Pith. sign in

REVIEW 1 major objections 5 minor 55 references

Estimating the quantile density by local polynomials yields an estimator that keeps its bias rate at the support boundaries, unlike classical kernel quantile-density estimators.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 21:37 UTC pith:LO5RXETC

load-bearing objection Solid bounded-support theory; unbounded-support boundary claim outruns the theorem. the 1 major comments →

arxiv 2607.16016 v1 pith:LO5RXETC submitted 2026-07-17 math.ST stat.MEstat.TH

Local polynomial estimation of quantile density functions

Niclas Jacobsen , Natalie Neumeyer This is my paper
classification math.ST stat.MEstat.TH MSC 62G0562G3062G0762G20
keywords quantile density functionsparsity functionlocal polynomial estimationboundary adaptationasymptotic normalityorder statisticsquantile processnonparametric estimation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proposes estimating the quantile density function q(u)=Q'(u) — the reciprocal of the density evaluated at the quantile — and its derivatives by local polynomial regression on the pairs (i/n, X_(i)), where X_(i) are the order statistics. The trick is the identity Q(u)=E[X_i | F(X_i)=u], so the empirical quantile curve can be smoothed exactly like a regression curve, even though the pseudo-covariates are dependent. The main result is asymptotic normality with explicit bias and variance, at both interior points and points within h of 0 or 1; at the boundary the bias rate is the same as in the interior, which is the property earlier kernel estimators lack. For distributions with unbounded support the paper proves ratio consistency of the estimator and its derivatives. The payoff is a simple, boundary-adaptive way to estimate sparsity and related quantities, which feed into confidence intervals for quantiles, hazard-quantile analysis, and density-mode location.

Core claim

Theorem 2.4: for polynomial order p, under a Lipschitz kernel and smoothness assumptions on Q, sqrt(nh^{2v-1})/q(u) times the estimation error of \hat Q^{(v)}_p(u) minus a deterministic bias h^{p+1-v}B_{p,v}(u) converges to a centered normal with asymptotic variance (v!)^2 e_v^T S_p^{-1} Γ_p S_p^{-1} e_v. The same statement holds for boundary sequences u=ch or u=1-ch, with the asymptotic variance unchanged and q and Q^{(p+1)} interpreted as one-sided limits at 0 or 1. For v=0 the estimator converges to the empirical quantile in the interior and has a slightly different boundary rate. Theorem 3.1 extends the method to unbounded support by showing \hat Q^{(v)}/Q^{(v)} → 1 in probability under

What carries the argument

The load-bearing device is the regression representation Q(u)=E[X_i | F(X_i)=u], which turns quantile-curve estimation into a local polynomial regression problem on pseudo-data (F_n(X_i), X_i), i.e., on order statistics with equidistant design i/n. Through this representation the estimator is an L-statistic whose stochastic behaviour is governed by the uniform and quantile processes; the proof uses a Bahadur-type approximation and strong approximations of these processes. The bias term B_{p,v} and variance V_{p,v} are expressed through the matrices S_p, Γ_p and vector c_p, with Γ_p built from integrals of (x∧t) times polynomial times kernel product; these matrix objects are the same in the b

Load-bearing premise

For the bounded-support theorem the load-bearing premise is that the support endpoints are finite and the quantile derivatives Q^{(v)} remain finite with q(0+)>0 and q(1-)>0; if the density vanishes at a boundary, the normal approximation has no basis.

What would settle it

Simulate a distribution whose density vanishes at a finite boundary, such as Beta(2,2), and evaluate \hat q(u_n) at u_n=h c with small c; the theorem predicts the normal limit fails because q(0+)=0. Also check the boundary bias rate for p=2 at a bounded-support distribution with q(0+)>0 — if the bias at u=ch does not decay like h^2, the paper's central rate claim is contradicted.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Plug-in confidence intervals for Q^{(v)}(u) can be built directly from the theorem, with q(u) replaced by \hat q(u) and the normalized estimator exhibiting standard-normal coverage in simulations.
  • The score-type functional q'(u)/q(u)^2 can be estimated at a O(h^p) bias rate with asymptotic normality, relevant to hazard and classification applications.
  • For p-v odd at interior points the leading bias vanishes, giving faster h^{p+2-v} rates; at the boundary the rate stays h^{p+1-v} for all parity choices.
  • For bounded-support distributions, the estimator remains well-behaved at u in [0,h) and (1-h,1], where earlier kernel quantile-density estimators are inconsistent.
  • For unbounded-support distributions, the ratio \hat Q^{(v)}/Q^{(v)} is consistent along sequences u_n → 0 or 1 that converge slowly enough, e.g., u_n=1-h^γ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural extension, already alluded to in the paper, is conditional quantile density estimation by local polynomial quantile regression; the same pseudo-data argument would apply with covariates.
  • The parity recommendation p-v odd gains an extra argument here: unlike in standard local polynomial regression, the higher-order bias term depends only on derivatives of Q, so the usual density-derivative complication disappears.
  • The paper's boundary adaptation suggests a testable practical recipe: for bounded Beta-type data, choosing p=2 and a slightly larger bandwidth than the usual h=0.15 reduces boundary MSE, as the simulations indicate.
  • Uniform convergence rates and simultaneous confidence bands are the obvious next step, and the authors state these are already in progress.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper proposes a local polynomial estimator of the quantile density function q(u)=Q'(u) and its derivatives Q^{(v)}(u), based on pseudo-data (F_n(X_i),X_i), equivalently a weighted regression on the order statistics. For bounded support (Assumption 2), Theorem 2.4 establishes pointwise asymptotic normality at interior and boundary points, with explicit bias constants and variance; the proof uses an L-statistic decomposition, Riemann-sum estimates, and strong approximations of the empirical/quantile process. For unbounded support, Theorem 3.1 gives ratio consistency for sequences u_n satisfying Assumption 5, which requires i_n^U(u)<n; the authors note this excludes the upper boundary sequences u=1−ch, c∈[0,1). The paper compares bias, variance, and boundary behaviour with kernel estimators, reports simulations for bounded and unbounded distributions, and concludes that the estimator has better boundary properties than classical estimators, including for unbounded supports.

Significance. The bounded-support result is a substantial contribution: it gives a unified, explicit asymptotic theory for a boundary-adaptive quantile-density estimator and its derivatives, and the proof is careful and largely self-contained. The variance and bias formulas are parameter-free in the sense that no constants are fitted, and the recommendation p−v odd at boundaries matches the local-polynomial folklore. The unbounded-support consistency result is also useful, but its scope is narrower than the paper's headline claims. If the claims are corrected or the missing boundary case for unbounded supports is proved, the paper will be a solid contribution to nonparametric quantile inference. The simulations provide credible evidence for bounded supports, but the unbounded 'boundary' simulations are run in a regime not covered by Theorem 3.1.

major comments (1)
  1. [§3.2, Theorem 3.1, Assumption 5] The advertised boundary adaptation for unbounded supports is not proven. Assumption 5 requires i_n^U(u)<n, which fails for the exact upper boundary sequences u=1−ch with c∈[0,1), because the kernel interval then includes i=n. The note after Theorem 3.1 concedes this, yet the Abstract, §1, §4 (Figure 3), and §5 state that the estimator has better boundary properties for unbounded supports. Example 3.2 treats u_n=1−h^γ with γ∈(0,1); since h^γ>h, this is an interior sequence (u_n<1−h), not a boundary sequence of Theorem 2.4's type. Figure 3 evaluates u up to 0.99 with h=0.19, which gives i_n^U=n and is outside Theorem 3.1's assumptions. The consistency result covers slowly converging sequences, but not the boundary points u=ch or 1−ch that define 'boundary adaptation'. Please either add a theorem covering these boundary sequences under additional tail conditions, or revise the claims and si
minor comments (5)
  1. [§3.2, Example 3.2] The statement 'this implies u_n > 1−h' is backwards. For γ∈(0,1), h^γ > h, so u_n = 1−h^γ < 1−h. The example is therefore an interior sequence, consistent with Assumption 5, but it does not illustrate boundary behaviour.
  2. [§2, MSE-rate table] The table titled 'MSE rates' actually reports the rate of the optimal bandwidth, not the MSE. For example, for v=1, p=2, p−v odd, the optimal bandwidth is n^{-1/5} but the resulting MSE is n^{-4/5}. The corresponding MSE rates are n^{-2(p+1−v)/(2p+1)} for p−v odd and for boundary p−v even, and n^{-2(p+2−v)/(2p+3)} for interior p−v even. Please correct the table or its label.
  3. [§4, Figure 3] For the unbounded cases, the caption says the plot is over 'x' on [0,1] but the estimator is defined on the quantile domain u. More importantly, the point u=0.99 with h=0.19 is in the boundary region (u>1−h) but is not covered by Theorem 3.1; the caption should state this limitation or the simulations should be redesigned to lie in the proven range.
  4. [§3.1, Epanechnikov comparison] The claim that for the Epanechnikov kernel the new estimator has a smaller bias but larger variance than tilde q_1 is correct: for the Epanechnikov kernel the bias coefficients are h^2 q''/(14) versus h^2 q''/(10), and the variance factors are 5/7 versus 3/5. No change is needed, but a short derivation of this comparison would help readers.
  5. [§2, Theorem 2.4] The notation V_{p,v} is defined in Theorem 2.4, but the proof first uses V_{p,v,h}(u) in (8). It would be clearer to define the h-dependent version explicitly before (8) and state its limit in Theorem 2.4.

Circularity Check

0 steps flagged

No circularity found: the asymptotic results are derived from first principles; the unbounded-support boundary limitation is a scope gap, not a definitional reduction.

full rationale

The derivation chain is self-contained in the relevant sense. Theorem 2.4 is proved by decomposing the local polynomial normal equations into A_n (Riemann-sum error), B_n (bias), and C_n (stochastic term). The bias term B_{p,v}(u) is obtained by a direct Taylor expansion of Q in Lemma A.2, not by fitting the limit to the estimator. C_n is handled by a Bahadur-type expansion, with external inputs being classical uniform quantile-process bounds (Csörgő–Mason, Csörgő–Révész) and Cattaneo et al.'s Lemma 3, all independent published results, not self-citations. The reference list contains no self-citations by the present authors, so no self-citation chain is load-bearing. The unbounded-support Theorem 3.1 is a consistency result under Assumption 5; the paper itself explicitly flags the limitation: 'But this assumption implies, that we cannot look at u in the upper boundary case as in section 2, meaning u=1−ch for c∈[0,1), as then i_n^U(u)=n.' That is an honest scope restriction rather than a circular reduction: the theorem does not assert the exact boundary case, and the broader boundary claim in the introduction is an overreach but not a circularity. Assumption 5 involves ratios of q and Q^{(v)}, but these are regularity conditions on the true Q, and the proof shows the remainder terms vanish under them; they are not the conclusion being derived. The simulations' plug-in bandwidth estimates q and q'' with the same estimator family, which is standard bandwidth-selection practice and does not make the limit theorems fitted. No step was found where an output is identical by construction to an input, and no fitted parameter is relabeled as a prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The central derivation relies on standard regularity assumptions, quantile-process strong approximation theorems, and Riemann-sum estimates; no fitted parameters or invented entities enter the theory. The only hand-chosen numbers are simulation bandwidths, which are not part of the theorems.

axioms (7)
  • domain assumption K is bounded, symmetric, Lipschitz-continuous density with support [-1,1] (Assumption 1).
    Needed for Riemann-sum approximations and covariance limits in the proofs of Lemmas A.1–A.3.
  • domain assumption Q is (p+1)-times continuously differentiable on [u−δ,u+δ] (or [0,δ]/[1−δ,1] at boundaries) and q>0 (Assumption 2).
    Gives Taylor expansions and bounded q in a neighborhood; fails at the endpoints for unbounded support.
  • domain assumption Bandwidth h→0, nh^{2p+1}=O(1), and nh^2/(log n)^2 log log n →∞ (Assumption 3).
    Balances bias and variance, and is used to make remainder terms and the Lyapunov condition vanish.
  • standard math Csörgő–Mason (1985) sup-norm bound on uniform quantile deviations O_P(n^{-1/2}) in the middle, and Csörgő–Révész (1978) Bahadur remainder bound O_P(log^{1/2}n loglog^{1/4}n/n^{3/4}).
    Used in Lemma A.3 and Lemma A.4 to replace empirical quantiles by empirical cdfs; cited theorems from the literature.
  • domain assumption Assumption 4: sup_t t(1−t)|f'(Q(t))|/f^2(Q(t)) ≤ γ, plus monotonicity of f at support boundaries if f=0 there.
    Controls the uniform quantile process on the tails for Theorem 3.1; not needed for the bounded-support theorem.
  • domain assumption Assumption 5: i_n^U(u)<n, Q smooth in a neighborhood, sup_{I_n(u)}|q|/Q^{(v)}(u)=o(h^v n^{1/2}), and h^{p+1} sup|Q^{(p+1)}|/Q^{(v)}(u)=o(h^v).
    Ensures the estimator's kernel neighborhood stays away from 1 and the tail terms are negligible; excludes the exact upper boundary u=1−ch.
  • standard math Riemann-sum approximations of integrals of K_h-weighted Lipschitz functions have error O(1/(nh)) (Lemmas A.1, A.2, A.4).
    Basis for replacing sums by integrals in the bias and covariance calculations.

pith-pipeline@v1.3.0-alltime-deepseek · 26797 in / 29934 out tokens · 285614 ms · 2026-08-01T21:37:46.187465+00:00 · methodology

0 comments
read the original abstract

A new approach for nonparametric estimation of the quantile density function (sparsity function) and its derivatives is suggested which is based on local polynomial estimation. The estimator has more advantageous properties at the boundaries than classical quantile density estimators. Asymptotic normality is shown and the bias, asymptotic variance as well as boundary properties are compared with other estimators.

Figures

Figures reproduced from arXiv: 2607.16016 by Natalie Neumeyer, Niclas Jacobsen.

Figure 1
Figure 1. Figure 1: Plots of the MSE (mean squared error) after 500 simulations using the Beta(0 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plots of the average of 10 simulations of sample size [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plots of the average of 10 simulations of sample size [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left is a plot of the number of times, that [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

55 extracted references · 22 canonical work pages

  1. [1]

    2025 , eprint =

    Jirak, Moritz and Kneip, Alois and Meister, Alexander and Pahl, Mario , title =. 2025 , eprint =

    2025

  2. [2]

    Wand, M. P. and Jones, M. C. , TITLE =. 1995 , PAGES =. doi:10.1007/978-1-4899-4493-1 , URL =

    doi:10.1007/978-1-4899-4493-1 1995

  3. [3]

    , TITLE =

    Cleveland, William S. , TITLE =. J. Amer. Statist. Assoc. , FJOURNAL =. 1979 , NUMBER =

    1979

  4. [4]

    , TITLE =

    Stone, Charles J. , TITLE =. Ann. Statist. , FJOURNAL =. 1977 , NUMBER =

    1977

  5. [5]

    Unnikrishnan and Sankaran, P

    Nair, N. Unnikrishnan and Sankaran, P. G. and Vinesh Kumar, B. , TITLE =. Statistics , FJOURNAL =. 2012 , NUMBER =. doi:10.1080/02331888.2011.555551 , URL =

    arXiv 2012

  6. [6]

    and Sheather, S

    Ruppert, D. and Sheather, S. J. and Wand, M. P. , TITLE =. J. Amer. Statist. Assoc. , FJOURNAL =. 1995 , NUMBER =

    1995

  7. [7]

    Staudte, R. G. , title =. Statistics , volume =. 2017 , publisher =. doi:10.1080/02331888.2016.1277225 , URL =

    arXiv 2017

  8. [8]

    and Staudte, Robert G

    Prendergast, Luke A. and Staudte, Robert G. , title =. Stat , volume =. doi:https://doi.org/10.1002/sta4.105 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/sta4.105 , year =

    work page doi:10.1002/sta4.105

  9. [9]

    , TITLE =

    Watson, Geoffrey S. , TITLE =. Sankhy\=a. 1964 , PAGES =

    1964

  10. [10]

    Smoothing techniques for curve estimation (

    Gasser, Theo and M\"uller, Hans-Georg , TITLE =. Smoothing techniques for curve estimation (. 1979 , ISBN =

    1979

  11. [11]

    Priestley, M. B. and Chao, M. T. , TITLE =. J. Roy. Statist. Soc. Ser. B , FJOURNAL =. 1972 , PAGES =

    1972

  12. [12]

    , TITLE =

    Nadaraya, .\ A. , TITLE =. Teor. Verojatnost. i Primenen. , FJOURNAL =. 1964 , PAGES =

    1964

  13. [13]

    Zhou, Yong and Yip, Paul S. F. , TITLE =. J. Nonparametr. Statist. , FJOURNAL =. 1999 , NUMBER =. doi:10.1080/10485259908832796 , URL =

    work page doi:10.1080/10485259908832796 1999

  14. [14]

    Xiang, Xiaojing , TITLE =. J. Nonparametr. Statist. , FJOURNAL =. 1995 , NUMBER =. doi:10.1080/10485259508832621 , URL =

    work page doi:10.1080/10485259508832621 1995

  15. [15]

    , TITLE =

    El Ghouch, Anouar and Genton, Marc G. , TITLE =. J. Amer. Statist. Assoc. , FJOURNAL =. 2009 , NUMBER =. doi:10.1198/jasa.2009.tm08400 , URL =

    work page doi:10.1198/jasa.2009.tm08400 2009

  16. [16]

    and Gijbels, I

    Adam, C. and Gijbels, I. , TITLE =. Ann. Inst. Statist. Math. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s10463-021-00799-y , URL =

    work page doi:10.1007/s10463-021-00799-y 2022

  17. [17]

    and Gijbels, I

    Fan, J. and Gijbels, I. , TITLE =. Journal of Computational and Graphical Statistics , VOLUME =. 1995 , NUMBER =

    1995

  18. [18]

    AIMS Math

    Bouanani, Oussama and Bouzebda, Salim , TITLE =. AIMS Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.3934/math.20241150 , URL =

    work page doi:10.3934/math.20241150 2024

  19. [19]

    and Chandak, Rajita and Jansson, Michael and Ma, Xinwei , TITLE =

    Cattaneo, Matias D. and Chandak, Rajita and Jansson, Michael and Ma, Xinwei , TITLE =. Bernoulli , FJOURNAL =. 2024 , NUMBER =. doi:10.3150/23-bej1711 , URL =

    work page doi:10.3150/23-bej1711 2024

  20. [20]

    Bernoulli , FJOURNAL =

    Bertin, Karine and Klutchnikoff, Nicolas and Ouimet, Fr\'ed\'eric , TITLE =. Bernoulli , FJOURNAL =. 2025 , NUMBER =. doi:10.3150/24-bej1802 , URL =

    work page doi:10.3150/24-bej1802 2025

  21. [21]

    and Dewan, Isha and Li, Jun , TITLE =

    Chaubey, Yogendra P. and Dewan, Isha and Li, Jun , TITLE =. Sankhya B , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s13571-020-00242-x , URL =

    work page doi:10.1007/s13571-020-00242-x 2021

  22. [22]

    Doosti, Hassan and Dewan, Isha and Talebian, Masoud , TITLE =. Econom. Lett. , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.econlet.2025.112670 , URL =

    arXiv 2025

  23. [23]

    Petersen, Alexander and M\"uller, Hans-Georg , TITLE =. Ann. Statist. , FJOURNAL =. 2016 , NUMBER =. doi:10.1214/15-AOS1363 , URL =

    doi:10.1214/15-aos1363 2016

  24. [24]

    Shakhgildyan, Ksenia , TITLE =. Rev. Econ. Des. , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s10058-022-00309-3 , URL =

    work page doi:10.1007/s10058-022-00309-3 2025

  25. [25]

    R: A Language and Environment for Statistical Computing , author =

  26. [26]

    , TITLE =

    O'Reilly, Neville E. , TITLE =. Ann. Probability , FJOURNAL =. 1974 , PAGES =. doi:10.1214/aop/1176996610 , URL =

    arXiv 1974

  27. [27]

    and Jansson, Michael and Ma, Xinwei , TITLE =

    Cattaneo, Matias D. and Jansson, Michael and Ma, Xinwei , TITLE =. J. Amer. Statist. Assoc. , FJOURNAL =. 2020 , NUMBER =. doi:10.1080/01621459.2019.1635480 , URL =

    arXiv 2020

  28. [28]

    Falk, Michael , TITLE =. Statist. Probab. Lett. , FJOURNAL =. 1986 , NUMBER =. doi:10.1016/0167-7152(86)90020-9 , URL =

    work page doi:10.1016/0167-7152(86)90020-9 1986

  29. [29]

    Duttweiler, D. L. , TITLE =. Ann. Statist. , FJOURNAL =. 1973 , PAGES =

    1973

  30. [30]

    On the asymptotic distribution of weighted uniform empirical and quantile processes in the middle and on the tails , JOURNAL =

    Cs. On the asymptotic distribution of weighted uniform empirical and quantile processes in the middle and on the tails , JOURNAL =. 1985 , NUMBER =. doi:10.1016/0304-4149(85)90381-3 , URL =

    work page doi:10.1016/0304-4149(85)90381-3 1985

  31. [31]

    Parzen, Emanuel , TITLE =. J. Amer. Statist. Assoc. , FJOURNAL =. 1979 , NUMBER =

    1979

  32. [32]

    Estimating densities, quantiles, quantile densities and density quantiles , author=. Ann. Inst. Statist. Math. , FJOURNAL =. 1992 , publisher=

    1992

  33. [33]

    Strong approximations of the quantile process , JOURNAL =

    Cs. Strong approximations of the quantile process , JOURNAL =. 1978 , NUMBER =

    1978

  34. [34]

    Soni, Pooja and Dewan, Isha and Jain, Kanchan , TITLE =. Comput. Statist. Data Anal. , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.csda.2012.04.014 , URL =

    work page doi:10.1016/j.csda.2012.04.014 2012

  35. [35]

    Chesneau, Christophe and Dewan, Isha and Doosti, Hassan , TITLE =. Comput. Statist. Data Anal. , FJOURNAL =. 2016 , PAGES =. doi:10.1016/j.csda.2015.08.006 , URL =

    work page doi:10.1016/j.csda.2015.08.006 2016

  36. [36]

    2021 , publisher=

    Introduction to matrix theory , author=. 2021 , publisher=

    2021

  37. [37]

    Parzen, Emanuel , TITLE =. Ann. Math. Statist. , FJOURNAL =. 1962 , PAGES =. doi:10.1214/aoms/1177704472 , URL =

    arXiv 1962

  38. [38]

    and Schmeiser, Bruce W

    Ramberg, John S. and Schmeiser, Bruce W. , TITLE =. Comm. ACM , FJOURNAL =. 1974 , PAGES =. doi:10.1145/360827.360840 , URL =

    arXiv 1974

  39. [39]

    and Dudewicz, Edward J

    Karian, Zaven A. and Dudewicz, Edward J. , TITLE =. 2000 , PAGES =. doi:10.1201/9781420038040 , URL =

    work page doi:10.1201/9781420038040 2000

  40. [40]

    Leadbetter, M. R. and Lindgren, Georg and Rootz\'en, Holger , TITLE =. 1983 , PAGES =

    1983

  41. [41]

    Acta Math

    Zhou, Yong and Sun, Liu-Quan , TITLE =. Acta Math. Appl. Sin. Engl. Ser. , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s10255-005-0239-6 , URL =

    work page doi:10.1007/s10255-005-0239-6 2005

  42. [42]

    , TITLE =

    Serfling, Robert J. , TITLE =. 1980 , PAGES =

    1980

  43. [43]

    Unnikrishnan and Sankaran, P

    Nair, N. Unnikrishnan and Sankaran, P. G. , TITLE =. Comm. Statist. Theory Methods , FJOURNAL =. 2009 , NUMBER =. doi:10.1080/03610920802187430 , URL =

    work page doi:10.1080/03610920802187430 2009

  44. [44]

    Sankaran, P. G. and Nair, N. Unnikrishnan , TITLE =. J. Nonparametr. Stat. , FJOURNAL =. 2009 , NUMBER =. doi:10.1080/10485250902919046 , URL =

    work page doi:10.1080/10485250902919046 2009

  45. [45]

    Economics Letters , year=2022, volume=

    Zhang, Yu Yvette , title=. Economics Letters , year=2022, volume=

    2022

  46. [46]

    Redivo, Edoardo and Viroli, Cinzia and Farcomeni, Alessio , TITLE =. Stat. Comput. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s11222-023-10224-4 , URL =

    work page doi:10.1007/s11222-023-10224-4 2023

  47. [47]

    and Gijbels, I

    Fan, J. and Gijbels, I. , TITLE =. 1996 , PAGES =

    1996

  48. [48]

    2003 , PAGES =

    Pepe, Margaret Sullivan , TITLE =. 2003 , PAGES =

    2003

  49. [49]

    and Marron, J

    Sheather, Simon J. and Marron, J. S. , TITLE =. J. Amer. Statist. Assoc. , FJOURNAL =. 1990 , NUMBER =

    1990

  50. [50]

    and Gastwirth, Joseph L

    Bloch, Daniel A. and Gastwirth, Joseph L. , TITLE =. Ann. Math. Statist. , FJOURNAL =. 1968 , PAGES =. doi:10.1214/aoms/1177698342 , URL =

    arXiv 1968

  51. [51]

    Bofinger, Eve , TITLE =. Austral. J. Statist. , FJOURNAL =. 1975 , NUMBER =

    1975

  52. [52]

    , TITLE =

    Reiss, R.-D. , TITLE =. Metrika , FJOURNAL =. 1978 , NUMBER =. doi:10.1007/BF02204347 , URL =

    work page doi:10.1007/bf02204347 1978

  53. [53]

    Welsh, A. H. , TITLE =. Statist. Probab. Lett. , FJOURNAL =. 1988 , NUMBER =. doi:10.1016/0167-7152(88)90103-4 , URL =

    work page doi:10.1016/0167-7152(88)90103-4 1988

  54. [54]

    , TITLE =

    Tukey, John W. , TITLE =. Proc. Nat. Acad. Sci. U.S.A. , FJOURNAL =. 1965 , PAGES =. doi:10.1073/pnas.53.1.127 , URL =

    work page doi:10.1073/pnas.53.1.127 1965

  55. [55]

    Stochastic Processes and their Applications , volume=

    On general quantile process in weighted sup-norm metrics , author=. Stochastic Processes and their Applications , volume=. 1982 , publisher=

    1982