substantially weaken the interpretation of the algorithm and remove or qualify the climatological claims.
  2. [Theorem 2.5(i)-(ii) and proof in Section 5.1.3] The strict inequalities in (5) and (6) are stronger than what the proof establishes. The Markov-bound argument gives log(1/HD(tx,P)) >= -log C_N + t sup_{h in N} <x,h> (respectively >= -log M + t <x,h*>), so dividing by t and taking lim inf yields only '>='. Equality is possible if the upper bound is asymptotically tight, and the manuscript provides no additional argument ruling out equality. The statements should be weakened to non-strict inequalities, or the proof supplemented to exclude equality. The same issue propagates to Theorem 3.8 and equation (11).
  3. [Theorem 3.5 and proof in Section 5.2.2] The proof of Theorem 3.5 displays the supremum over ||x|| = epsilon, while the statement requires a supremum over ||x|| = 1. Theorem 3.1 is formulated for a fixed epsilon-ball, so the proof needs to show explicitly that Condition (C2) can be verified on a set containing the unit sphere (for example by taking epsilon >= 1 and checking the condition uniformly on the unit ball) and that Proposition A.1 supplies the required uniform bound on that set. The current switch between epsilon and 1 leaves a gap that should be closed by a precise argument.
  4. [Lemma 3.3 and Proposition 5.1, proofs in Section 5.2.1] The proofs of Lemma 3.3 and Proposition 5.1 assume the existence of halfspaces attaining the infimum in the halfspace depth for an arbitrary probability measure P and for the empirical measure P_n. For a general probability measure the infimum over the family of halfspaces containing a given point need not be attained, because the map from the sphere to halfspace probabilities is only continuous under additional assumptions such as absolute continuity of P. The proofs should be reworked with minimizing sequences and an epsilon argument, or the result restricted to cases where attainment is guaranteed (e.g., absolutely continuous P and finite-support P_n).
minor comments (6)
  1. [Section 4.3] The word 'bevaiour' should be 'behaviour'.
  2. [Remark 3.6] The word 'emcompasses' should be 'encompasses'.
  3. [Appendix B] The heading 'V apnik–˘Cervonenkis' contains a formatting artifact and should be 'Vapnik–Chervonenkis'.
  4. [References and Introduction] The author 'Nagy' is cited inconsistently as 'Nagy, N. (2021)' in the Introduction and as 'Nagy, S.' in the reference list; the name should be normalized.
  5. [Algorithm 1 and Proposition 2.6] The classification criteria in Algorithm 1 would benefit from a short explanation connecting the branches to Proposition 2.6, since the case y_n -> c > 0 corresponds to exponential decay, while the case y_n -> 0 requires the second-level w_n comparison to separate subexponential from polynomial behaviour.
  6. [Figure 2 caption and Section 4.2] The caption says 'QQ-plot of the marginals X_i' and the text states that the QQ-plot shows X_1, X_2, X_3 exhibit heavy-tailed behaviour; clarifying that the reference distribution is standard Laplace and that 'heavy-tailed' means deviation from Laplace would make the discussion easier to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's theorem chain is self-contained and rests on external, non-author results such as Alexander (1987).

full rationale

The paper's central claim, Theorem 3.1, is proved by combining Lemma 3.3/Proposition 5.1 with the external weighted empirical process theorem of Alexander (1987); neither the theorem nor its proof fits parameters to the data whose tail behavior is later classified. The population results (Theorems 2.2, 2.4, 2.5) are derived directly from the definition of halfspace depth, Markov/Chernoff estimates, and standard tail assumptions; they do not assume the empirical depth decay they are used to explain. The empirical theorems (3.5, 3.8) are deductions from Theorem 3.1 together with the population bounds, and the sequence t_n is chosen from population depth lower bounds, which the paper explicitly supplies in Remark 3.2 rather than fitting from the sample depth. Algorithm 1 is introduced after and derived from these results, not used to reverse-engineer them. The skeptic's counterexample shows a genuine inferential gap in Algorithm 1's per-coordinate reading of Proposition 2.3: the upper bound HD(tx,P) ≤ min marginal tail probabilities is not an equivalence, so a tilted halfspace can make HD(t e_k,P) decay faster than the k-th marginal. That is a correctness/validity concern about the algorithm's interpretation, not a circularity: no claimed theorem is equivalent by construction to its own input, no fitted parameter is renamed as a prediction, and no load-bearing self-citation or imported uniqueness theorem is present. The paper's own Section 4.2 discussion even concedes the directional-coefficient effect, which further shows the authors are not smuggling the conclusion into the assumptions.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central proofs rely on Alexander's weighted empirical process theorem and standard VC theory, plus explicit tail assumptions on P. No free parameter is fitted to make the theory match data; the simulation uses the Dyckerhoff (2004) approximate depth as an unverified proxy for exact depth in the tail.

assumptions (6)
  • standard math Alexander (1987) Theorem 5.1 on weighted empirical processes indexed by VC classes holds for the class of halfspaces with capacity function g_c
    Core tool used in the proof of Theorem 3.1 (Section 5.2.1, Step 3; Appendix B).
  • standard math The class of all halfspaces in R^d is a VC class
    Invoked to apply Alexander's theorem; standard result.
  • domain assumption The tail lower-bound and moment generating function conditions on P in Theorems 2.4, 2.5, 3.8 are satisfied by the distributions under study
    These are explicit conditions on P stated in the theorems.
  • ad hoc to paper Optimal halfspaces attaining the infimum in HD exist in Lemma 3.3
    The proof assumes minimizers eH and H* exist; for arbitrary P with atoms this is not guaranteed, though the argument can be patched with an epsilon approximation.
  • domain assumption The limit measure nu in Theorem 3.5 assigns positive mass to every set containing a halfspace
    Condition 1 of Theorem 3.5 is used to ensure HD(x,nu) is positive and the convergence in Proposition A.1 is meaningful.
  • ad hoc to paper The Dyckerhoff (2004) approximate halfspace depth is a faithful proxy for exact depth in the tail region explored in the simulations
    The large-sample plots use the approximate depth library, but the paper does not bound the approximation error for points far from the data hull, where exact empirical depth can be zero.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Discriminating Tail Behavior Using Halfspace Depths: Population and Empirical Perspectives." pith.science (2026). https://pith.science/paper/VWFK6H4G

@misc{pith2026250601126,
  author       = {Pith},
  title        = {Pith review of: Discriminating Tail Behavior Using Halfspace Depths: Population and Empirical Perspectives},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VWFK6H4G}},
  note         = {Machine review of arXiv:2506.01126}
}
read the original abstract

We study the empirical version of halfspace depths with the objective of establishing a connection between the rates of convergence and the tail behaviour of the corresponding underlying distributions. The intricate interplay between the sample size and the parameter driving the tail behaviour forms one of the main results of this analysis. The chosen approach is mainly based on weighted empirical processes indexed by sets by Alexander (1987), which leads to relatively direct and elegant proofs, regardless of the nature of the tail. This method is further enriched by our findings on the population version, which also enable us to distinguish between light and heavy tails. These results lay the foundation for our subsequent analysis of the empirical versions. Building on these theoretical insights, we propose a methodology to assess the tail behaviour of the underlying multivariate distribution of a sample, which we illustrate on simulated data. The study concludes with an application to a real-world dataset.

Figures

Figures reproduced from arXiv: 2506.01126 by the authors.

Figure 1
Figure 1. Representation of the Tukey depth contours for 6 different depths, considering a sample of 1000 observations (black points) from a mean zero Gaussian distribution with covariance diag(1, 100). Decay rate of halfspace depth: Recall that our motivation is to understand the connection between the extremal behaviour of a probability measure and the asymptotics of halfspace depth functions. Specifically, we question the … view at source ↗
Figure 2
Figure 2. Left plot: yn := log  (HD(tnx, Pn))−1  /tn against tn = n/1000 (with n = k103 , for k = 10, . . . , 100) in e1, e2 and e3 directions. Right plot: QQ-plot of the marginals Xi , i = 1, 2, 3 (where X = AT Y), w.r.t. the standard Laplace quantiles. tailed behaviour. In contrast, the halfspace depth plots in the directions e1, e2 and e3 (left plot) exhibit mixed behaviour. For the e1 direction, no defini￾tive conclusio… view at source ↗
Figure 3
Figure 3. The left and middle plots correspond to yn := log  (HD(tnx, Pn))−1  /tn plotted for the OLR data in −e1, −e2, −e3 and −e4 directions against tn, with tn = 6n/100 for the left plot, and tn = n/10 for the middle plot, for n = ⌊17.08k⌋, for k = 10, . . . , 100. The right most figure depicts the QQ plot of all the four marginals w.r.t. univariate standard Laplace distribution. empirical halfspace depths are presented … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Tukey depths are computed at points in direction x = (1, 1) and given in terms of (tn) growing linearly in n (tn = 1.8 + n.10−4 , with n = 105k/50, k = 1, 2 · · · , 50). Samples are taken from independent bivariate Pareto with parameter 2.2 and 3.2, respectively, and G…
Figure 5
Figure 5. Figure 5: Tukey depths are computed at points in direction x = (1, 1) given in terms of (tn). Samples are taken from independent bivariate Pareto with pa￾rameter 1.9, 2.2 and 3.2, respectively, and Gaussian distribution with diagonal covariance matrix diag(2, 2). Number of obser…
Figure 6
Figure 6. Figure 6: The left and right plots correspond to yn := log  (HD(tnx, Pn))−1  /tn plotted for the OLR data in e1, e2, e3 and e4 directions against tn, with tn = 5n/100 for the left plot, and tn = 3n/100 for the right plot, for n = ⌊17.08k⌋, for k = 10, . . . , 100. 39 [PITH_FU…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

  1. [1]

    Agostinelli, C. and M. Romanazzi (2011). Local depth. Journal of Statistical Planning and Inference\/ 141\/ (2), 817--830

    work page 2011

  2. [2]

    Alexander, K. (1987). Rates of growth and sample moduli for weighted empirical processes indexed by sets. Probability Theory and Related Fields\/ 75\/ (3), 379--423

    work page 1987

  3. [3]

    Barber, C. and P. Mozharovskyi (2022). Tukey R egion: Tukey region and median. R package version

    work page 2022

  4. [4]

    Barnett, V. (1976). The ordering of multivariate data. Journal of the Royal Statistical Society A\/ 139\/ (3), 318--344

    work page 1976

  5. [5]

    Burr, M. and R. Fabrizio (2017). Uniform convergence rates for halfspace depth. Statistics & Probability Letters\/ 124 , 33--40

    work page 2017

  6. [6]

    Chaudhury, P. (1996). Multivariate location estimation using extension of R -estimates through U -statistics type approach. Journal of the American Statistical Association\/ 91 , 862--872

    work page 1996

  7. [7]

    Galichon, M

    Chernozhukov, V., A. Galichon, M. Hallin, and M. Henry (2017). Monge- K antorovich depth, quantiles, ranks and signs. Annals of Statistics\/ 45\/ (1), 223--256

    work page 2017

  8. [8]

    Donoho, D. and M. Gasko (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Annals of Statistics\/ 20\/ (4), 1803--1827

    work page 1992

Show all 53 references
  1. [9]

    Dudley, R. (1984). A course on empirical processes. In Ecole d' \'e t \'e de probabilit \'e s de Saint-Flour XII-1982 , pp.\ 1--142. Springer

    1984

  2. [10]

    Dudley, R. and V. Koltchinskii (1992). The spatial quantiles. Unpublished manuscript

    1992

  3. [11]

    Dutta, S. and A. Ghosh (2012). On robust classification using projection depth. Annals of the Institute of Statistical Mathematics\/ 64 , 657--676

    2012

  4. [12]

    Kiefer, and J

    Dvoretzky, A., J. Kiefer, and J. Wolfowitz (1953). Sequential decision problems for processes with continuous time parameter. testing hypotheses. Annals of Mathematical Statistics\/ 24\/ (2), 254--264

    1953

  5. [13]

    Dyckerhoff, R. (2004). Data depths satisfying the projection property. Allgemeines Statistisches Archiv\/ 88 , 163--190

    2004

  6. [14]

    Mosler, and G

    Dyckerhoff, R., K. Mosler, and G. Koshevoy (1996). Zonoid data depth: Theory and computation. In COMPSTAT: Proceedings in Computational Statistics 12th Symposium, Barcelona, Spain, 1996 , pp.\ 235--240. Springer

    1996

  7. [15]

    Mozharovskyi, and S

    Dyckerhoff, R., P. Mozharovskyi, and S. Nagy (2021). Approximate computation of projection depths. Computational Statistics & Data Analysis\/ 157 , 107166

    2021

  8. [16]

    Eddy, W. F. (1982). Convex hull peeling. In COMPSTAT 1982 5th Symposium held at Toulouse 1982: Part I: Proceedings in Computational Statistics , pp.\ 42--47. Springer

    1982

  9. [17]

    Li, and R

    Einmahl, J., J. Li, and R. Y. Liu (2015). Bridging centrality and extremity: refining empirical data depth using extreme value statistics. Annals of Statistics\/ 43\/ (6), 2738--2765

    2015

  10. [18]

    Laketa, P

    Fojtík, V., P. Laketa, P. Mozharovskyi, and S. Nagy (2024). On exact computation of tukey depth central regions. Journal of Computational and Graphical Statistics\/ 33\/ (2), 699--713

    2024

  11. [19]

    Mass\'e, and J.-F

    Genest, M., J.-C. Mass\'e, and J.-F. Plante (2019). Depth: Nonparametric depth functions for multivariate analysis. R package version

    2019

  12. [20]

    Gin \'e , E. and V. Koltchinskii (2006). Concentration inequalities and asymptotic results for ratio type empirical processes. Annals of Probability\/ 34\/ (3), 1143--1216

    2006

  13. [21]

    Paindaveine, and M

    Hallin, M., D. Paindaveine, and M. Siman (2010). Multivariate quantiles and multiple-output regression quantiles: From l_1 optimization to halfspace depth. Annals of Statistics\/ 38\/ (2), 635--669

    2010

  14. [22]

    He, Y. and U. Einmahl (2017). Estimation of extreme depth-based quantile regions. Journal of the Royal Statistical Society B\/ 79\/ (2), 449--461

    2017

  15. [23]

    Koltchinskii, V. (1997). M--estimation, convexity and quantiles. Annals of Statistics\/ 25\/ (2), 435--477

    1997

  16. [24]

    Koshevoy, G. (2003). Lift-zonoid and multivariate depths. In Developments in Robust Statistics: International Conference on Robust Statistics 2001 , pp.\ 194--202. Springer

    2003

  17. [25]

    Koshevoy, G. and K. Mosler (1997). Zonoid trimming for multivariate distributions. Annals of Statistics\/ 25\/ (5), 1998--2017

    1997

  18. [26]

    Kuelbs, J. and J. Zinn (2016). Convergence of quantile and depth regions. Stochastic Processes and their Applications\/ 126 , 3681--3700

    2016

  19. [27]

    Liu, R. (1990). On a notion of data depth based on random simplices. Annals of Statistics\/ 18 , 405--414

    1990

  20. [28]

    Liu, R. and K. Singh (1993). A quality index based on data depth and multivariate rank tests. Journal of the American Statistical Association\/ 88\/ (421), 252--260

    1993

  21. [29]

    Mosler, and P

    Liu, X., K. Mosler, and P. Mozharovskyi (2019). Fast computation of tukey trimmed regions and median in dimension p>2 . Journal of Computational and Graphical Statistics\/ 28\/ (3), 682--697

    2019

  22. [30]

    Liu, X. and Y. Zuo (2015). Comp PD : A MATLAB package for computing projection depth

    2015

  23. [31]

    on the generalised distance in statistics

    Mahalanobis, P. (2018). Reprint of: Mahalanobis, p.c. (1936) "on the generalised distance in statistics.". Sankhya A\/ 80 , 1--7

    2018

  24. [32]

    Mahalanobish, O. and S. Karmakar (2015). Depth. plot: Multivariate analogy of quantiles. R package version 0.1

    2015

  25. [33]

    Mosler, K. (2002). Multivariate dispersion, central regions and depth: The lift zonoid approach , Volume 165. Berlin: Springer-Verlag

    2002

  26. [34]

    Mosler, K. (2013). Depth statistics. In Robustness and complex data structures\/ (Becker, C. and Fried, R. and Kuhnt, S. ed.)., Heidelberg, pp.\ 17--34. Springer

    2013

  27. [35]

    Mosler, K. and P. Mozharovskyi (2021). Choosing among notions of multivariate depth statistics. Statistical Science\/ 37\/ (3), 348--368

    2021

  28. [36]

    Oja, and R

    M \"o tt \"o nen, J., H. Oja, and R. Serfling (2005). Multivariate generalized spatial signed-rank methods. Journal of Statistical Research\/ 39\/ (1), 19--42

    2005

  29. [37]

    Nagy, N. (2021). Halfspace depth does not characterize probability distributions. Statistical Papers\/ 62 , 1135--1139

    2021

  30. [38]

    Dyckerhoff, and P

    Nagy, N., R. Dyckerhoff, and P. Mozharovskyi (2020). Uniform convergence rates for the approximated halfspace and projection depth. Electronic Journal of Statistics\/ 14\/ (2), 3939--3975

    2020

  31. [39]

    Helander, G

    Nagy, N., S. Helander, G. Van Bever, L. Viitasaari, and P. Ilmonen (2021). Flexible integrated functional depths. Bernoulli\/ 27\/ (1), 673--701

    2021

  32. [40]

    Nagy, S. (2022). Halfspace depth: Theory and computation. Habilitation thesis, available at https://chres.is.cuni.cz/media/documents/2023/01/09/Nagy_ThesisShort.pdf

    2022

  33. [41]

    Nagy, S. and P. Laketa (2025). Theoretical properties of angular halfspace depth. Bernoulli\/ 31\/ (2), 1007--1031

    2025

  34. [42]

    Oja, H. (1983). Descriptive statistics for multivariate distributions. Statistics & Probability Letters\/ 1\/ (6), 327--332

    1983

  35. [43]

    Paindaveine, D. and G. Van Bever (2013). From depth to local depth: a focus on centrality. Journal of the American Statistical Association\/ 108\/ (503), 1105--1119

    2013

  36. [44]

    Mozharovskyi, and R

    Pokotylo, O., P. Mozharovskyi, and R. Dyckerhoff (2019). Depth and depth-based classification with R package ddalpha

    2019

  37. [45]

    Shorack, G. and J. Wellner (2009). Empirical processes with applications to statistics . SIAM edition

    2009

  38. [46]

    Struyf, A. and P. Rousseeuw (1999). Halfspace depth and regression depth characterize the empirical distribution. Journal of Multivariate Analysis\/ 69\/ (1), 135--153

    1999

  39. [47]

    Talagrand, M. (2003). Vapnik-- C hervonenkis type conditions and uniform D onsker classes of functions. Annals of Probability\/ 31\/ (3), 1565--1582

    2003

  40. [48]

    Tukey, J. (1975). Mathematics and picturing data. In Proceedings of the International Congress on Mathematics\/ (James, R.D. ed.), Volume 2, pp.\ 523--531

    1975

  41. [49]

    Vapnik, V. N. and A. Chervonenkis (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications\/ 16\/ (2), 264--280

    1971

  42. [50]

    and C.-H

    Vardi, Y. and C.-H. Zhang (2000). The multivariate l_1 -median and associated data depth. Proceedings of the National Academy of Sciences\/ 97\/ (4), 1423--1426

    2000

  43. [51]

    Wellner, J. (1992). Empirical processes in action: A review. International Statistical Review\/ 60 , 247--269

    1992

  44. [52]

    Zuo, Y. (2003). Projection-based depth functions and associated medians. Annals of Statistics\/ 31\/ (5), 1460--1490

    2003

  45. [53]

    Zuo, Y. and R. Serfling (2000). General notions of statistical depth function. Annals of Statistics\/ 28\/ (1), 461--482

    2000

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.