pith. sign in

arxiv: 2606.30864 · v1 · pith:CCETKTYInew · submitted 2026-06-29 · 🧮 math.ST · stat.TH

Analysis of gradual changes in nonparametric regression based on a new optimization method in the non-unique case

Marie Hu\v{s}kov\'a , Natalie Neumeyer , Leonie Selk This is my paper

Pith reviewed 2026-07-01 01:23 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords nonparametric regressiongradual change pointoptimization methodnon-unique minimizerconsistent estimatorsbootstrap approximationtwo-sample problem
0
0 comments X

The pith

A new optimization method enables consistent estimation of the largest minimizer for gradual change points in nonparametric regression even when the minimizer is not unique.

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

The paper aims to estimate an unknown gradual change point in a nonparametric regression model with a continuous regression function that begins at zero from the left of the covariate support. It introduces a general optimization method to find the largest minimization point of an objective function in cases where this point may not be unique, and defines various consistent estimators based on this method. A sympathetic reader would care because this approach handles situations where standard methods might fail due to non-uniqueness, allowing reliable change point detection and further estimation of the regression function. The work also covers rates of convergence, bootstrap methods, and extensions to two-sample problems.

Core claim

We define and compare various consistent estimators based on a new general optimization method in the case where the aim is to estimate the largest minimization point of some objective function. We discuss rates of convergence and estimating the regression function based on the gradual change structure. Bootstrap bias approximation is discussed. Further applications in a two sample case are considered, where two continuous regression functions first equal and then change at some point of interest.

What carries the argument

A new general optimization method for estimating the largest minimization point of an objective function in the non-unique case.

If this is right

  • Consistent estimators for the gradual change point can be constructed using the optimization method.
  • Rates of convergence for these estimators can be analyzed.
  • The regression function can be estimated incorporating the gradual change structure.
  • Bootstrap methods can approximate bias in the estimators.
  • The approach applies to two-sample problems with change points in regression functions.

Where Pith is reading between the lines

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

  • This optimization method could potentially be adapted to other statistical estimation problems involving non-unique optima.
  • Applications might extend to time series or spatial data with gradual transitions.
  • Comparing the estimators in finite samples could reveal practical advantages of the new method.
  • Integration with machine learning techniques for nonparametric regression might enhance performance.

Load-bearing premise

The regression function is continuous, starts at zero from the left, changes at an unknown point, and the optimization method consistently estimates the largest minimizer without needing uniqueness or specific details on the objective function.

What would settle it

A simulation study or real data example where the estimated change point fails to converge to the true value as sample size increases, despite the function starting at zero and being continuous.

Figures

Figures reproduced from arXiv: 2606.30864 by Leonie Selk, Marie Hu\v{s}kov\'a, Natalie Neumeyer.

Figure 1
Figure 1. Figure 1: Top left: scatter plot of observations and true regression function; top right: population objective function M; bottom left: empirical objective function Mn; bottom right: increasing rearrangement of Mn and horizontal line tn. While the graphics in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Top left: scatter plot of observations and true regression function; top right: box￾plots (left ϑˆ n, right argmin ϑ¯ n); bottom left: estimated density of ϑˆ n; bottom right: estimated density of the argmin estimator ϑ¯ n. 3.1.2 Method (2): non-zero regression function Assume that the P X-measure of {x ∈ (ϑ0, ϑ0+∆) : m(x) = 0} is zero. Let ˆmn be a uniformly consistent nonparametric estimator for the regr… view at source ↗
Figure 3
Figure 3. Figure 3: First example; left: scatterplot of observations and true regression function; mid￾dle: boxplots (left ϑˆ n, middle median-based bias reduction estimator, right mean-based bias reduction restimator), right: 1 density ϑˆ n, 2 density median-based bias reduction estimator, 3 mean-based bias reduction estimator. point easier to estimate. We used tn = 0.5σ/n0.49. In both cases the change point is ϑ0 = 0.4 and … view at source ↗
Figure 4
Figure 4. Figure 4: Second example; left: scatterplot of observations and true regression function; mid￾dle: boxplots (left ϑˆ n, middle median-based bias reduction estimator, right mean-based bias reduction restimator), right: 1 density ϑˆ n, 2 density median-based bias reduction estimator, 3 mean-based bias reduction estimator. the data that Mn is based on (e. g. (Y1, . . . , Yn) for method (1) and ( ˆmn(X1) 2 , . . . , mˆ … view at source ↗
Figure 5
Figure 5. Figure 5: Results of change point estimators in 300 independent repetitions. The horizontal dashed line marks the true change point ϑ0 = 0.4. 0.0 0.2 0.4 0.6 0.8 1.0 Meth. (1) Meth. (2) Meth. (3) Meth. (1) Meth. (2) Meth. (3) Meth. (1) Meth. (2) Meth. (3) Meth. (1) Meth. (2) Meth. (3) c=0.2 c=0.7 c=1.0 c=1.5 Change point estimates without (left) and with (right) bias reduction [PITH_FULL_IMAGE:figures/full_fig_p018… view at source ↗
Figure 6
Figure 6. Figure 6: Results of change point estimators in 300 independent repetitions with different threshold tn. The horizontal dashed line marks the true change point ϑ0 = 0.4. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Estimation results for mˆ n as proposed in Section 4 for four independent runs. The solid line is the true regression function. The dashed line is our estimator mˆ n (which depends on ϑˆ n) where ϑˆ n is estimated by method (3) with bias reduction based on 300 bootstrap repetitions. As competitors we compare to a classic local linear (dotted line) and a local constant (dashed-dotted line) regression estima… view at source ↗
Figure 9
Figure 9. Figure 9: The graph shows the age and height of 781 pupils, as well as the estimated regression function (local linear estimator). The circles and dashed line represent data from girls, and the crosses and solid line data from boys. 6 8 10 12 14 16 18 0.00 0.02 0.04 0.06 0.08 0.10 Age Density [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The graph shows the estimated density of the age of girls (dashed line) and boys (solid line). 23 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
read the original abstract

Consider a nonparametric regression model with one-dimensional covariates and a continuous regression function. Assume that the regression function from the left of the covariate support starts equal to zero and then changes at some unknown point. Our aim is to estimate this gradual change point. We define and compare various consistent estimators based on a new general optimization method in the case where the aim is to estimate the largest minimization point of some objective function. We discuss rates of convergence and estimating the regression function based on the gradual change structure. Bootstrap bias approximation is discussed. Further applications in a two sample case are considered, where two continuous regression functions first equal and then change at some point of interest.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper considers a nonparametric regression model with one-dimensional covariates and a continuous regression function that equals zero to the left of the covariate support before changing at an unknown point. It defines and compares various consistent estimators for this gradual change point using a new general optimization method for estimating the largest minimization point of an objective function in the non-unique case, discusses rates of convergence, estimation of the regression function exploiting the change structure, bootstrap bias approximation, and extensions to two-sample problems where two regression functions coincide before diverging at a point of interest.

Significance. If the proposed optimization method can be rigorously shown to yield consistent estimators for the largest minimizer under verifiable conditions, the work could offer a flexible framework for change-point estimation in nonparametric settings that handles non-uniqueness, with potential extensions to two-sample comparisons. The emphasis on rates, bootstrap, and structural exploitation of the zero-left-tail assumption adds practical value, but the significance is currently constrained by the lack of explicit regularity conditions supporting the central consistency claims.

major comments (3)
  1. [Definition of optimization method and consistency results] The central consistency claims for the estimators rest on the new optimization method targeting the largest minimizer in the non-unique case, yet no explicit conditions on the objective function, the structure of its minimizer set, or the resolution rule for non-uniqueness are stated to guarantee recovery of the largest point (see the section defining the general optimization method and the subsequent consistency theorems).
  2. [Rates of convergence and bootstrap bias] The rates of convergence and bootstrap bias approximation are asserted to follow from the method, but without the regularity conditions on the objective or the regression function beyond continuity and the left-zero assumption, it is unclear whether the claimed rates hold or reduce to tautological statements (see the sections on rates of convergence and bootstrap bias).
  3. [Two-sample case applications] The two-sample extension assumes the same optimization method applies when two continuous regression functions coincide before a change point, but again lacks specification of how non-uniqueness is handled to ensure the largest minimizer is recovered, undermining the claim that the method is general (see the section on further applications in the two-sample case).
minor comments (2)
  1. [Abstract and introduction] The abstract and introduction would benefit from a brief statement of the key regularity conditions required for the optimization method to achieve consistency.
  2. [Model setup] Notation for the objective function and the change point should be introduced more explicitly before the method is applied to the regression model.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, agreeing that additional explicit conditions are needed to support the claims, and will revise accordingly.

read point-by-point responses

  1. Referee: [Definition of optimization method and consistency results] The central consistency claims for the estimators rest on the new optimization method targeting the largest minimizer in the non-unique case, yet no explicit conditions on the objective function, the structure of its minimizer set, or the resolution rule for non-uniqueness are stated to guarantee recovery of the largest point (see the section defining the general optimization method and the subsequent consistency theorems).

    Authors: We agree that the manuscript would benefit from more explicit conditions. In the revised version, we will introduce a new subsection detailing the assumptions: the objective function is continuous, the minimizer set is a closed interval, and the method selects the rightmost point. These will be used to establish consistency in the theorems. revision: yes

  2. Referee: [Rates of convergence and bootstrap bias] The rates of convergence and bootstrap bias approximation are asserted to follow from the method, but without the regularity conditions on the objective or the regression function beyond continuity and the left-zero assumption, it is unclear whether the claimed rates hold or reduce to tautological statements (see the sections on rates of convergence and bootstrap bias).

    Authors: We agree that further regularity conditions are required. The revision will add assumptions including Lipschitz continuity of the regression function and quadratic growth of the objective away from the minimizer set. Under these, explicit rates will be derived and the bootstrap justification strengthened. revision: yes

  3. Referee: [Two-sample case applications] The two-sample extension assumes the same optimization method applies when two continuous regression functions coincide before a change point, but again lacks specification of how non-uniqueness is handled to ensure the largest minimizer is recovered, undermining the claim that the method is general (see the section on further applications in the two-sample case).

    Authors: The two-sample case applies the same general method. We will explicitly restate that the continuity and closed-interval minimizer conditions carry over to the two-sample objective, ensuring recovery of the largest minimizer and clarifying generality. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation chain not exhibited in visible text

full rationale

The provided abstract and description define consistent estimators via a new optimization method targeting the largest minimizer in the non-unique case, with discussion of rates and bootstrap. No equations, explicit derivations, self-citations, or load-bearing premises are quoted that reduce any claim to its own inputs by construction. The reader's note confirms no equations are visible, precluding any specific reduction. Per hard rules, circularity requires quotable evidence of equivalence to inputs; absent that, the paper is treated as self-contained with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5645 in / 1088 out tokens · 28845 ms · 2026-07-01T01:23:42.376496+00:00 · methodology

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

40 extracted references · 31 canonical work pages

  1. [1]

    Akritas and Ingrid Van Keilegom

    Michael G. Akritas and Ingrid Van Keilegom. Non-parametric estimation of the residual distribution. Scand. J. Statist., 28 0 (3): 0 549--567, 2001. ISSN 0303-6898,1467-9469. doi:10.1111/1467-9469.00254. URL https://doi.org/10.1111/1467-9469.00254

    work page doi:10.1111/1467-9469.00254 2001

  2. [2]

    Limit properties of the monotone rearrangement for density and regression function estimation

    Dragi Anevski and Anne-Laure Foug\`eres. Limit properties of the monotone rearrangement for density and regression function estimation. Bernoulli, 25 0 (1): 0 549--583, 2019. ISSN 1350-7265,1573-9759. doi:10.3150/17-bej998. URL https://doi.org/10.3150/17-bej998

    work page doi:10.3150/17-bej998 2019

  3. [3]

    Change-point problem and bootstrap

    Jarom\'ir Antoch and Marie Hu s kov\' a . Change-point problem and bootstrap. J. Nonparametr. Statist., 5 0 (2): 0 123--144, 1995. ISSN 1048-5252,1029-0311. doi:10.1080/10485259508832639. URL https://doi.org/10.1080/10485259508832639

    work page doi:10.1080/10485259508832639 1995

  4. [4]

    Tests for continuity of regression functions

    Jarom\'ir Antoch, Gerard Gregoire, and Marie Hu s kov\' a . Tests for continuity of regression functions. J. Statist. Plann. Inference, 137 0 (3): 0 753--777, 2007. ISSN 0378-3758,1873-1171. doi:10.1016/j.jspi.2006.06.007. URL https://doi.org/10.1016/j.jspi.2006.06.007

    work page doi:10.1016/j.jspi.2006.06.007 2007

  5. [5]

    Gradual changes in functional time series

    Patrick Bastian and Holger Dette. Gradual changes in functional time series. J. Time Ser. Anal., 2025. doi:10.1111/jtsa.12809. URL https://doi.org/10.1111/jtsa.12809

    work page doi:10.1111/jtsa.12809 2025

  6. [6]

    Estimation of semiparametric models when the criterion function is not smooth

    Xiaohong Chen, Oliver Linton, and Ingrid Van Keilegom. Estimation of semiparametric models when the criterion function is not smooth. Econometrica, 71 0 (5): 0 1591--1608, 2003. ISSN 0012-9682,1468-0262. doi:10.1111/1468-0262.00461. URL https://doi.org/10.1111/1468-0262.00461

    work page doi:10.1111/1468-0262.00461 2003

  7. [7]

    Holger Dette, Natalie Neumeyer, and Kay F. Pilz. A simple nonparametric estimator of a strictly monotone regression function. Bernoulli, 12 0 (3): 0 469--490, 2006. ISSN 1350-7265,1573-9759. doi:10.3150/bj/1151525131. URL https://doi.org/10.3150/bj/1151525131

    work page doi:10.3150/bj/1151525131 2006

  8. [8]

    Asymmetric cusp estimation in regression models

    Maik D\"oring. Asymmetric cusp estimation in regression models. Statistics, 49 0 (6): 0 1279--1297, 2015 a . ISSN 0233-1888,1029-4910. doi:10.1080/02331888.2015.1016027. URL https://doi.org/10.1080/02331888.2015.1016027

    work page doi:10.1080/02331888.2015.1016027 2015

  9. [9]

    Rate of convergence of a change point estimator in a misspecified regression model

    Maik D\"oring. Rate of convergence of a change point estimator in a misspecified regression model. Stochastic Models, Statistics and Their Applications: Wroclaw, Poland, Springer, pages 49--55, 2015 b

    2015

  10. [10]

    Smooth change point estimation in regression models with random design

    Maik D\"oring and Uwe Jensen. Smooth change point estimation in regression models with random design. Ann. Inst. Statist. Math., 67 0 (3): 0 595--619, 2015. ISSN 0020-3157,1572-9052. doi:10.1007/s10463-014-0467-8. URL https://doi.org/10.1007/s10463-014-0467-8

    work page doi:10.1007/s10463-014-0467-8 2015

  11. [11]

    A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes

    Dietmar Ferger. A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes. Kybernetika (Prague), 57 0 (3): 0 426--445, 2021. ISSN 0023-5954,1805-949X. doi:10.14736/kyb-2021-3-0426. URL https://doi.org/10.14736/kyb-2021-3-0426

    work page doi:10.14736/kyb-2021-3-0426 2021

  12. [12]

    On semi-continuity and continuity of the smallest and largest minimizing point of real convex functions with applications in probability and statistics

    Dietmar Ferger. On semi-continuity and continuity of the smallest and largest minimizing point of real convex functions with applications in probability and statistics. Eur. J. Math., 10 0 (1): 0 Paper No. 14, 18, 2024. ISSN 2199-675X,2199-6768. doi:10.1007/s40879-024-00728-2. URL https://doi.org/10.1007/s40879-024-00728-2

    work page doi:10.1007/s40879-024-00728-2 2024

  13. [13]

    Nonparametric Functional Data Analysis

    Frederic Ferraty and Philippe Vieu. Nonparametric Functional Data Analysis. Springer Series in Statistics. Springer, New York, 2006. ISBN 978-0387-30369-7

    2006

  14. [14]

    Nonparametric regression analysis of growth curves

    Theo Gasser, Hans-Georg M\" u ller, Walter Kohler, Luciano Molinari, and Andrea Prader. Nonparametric regression analysis of growth curves. Ann. Statist., 12 0 (1): 0 210--229, 1984. URL https://doi.org/10.1214/aos/1176346402

    work page doi:10.1214/aos/1176346402 1984

  15. [15]

    Data-driven discontinuity detection in derivatives of a regression function

    Irene Gijbels and Anne-C\' e cile Goderniaux. Data-driven discontinuity detection in derivatives of a regression function. Comm. Statist. Theory Methods, 33 0 (4): 0 851--871, 2004. ISSN 0361-0926,1532-415X. doi:10.1081/STA-120028730. URL https://doi.org/10.1081/STA-120028730

    work page doi:10.1081/sta-120028730 2004

  16. [16]

    Nonparametric estimation under shape constraints, volume 38 of Cambridge Series in Statistical and Probabilistic Mathematics

    Piet Groeneboom and Geurt Jongbloed. Nonparametric estimation under shape constraints, volume 38 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York, 2014. ISBN 978-0-521-86401-5. doi:10.1017/CBO9781139020893. URL https://doi.org/10.1017/CBO9781139020893. Estimators, algorithms and asymptotics

    work page doi:10.1017/cbo9781139020893 2014

  17. [17]

    Bruce E. Hansen. Uniform convergence rates for kernel estimation with dependent data. Econometric Theory, 24 0 (3): 0 726--748, 2008. ISSN 0266-4666,1469-4360. doi:10.1017/S0266466608080304. URL https://doi.org/10.1017/S0266466608080304

    work page doi:10.1017/s0266466608080304 2008

  18. [18]

    Inequalities

    Godfrey Harold Hardy, John Edensor Littlewood, and George P\'olya. Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988. ISBN 0-521-35880-9. Reprint of the 1952 edition

    1988

  19. [19]

    Two-sample gradual change analysis

    Zden e k Hl\' a vka and Marie Hu s kov\' a . Two-sample gradual change analysis. REVSTAT, 15 0 (3): 0 355--372, 2017. ISSN 1645-6726,2183-0371

    2017

  20. [20]

    Kernel type estimators for gradual changes

    Lajos Horv\' a th and Marie Hu s kov\' a . Kernel type estimators for gradual changes. In Limit theorems in probability and statistics, V ol. II ( B alatonlelle, 1999) , pages 175--187. J\'anos Bolyai Math. Soc., Budapest, 2002. ISBN 963-9453-013

    1999

  21. [21]

    Estimators in the location model with gradual changes

    Marie Hu s kov\' a . Estimators in the location model with gradual changes. Comment. Math. Univ. Carolin., 39 0 (1): 0 147--157, 1998. ISSN 0010-2628,1213-7243

    1998

  22. [22]

    Limit theorems for a class of tests of gradual changes

    Marie Hu s kov\' a and Josef Steinebach. Limit theorems for a class of tests of gradual changes. J. Statist. Plann. Inference, 89 0 (1-2): 0 57--77, 2000. ISSN 0378-3758,1873-1171. doi:10.1016/S0378-3758(00)00094-X. URL https://doi.org/10.1016/S0378-3758(00)00094-X

    work page doi:10.1016/s0378-3758(00)00094-x 2000

  23. [23]

    Asymptotic tests for gradual changes

    Marie Hu s kov\' a and Josef Steinebach. Asymptotic tests for gradual changes. Statist. Decisions, 20 0 (2): 0 137--151, 2002. ISSN 0721-2631

    2002

  24. [24]

    Ioannides

    Dimitrios A. Ioannides. Consistent nonparametric regression: some generalizations in the fixed design case. J. Nonparametr. Statist., 2 0 (3): 0 203--213, 1993. ISSN 1048-5252,1029-0311. doi:10.1080/10485259308832553. URL https://doi.org/10.1080/10485259308832553

    work page doi:10.1080/10485259308832553 1993

  25. [25]

    Testing appearance of polynomial trend

    Daniela Jaru s kov\' a . Testing appearance of polynomial trend. Extremes, 2 0 (1): 0 25--37, 1999. ISSN 1386-1999,1572-915X. doi:10.1023/A:1009951807623. URL https://doi.org/10.1023/A:1009951807623

    work page doi:10.1023/a:1009951807623 1999

  26. [26]

    Fully functional weighted testing for abrupt and gradual location changes in functional time series

    Claudia Kirch, Hedvika Rano s ov\' a , and Martin Wendler. Fully functional weighted testing for abrupt and gradual location changes in functional time series. ArXiv preprint arXiv: 2601.16058v1, 2026+

    work page arXiv 2026

  27. [27]

    Michael R. Kosorok. Introduction to empirical processes and semiparametric inference. Springer Series in Statistics. Springer, New York, 2008. ISBN 978-0-387-74977-8. doi:10.1007/978-0-387-74978-5. URL https://doi.org/10.1007/978-0-387-74978-5

    work page doi:10.1007/978-0-387-74978-5 2008

  28. [28]

    A note on uniform consistency of monotone function estimators

    Natalie Neumeyer. A note on uniform consistency of monotone function estimators. Statist. Probab. Lett., 77 0 (7): 0 693--703, 2007. ISSN 0167-7152,1879-2103. doi:10.1016/j.spl.2006.11.004. URL https://doi.org/10.1016/j.spl.2006.11.004

    work page doi:10.1016/j.spl.2006.11.004 2007

  29. [29]

    Simulation and the asymptotics of optimization estimators

    Ari\' e l Pakes and David Pollard. Simulation and the asymptotics of optimization estimators. Econometrica, 57 0 (5): 0 1027--1057, 1989. ISSN 0012-9682,1468-0262. doi:10.2307/1913622. URL https://doi.org/10.2307/1913622

    work page doi:10.2307/1913622 1989

  30. [30]

    Minimax estimation of sharp change points

    Marc Raimondo. Minimax estimation of sharp change points. Ann. Statist., 26 0 (4): 0 1379--1397, 1998. ISSN 0090-5364,2168-8966. doi:10.1214/aos/1024691247. URL https://doi.org/10.1214/aos/1024691247

    work page doi:10.1214/aos/1024691247 1998

  31. [31]

    Relative rearrangement

    Jean Michel Rakotoson. Relative rearrangement. Lecture Notes in Mathematics. Springer Nature Switzerland, 2025. doi:10.1007/978-3-032-02228-8. URL https://doi.org/10.1007/978-3-032-02228-8

    work page doi:10.1007/978-3-032-02228-8 2025

  32. [32]

    Designs for regression problems with correlated errors

    Jerome Sacks and Donald Ylvisaker. Designs for regression problems with correlated errors. III . Ann. Math. Statist., 41: 0 2057--2074, 1970. ISSN 0003-4851. doi:10.1214/aoms/1177696705. URL https://doi.org/10.1214/aoms/1177696705

    work page doi:10.1214/aoms/1177696705 2057

  33. [33]

    Limit theorems for rank statistics detecting gradual changes

    Ale s Slab\' y . Limit theorems for rank statistics detecting gradual changes. Comment. Math. Univ. Carolin., 42 0 (3): 0 591--600, 2001. ISSN 0010-2628,1213-7243

    2001

  34. [34]

    Charles J. Stone. Optimal global rates of convergence for nonparametric regression. Ann. Statist., 10 0 (4): 0 1040--1053, 1982. ISSN 0090-5364,2168-8966. URL https://www.jstor.org/stable/2240707

    work page arXiv 1982

  35. [35]

    Tsybakov

    Alexandre B. Tsybakov. Introduction to nonparametric estimation. Springer Series in Statistics. Springer, New York, 2009. ISBN 978-0-387-79051-0. doi:10.1007/b13794. URL https://doi.org/10.1007/b13794. Revised and extended from the 2004 French original, Translated by Vladimir Zaiats

    work page doi:10.1007/b13794 2009

  36. [36]

    van der Vaart

    Aad W. van der Vaart. Asymptotic statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 1998. ISBN 0-521-49603-9; 0-521-78450-6. doi:10.1017/CBO9780511802256. URL https://doi.org/10.1017/CBO9780511802256

    work page doi:10.1017/cbo9780511802256 1998

  37. [37]

    van der Vaart and Jon A

    Aad W. van der Vaart and Jon A. Wellner. Weak convergence and empirical processes. Springer Series in Statistics. Springer, New York, 1996. ISBN 0-387-94640-3. doi:10.1007/978-1-4757-2545-2. URL https://doi.org/10.1007/978-1-4757-2545-2. With applications to statistics

    work page doi:10.1007/978-1-4757-2545-2 1996

  38. [38]

    Detecting gradual changes in locally stationary processes

    Michael Vogt and Holger Dette. Detecting gradual changes in locally stationary processes. Ann. Statist., 43 0 (2): 0 713--740, 2015. ISSN 0090-5364,2168-8966. doi:10.1214/14-AOS1297. URL https://doi.org/10.1214/14-AOS1297

    work page doi:10.1214/14-aos1297 2015

  39. [39]

    Matyskova, Zdenek Hl\' a vka, L

    Zden e k S umn\' i k, J. Matyskova, Zdenek Hl\' a vka, L. Durdilova, O. Sou c ek, and Dana Zemkov\' a . Reference data for jumping mechanography in healthy children and adolescents aged 6-18 years. Journal of musculoskeletal & neuronal interactions, 13: 0 259--73, 09 2013

    2013

  40. [40]

    Gradual changes in long memory processes with applications

    Lihong Wang. Gradual changes in long memory processes with applications. Statistics, 41 0 (3): 0 221--240, 2007. ISSN 0233-1888,1029-4910. doi:10.1080/02331880701223555. URL https://doi.org/10.1080/02331880701223555

    work page doi:10.1080/02331880701223555 2007