Pith. sign in

REVIEW 3 major objections 4 minor 30 references

Identifying shifts between two regression curves

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read A shift between two convex regression curves reduces to a derivative identity

desk verdict The core inverse-derivative idea is elementary but the extension to dependent non-stationary errors makes it worth a look; however, the bootstrap test's level is unproven and the simulation nulls fall outside the paper's own c>0 restriction. read the letter →

arxiv 1908.04328 v1 pith:LGY3Z6SQ submitted 2019-08-12 math.ST stat.TH

Holger Dette , Subhra Sankar Dhar , Weichi Wu This is my paper · ORCID
classification math.STstat.TH MSC 62G0862G1062G20
keywords comparisonofcurvesnonparametricregressionhypothesistestingshapeinvariantmodelshorizontalshiftverticallocallystationaryprocessesbootstraptest
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 asks when two nonparametric regression curves differ only by a horizontal shift $c$ and a vertical shift $d$, so that $m_1(t)=m_2(t+c)+d$ for all $t$ in an interval. Its first claim is an equivalence: under this hypothesis the functions $((m_1')^{-1})'$ and $((m_2')^{-1})'$ coincide, and conversely. This turns a shape-invariant comparison into a problem of comparing two curves that can be estimated directly. On that basis the paper constructs a graphical diagnostic and an $L^2$ test statistic, derives its asymptotic normal distribution under local alternatives, and proposes a bootstrap version that avoids estimating the complicated bias and variance. The intended payoff is a shift test that works for dependent and non-stationary errors, where most earlier shape-invariant methodology requires independence.

What carries the argument

The load-bearing object is $f_s=((m_s')^{-1})'$, the derivative of the inverse of the derivative of the regression function. A horizontal shift $c$ in the original curves becomes an additive constant in the inverse derivatives, so the derivative cancels it and $f_1-f_2$ vanishes exactly under the null. The paper estimates $f_s$ without explicit inversion by noting that if $U$ is uniform, $f_s$ is the density of $m_s'(U)$; hence $\hat f_s(t)=(Nh_{d,s})^{-1}\sum_{i=1}^N K_d((\hat m_s'(i/N)-t)/h_{d,s})$, with $\hat m_s'$ a local linear estimate. The test statistic is the integrated squared difference of $\hat f_1$ and $\hat f_2$, and the asymptotic argument combines a Gaussian approximation for locally stationary error processes with a central limit theorem for quadratic forms.

What would settle it

Simulate two samples with convex curves satisfying $m_1(t)=m_2(t+c)+d$, generating errors from a locally stationary process with polynomial decay such as $\delta_4(k)=O(k^{-2})$ instead of $O(\rho^k)$, and record the empirical rejection rate of the bootstrap test at nominal 5%; a rate clearly above 5% as $n$ grows would show the geometric-decay assumption is load-bearing.

Watch

Extended reading notes

Core claim

The central discovery is Lemma 2.1: for regression functions with strictly increasing first derivative, $m_1(t)=m_2(t+c)+d$ on $(0,1-c)$ holds if and only if $((m_1')^{-1})'(u)=((m_2')^{-1})'(u)$ for $u$ in $(m_1'(0),\,m_1'(1-c))$. The paper then estimates $f_s=((m_s')^{-1})'$ by a kernel density estimate of $\hat m_s'(U)$ rather than by inverting a non-monotone estimate, and forms the statistic $T_{n_1,n_2}=\int(\hat f_1-\hat f_2)^2\hat w(t)\,dt$. Theorem 3.2 shows that under Assumptions 3.1–3.4 and local alternatives with $((m_1')^{-1})'-((m_2')^{-1})'=\rho_n g+o(\rho_n)$, $\rho_n=(n_1 b_{n,1}^{9/2})^{-1/2}$, the standardized statistic $n_1 b_{n,1}^{9/2}T_{n_1,n_2}-B_n(g)$ converges weakly to $N(0,V_T)$, with explicit formulas for $B_n(g)$ and $V_T$ in terms of long-run variances and second derivatives of the regression functions. Because $B_n$ and $V_T$ are hard to estimate, the paper proposes a bootstrap test, whose finite-sample level and power it studies by simulation and on infant growth data.

Load-bearing premise

The argument assumes the errors form a locally stationary process whose dependence on a single past innovation decays geometrically fast; if the true errors have only polynomial dependence or are not locally stationary, the claimed null distribution and bootstrap calibration have no stated justification.

Editorial extensions

If this is right

  • Testing the shift hypothesis does not require first estimating $c$ and $d$; the same statistic and graphical device can be used for any convex regression pair.
  • The test controls asymptotic level for locally stationary dependent errors, so it applies to time series settings where independence-based shape-invariant tests are not justified.
  • Under local alternatives the test detects deviations of order $(n_1 b_{n,1}^{9/2})^{-1/2}$, and the asymptotic power is approximately $\Phi(\int g^2 w / V_T^{1/2}-z_{1-\alpha})$.
  • The bootstrap version avoids direct estimation of the bias term of order $1/\sqrt{b_{n,1}}$ and the variance involving long-run variances and second derivatives, and the simulation study reports levels close to nominal for $n=100,200,500$.
  • For concave regression functions the same methodology applies after negating the responses.

Reading between the lines

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

  • The equivalence may extend to testing whether two monotone regression functions coincide up to any increasing transformation of the covariate axis, not just an additive shift, by comparing derivative-inverse derivatives after a suitable transformation; the paper does not pursue this.
  • Because the construction only uses monotonicity of $m'$, the same estimator could be adapted to higher-dimensional index sets by comparing densities of gradient images, though the asymptotic theory would need new concentration arguments.
  • A practical robustness question the paper leaves open is whether the bootstrap still controls level under polynomially decaying dependence; a natural extension would be a block or sieve bootstrap under weaker conditions.
  • The graphical device could be used as a diagnostic before fitting parametric shape-invariant models, potentially improving model selection in growth-curve and production-function applications.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the problem of testing whether two convex (or concave) regression functions differ by a horizontal shift c and a vertical shift d, i.e., m1(t)=m2(t+c)+d. A key observation is that under this null hypothesis the derivatives of the inverse derivatives, ((m1')^{-1})' and ((m2')^{-1})', coincide. The authors propose a graphical device based on estimating these quantities by kernel density estimators of local linear derivative estimates, and a formal L2 test statistic T_{n1,n2} constructed from the squared difference of the two estimated curves. For locally stationary, dependent errors, they derive asymptotic normality of n1 b_{n,1}^{9/2} T_{n1,n2} under the null and local alternatives. Because the asymptotic bias and variance are complicated, they propose a bootstrap procedure (Algorithm 4.1) and illustrate it with simulations and a real-data application on infant growth curves.

Significance. The theoretical result in Theorem 3.2, if correct, is a substantial contribution: it gives a nonparametric test for shift-invariant curve comparison under a general locally stationary error structure, going beyond the independence assumptions common in the shape-invariant-model literature. The inverse-derivative characterization in Lemma 2.1 is elegant and leads to a simple graphical tool. The paper also makes good use of existing Gaussian approximation results (Wu and Zhou, 2011) and inverse-regression estimation (Dette et al., 2006). However, the practical test advocated in the paper, the bootstrap in Algorithm 4.1, is not accompanied by a consistency theorem, and the simulation study does not evaluate the null hypothesis as defined because the two null models use a negative shift c, violating the c in (0,1) restriction in (2.2). These gaps substantially weaken the support for the paper's central practical claim of a test with controlled level.

major comments (3)
  1. [Section 4.1, Algorithm 4.1] There is no theorem establishing consistency of the bootstrap test. The heuristic preceding Algorithm 4.1 connects W_B to the Gaussian process U_n used in the proof of Theorem 3.2, but no argument shows that the bootstrap quantile W_{⌊B(1−α)⌋} converges to the null distribution of T_{n1,n2}. In particular, replacing the unknown functions m_s' and σ_s in U_n by the estimates \mhat{m}_s' and \mhat{σ}_s requires a uniform-in-bandwidth or continuity argument that the paper does not supply. Without such a result, the level control of the bootstrap test is unproven.
  2. [Section 4.1, Algorithm 4.1(b)] As written, the bootstrap statistic W_B uses the deterministic weight function w(t) from (2.16), which depends on the unknown quantities a=m_1'(0) and b=m_1'(1−c). The test statistic T_{n1,n2} in (2.15) uses the estimated weight \mhat{w}(t). The algorithm does not state how w(t) is to be obtained in practice; taken literally, the procedure is infeasible. The authors should either replace w(t) by \mhat{w}(t) and prove that the replacement is asymptotically negligible in the bootstrap, or provide a feasible construction of the weight.
  3. [Section 4.2, models (4.4) and (4.5)] The two null models used in the simulation study satisfy m1(t)=m2(t−0.1)+d, i.e., c=−0.1, which violates the assumption c∈(0,1) in the null hypothesis (2.2). The construction of the test, including the estimate \mhat{c} in (2.9) and the domain of the weight function in (2.16), is explicitly based on c>0. Therefore the reported empirical sizes in Table 1 do not provide evidence that the test controls the level under the null hypothesis as formulated. The simulations should be rerun with c∈(0,1), or the paper should justify that the procedure is invariant under the sign of c.
minor comments (4)
  1. [Throughout, Section 2] The paper repeatedly calls c the 'vertical shift' (e.g., in Section 2.1 'let \mhat{c} be an estimate of the vertical shift c' and in the sentence after (2.10)), but c is the horizontal shift in (2.2). This terminology should be corrected.
  2. [Section 4.1, first paragraph] The text says the bootstrap 'does not require the estimation of the derivatives', but Algorithm 4.1(a) requires estimates of m_1' and m_2' via (2.6), which are then plugged into Ξ_1^{(B)} and Ξ_2^{(B)}. This statement is misleading and should be corrected.
  3. [Section 3.1, Assumption 3.1(b)] Assumption 3.1(b), sup_{0≤t<s≤1} ||G(t,F0)−G(s,F0)||_4<∞, is automatically implied by Assumption 3.1(a) and the triangle inequality; if a stronger smoothness condition is needed, it should be stated explicitly (e.g., a Hölder or Lipschitz condition in t).
  4. [References] The reference 'Silverman (1998)' is incomplete; in the reference list it appears only as 'CHAPMAN & HALL, London.' The correct reference is B. W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman and Hall, 1986.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inverse-derivative characterization is proved, the test statistic is defined directly from it, and the asymptotic distribution follows from external Gaussian approximation and quadratic-form limit theorems.

full rationale

The paper's derivation chain is self-contained in the relevant sense. Lemma 2.1 proves that the null hypothesis m1(t)=m2(t+c)+d is equivalent to equality of the functions ((m1')^{-1})' and ((m2')^{-1})' on the common domain; this is a mathematical equivalence, not an assumption of the conclusion. The test statistic T_{n1,n2} in (2.15) is then defined as an L2 distance between kernel estimators of those two functions, so the null behavior is a direct consequence of the lemma. Theorem 3.2 computes the asymptotic bias and variance from the Gaussian approximation of Wu and Zhou (2011), the quadratic-form CLT of de Jong (1987), and kernel expansions; no parameter is fitted to the data and then renamed as a prediction. The citations to Dette et al. (2006) for monotone-rearrangement estimators and to Dette and Wu (2019) for uniform bounds and long-run variance estimation are technical lemmas with stated assumptions that do not include the shift hypothesis; although the author sets overlap, this is independent support rather than circularity. Two genuine concerns remain but they are not circularity: Algorithm 4.1(b) as written uses the deterministic weight w(t) depending on unknown a and b, and no theorem proves consistency of the bootstrap quantile W_{floor(B(1-alpha))}; moreover the null simulations in (4.4)-(4.5) use c=-0.1, outside the c in (0,1) restriction in (2.2). These are correctness or implementation gaps, not reductions of the claimed result to its own inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles, forces, or unobserved quantities. Its load-bearing inputs are a locally stationary dependence model, smooth convex regression functions, and a set of bandwidth conditions. The implementation also needs several tuning parameters, but they are standard and not fitted to force the conclusion.

free parameters (4)
  • Local linear bandwidths b_{n,s} = Selected by generalized cross-validation (Zhou and Wu 2010)
    The test statistic and derivative estimates depend on bandwidths; theory requires the Assumption 3.4 rate conditions, while implementation chooses them by GCV.
  • Density bandwidth h_{d,s} = n_s^{-1/3} by rule of thumb
    Used in the kernel density estimators (2.7) and (2.8); the paper states the choice has negligible impact if small.
  • Trimming constant eta = 0.01 in simulations, 0.001 in the data example
    Defines the weight function and the domain of the graphical device; it is chosen by hand to avoid boundary effects.
  • Block size m in long-run variance estimator (4.1) = Not specified in the paper
    The difference-based estimator of sigma^2_s requires a block length m; its value is left unspecified, which affects implementation.
assumptions (5)
  • domain assumption Assumption 3.1: the error vector is a locally stationary process in the sense of Zhou and Wu (2009) with zero mean, finite fourth moment, and geometrically decaying physical dependence measure.
    This is the dependence structure that makes Proposition 5.1 and the Gaussian approximation valid; the central limit theorem in Theorem 3.2 rests on it.
  • domain assumption Assumption 3.1(e): the long-run covariance matrix Sigma^2(t) is diagonal with components sigma_1^2(t) and sigma_2^2(t).
    Diagonal long-run variances simplify the cross-covariance term D3 in the proof and are not implied by the other assumptions.
  • domain assumption Assumption 3.3: the regression functions m1 and m2 are in C^{2,1}[0,1] with strictly increasing first derivatives.
    Convexity with a strictly increasing derivative is needed for Lemma 2.1 and for the inverse derivative calculations.
  • domain assumption Assumption 3.4: bandwidth conditions linking b_n, h_d, N, and n, including nb_n^4 log n times a squared error term going to zero.
    These technical rate conditions ensure that estimation errors in the derivatives and density estimators vanish fast enough for the asymptotic normality result.
  • domain assumption The null hypothesis restricts the horizontal shift to c in (0,1) with known sign.
    The theory is stated only for positive c, while the simulation null models use c = -0.1, exposing a mismatch.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Identifying shifts between two regression curves." pith.science (2026). https://pith.science/paper/LGY3Z6SQ

@misc{pith2026190804328,
  author       = {Pith},
  title        = {Pith review of: Identifying shifts between two regression curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LGY3Z6SQ}},
  note         = {Machine review of arXiv:1908.04328}
}
read the original abstract

This article studies the problem whether two convex (concave) regression functions modelling the relation between a response and covariate in two samples differ by a shift in the horizontal and/or vertical axis. We consider a nonparametric situation assuming only smoothness of the regression functions. A graphical tool based on the derivatives of the regression functions and their inverses is proposed to answer this question and studied in several examples. We also formalize this question in a corresponding hypothesis and develop a statistical test. The asymptotic properties of the corresponding test statistic are investigated under the null hypothesis and local alternatives. In contrast to most of the literature on comparing shape invariant models, which requires independent data the procedure is applicable for dependent and non-stationary data. We also illustrate the finite sample properties of the new test by means of a small simulation study and a real data example.

Figures

Figures reproduced from arXiv: 1908.04328 by the authors.

Figure 1
Figure 1. Plots of the set Cn1,n2 for different examples. The panels on the left correspond to the models (2.11) and (2.13) (null hypothesis) and the panels on the right correspond to the models (2.12) and (2.14) (alternative). 9 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Plots of the length of the male (right part) and female (left parts) infants for different age. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -4 -2 0 2 4 t [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Plots of Cn1,n2 for the real data described in Section 4.3. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    and Duarte, J

    Ait-Sahalia, Y. and Duarte, J. (2003). Nonparametric option pricing under shape restrictions. Journal of Econometrics , 116:9–47

    work page 2003

  2. [2]

    Carroll, R. J. and Hall, P. (1993). Semiparametric comparison of regression curves via normal likelihoods. Australian Journal of Statistics , 34(3):471–487

    work page 1993

  3. [3]

    and Dalalyan, A

    Collier, O. and Dalalyan, A. S. (2015). Curve registration by nonparametric goodness-of-fit testing. Journal of Statistical Planning and Inference , 162:20–42. 32

    work page 2015

  4. [4]

    Dahlhaus, R. (1997). Fitting time series models to nonstationary processes.The Annals of Statistics , 25(1):1–37. de Jong, P. (1987). A central limit theorem for generalized quadratic forms. Probability Theory and Related Fields , 75(2):261–277

    work page 1997

  5. [5]

    Degras, D., Xu, Z., Zhang, T., and Wu, W. B. (2012). Testing for parallelism among trends in multiple time series. IEEE Transactions on Signal Processing , 60(3):1087–1097

    work page 2012

  6. [6]

    and Munk, A

    Dette, H. and Munk, A. (1998). Nonparamatric comparison of several regression curves: exact and asymptotic theory. Annals of Statistics , 26(6):2339–2368

    work page 1998

  7. [7]

    and Neumeyer, N

    Dette, H. and Neumeyer, N. (2001). Nonparametric analysis of covariance. Annals of Statistics , 29(5):1361–1400

    work page 2001

  8. [8]

    Dette, H., Neumeyer, N., and Pilz, K. F. (2006). A simple nonparametric estimator of a strictly monotone regression function. Bernoulli, 12:469–490

    work page 2006

Show all 30 references
  1. [9]

    and Wu, W

    Dette, H. and Wu, W. (2019+). Detecting relevant changes in the mean of non-stationary processes - a mass excess approach. The Annals of Statistics, to appear

    2019

  2. [10]

    Durot, C., Groeneboom, P., and Lopuha¨ a, H. P. (2013). Testing equality of functions under monotonicity constraints. Journal of Nonparametric Statistics , 25(4):939–970

    2013

  3. [11]

    Gamboa, F., Loubes, J., and Maza, E. (2007). Semi-parametric estimation of shifts. Electronic Journal of Statistics , 1:616–640

    2007

  4. [12]

    and Hart, J

    Hall, P. and Hart, J. D. (1990). Bootstrap test for difference between means in nonparametric regression. Journal of the American Statistical Association , 85(412):1039–1049

    1990

  5. [13]

    and Huang, L.-S

    Hall, P. and Huang, L.-S. (2001). Nonparametric kernel regression subject to monotonicity con- straints. Annals of Statistics , 29(3):624–647. H¨ ardle, W. and Marron, J. S. (1990). Semiparametric comparison of regression curves. Annals of Statistics, 18(1):63–89

    2001

  6. [14]

    Maity, A. (2012). A powerful test for comparing multiple regression functions. Journal of Non- parametric Statistics, 24(3):563–576

    2012

  7. [15]

    Mallat, S., Papanicolaou, G., and Zhang, Z. (1998). Adaptive covariance estimation of locally stationary processes. The Annals of Statistics , 26(1):1–47

    1998

  8. [16]

    Mammen, E. (1991). Estimating a smooth monotone regression function. Annals of Statistics , 19(2):724–740

    1991

  9. [17]

    Matzkin, R. L. (1991). Semiparametric estimation of monotone and concave utility functionsfor polychotomous choice models. Econometrica, 59(5):1315–1327

    1991

  10. [18]

    P., Von Sachs, R., and Kroisandt, G

    Nason, G. P., Von Sachs, R., and Kroisandt, G. (2000). Wavelet processes and adaptive estima- tion of the evolutionary wavelet spectrum. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 62(2):271–292

    2000

  11. [19]

    and Dette, H

    Neumeyer, N. and Dette, H. (2003). Nonparametric comparison of regression curves - an empirical 33 process approach. Annals of Statistics , 31(3):880–920

    2003

  12. [20]

    and Pardo-Fern´ andez, J

    Neumeyer, N. and Pardo-Fern´ andez, J. C. (2009). A simple test for comparing regression curves versus one-sided alternatives. Journal of Statistical Planning and Inference , 139(12):4006–4016

    2009

  13. [21]

    Ombao, H., Von Sachs, R., and Guo, W. (2005). Slex analysis of multivariate nonstationary time series. Journal of the American Statistical Association , 100(470):519–531

    2005

  14. [22]

    Park, C., Hannig, J., and Kang, K.-H. (2014). Nonparametric comparison of multiple regression curves in scale-space. Journal of Computational and Graphical Statistics , 23(3):657–677. Rønn, B. (2001). Nonparametric maximum likelihood estimation for shifted curves. Journal of t...

    2014

  15. [23]

    Silverman, B. W. (1998). CHAPMAN & HALL, London

    1998

  16. [24]

    Varian, H. R. (1984). The nonparametric approach to production analysis. Econometrica, 52(3):579–597. Vilar-Fern´ andez, J. M., Vilar-Fern´ andez, J. A., and Gonz´ alez-Manteiga, W. (2007). Bootstrap tests for nonparametric comparison of regression curves with dependent errors...

    1984

  17. [25]

    Vimond, M. (2010). Efficient estimation for a subclass of shape invariant models. Annals of Statistics, 38(3):1885–1912

    2010

  18. [26]

    Vogt, M. (2012). Nonparametric regression for locally stationary time series. The Annals of Statistics, 40(5):2601–2633

    2012

  19. [27]

    Wu, W. B. and Zhou, Z. (2011). Gaussian approximations for non-stationary multiple time series. Statistica Sinica, 21(3):1397–1413

    2011

  20. [28]

    Zhou, Z. (2010). Nonparametric inference of quantile curves for nonstationary time series. The Annals of Statistics , 38(4):2187–2217

    2010

  21. [29]

    and Wu, W

    Zhou, Z. and Wu, W. B. (2009). Local linear quantile estimation for nonstationary time series. The Annals of Statistics , 37(5B):2696–2729

    2009

  22. [30]

    and Wu, W

    Zhou, Z. and Wu, W. B. (2010). Simultaneous inference of linear models with time varying coeffi- cients. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 72(4):513–531. 34

    2010

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.