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On rank estimators in increasing dimensions

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Pith's one-line read This paper establishes the first high-dimensional theory for rank estimators such as Han’s maximum rank correlation estimator, showing the estimator stays accurate at the minimax rate but its normal approximation requires a much stronger…

desk verdict The qualitative message is solid, but the advertised Bahadur-type bounds are off by a square root — the proof only supports the square root of the stated rate. read the letter →

arxiv 1908.05255 v1 pith:LVOWM5JZ submitted 2019-08-14 math.ST econ.EMstat.TH

Yanqin Fan , Fang Han , Wei Li , Xiao-Hua Zhou This is my paper · ORCID
classification math.STecon.EMstat.TH MSC 62F1260F1762G20
keywords rankestimatorsmaximumcorrelationincreasingdimensionU-processesBahadur-typeboundsnormalapproximationasymptoticcovarianceestimationsemiparametricindexmodels
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

We now know what happens to rank estimators (like Han's maximum rank correlation estimator) when the number of coefficients grows with the sample size. The paper proves that estimation remains attractive: as long as the parameter count $p_n$ grows more slowly than the sample size $n$, the estimator converges at the minimax-optimal $(p_n/n)^{1/2}$ rate. Inference is another story, because the paper shows that a normal approximation for these estimators requires a much stronger scaling than $p_n^2/n \to 0$; the Bahadur-type bound and the consistency of the covariance estimator both demand stricter conditions. This matters because rank estimators are popular for semiparametric models with many covariates, and the results say the usual $p_n^2/n$ rule used for smooth M-estimators is not enough when the objective function is discontinuous. The paper's simulations show normal confidence intervals losing coverage quickly as $p_n$ grows, even for modest $p_n$.

What carries the argument

The central object is the U-process objective $\Gamma_n(\theta) = \frac{1}{n(n-1)}\sum_{i\neq j} f(Z_i,Z_j;\theta)$ and its Hoeffding decomposition $\Gamma_n(\theta) = \Gamma(\theta) + P_n g(\cdot;\theta) + U_n h(\cdot,\cdot;\theta)$. The key technical contribution is a new maximal inequality for degenerate U-processes in increasing dimensions, which controls the uniform decay of the remainder $\sup_{\theta\in B(\theta_0,r_n)}|U_n h(\cdot,\cdot;\theta)|$ in terms of the VC dimension $\nu_n$, the radius $r_n$, and a variance proxy $\tilde\epsilon_n$. This inequality makes possible a Bahadur-type representation for $\hat\theta_n$ by transferring the analysis to the smoothed objective $\tilde\Gamma_n(\theta) = \Gamma(\theta) + P_n \tau(\cdot;\theta)$, whose theoretical properties are handled via Assumption 3 and exponential-moment bounds. The machinery is what turns the discontinuous loss into a tractable smooth one while keeping track of how $p_n$ affects all rates.

What would settle it

Simulate Han's MRC estimator with heavy-tailed covariates (for example, $t$-distributed with few degrees of freedom) while keeping $p_n$ and $n$ within the paper's scaling regime, and examine whether the coverage probability of the normal confidence interval for a fixed projection deteriorates substantially faster than in the Gaussian-design simulations, or whether the Bahadur expansion's error term grows at a rate larger than the paper's bound.

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Extended reading notes

Core claim

The paper's central claim is that for M-estimators whose objective functions are U-processes and possibly discontinuous, in the increasing-dimension regime, estimation still achieves the minimax-optimal $(p_n/n)^{1/2}$ rate, but normal approximation demands far more stringent scaling. Specifically, for Han's maximum rank correlation estimator under Assumptions 4–6, the paper establishes $\|\hat\theta_n^H - \theta_0 + (V^H)^{-1}P_n\nabla_1\tau^H(\cdot;\theta_0)\|_2 = O_P\bigl(\log(n/p_n^2)p_n^{3/2}/n^{5/4}\bigr)$ whenever $p_n^2/n = o(1)$ and $\log(n/p_n^2)p_n^{3/2}/n^{5/4} = o(1)$. It further shows that $\sqrt{n}\gamma^T(\hat\theta_n^H - \theta_0) / (\gamma^T (V^H)^{-1}\Delta^H (V^H)^{-1} \gamma)^{1/2} \Rightarrow N(0,1)$ under the stronger condition $\log(n/p_n^2)p_n^{3/2}/n^{1/4} = o(1)$. The same pattern holds for the other three rank estimators: a minimax estimation rate, followed by a much more demanding condition for valid normal inference. The paper also proves that the numerical-derivative covariance estimator is consistent only if the step size is tuned with respect to $p_n$, not just $n$.

Load-bearing premise

The results for Han's estimator require an exponential moment bound (Assumption 3(v) under Conditions 1–3), which is verified only for subgaussian designs with smooth conditional densities and bounded second derivatives of the link function; if the covariates are heavy-tailed or the conditional density is not smooth, the Bahadur bound and normal approximation may fail.

Editorial extensions

If this is right

  • For semiparametric index models with many covariates, rank estimators remain usable for point estimation under only $p_n/n \to 0$, matching the minimax-optimal $(p_n/n)^{1/2}$ rate.
  • The usual scaling condition $p_n^2/n \to 0$ used for smooth M-estimators is generally insufficient for normal approximation of rank estimators; inference requires the stronger $\log(n/p_n^2)p_n^{3/2}/n^{1/4} = o(1)$ in the Han example.
  • To obtain consistent covariance matrices by numerical differentiation, the step size must shrink with $p_n$: the paper shows consistency under $\varepsilon_n\sqrt{p_n} = o(1)$ and $\varepsilon_n^{-2}p_n/\sqrt{n} = o(1)$, so the recommended $\varepsilon_n \asymp (p_n/n)^{1/6}$ depends on the dimension.
  • Confidence intervals based on normal approximation deteriorate quickly as $p_n$ increases for fixed $n$, as confirmed by the paper's simulations where even $p_n=3$ or $4$ shows severe coverage distortion at moderate sample sizes.

Reading between the lines

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

  • A likely consequence is that practitioners who want valid confidence intervals with many regressors should shift to alternative inferential methods for rank estimators, such as resampling or bootstrap calibrations, though the paper does not analyze those procedures here.
  • The maximal inequality for degenerate U-processes is a general tool that could be applied to other non-smooth econometric estimators beyond rank correlations, such as maximum score or other pairwise-comparison estimators in increasing dimensions.
  • A natural testable extension is to check whether the conditions imply that the bootstrap, or subsampling, can restore valid coverage under weaker scaling than the normal approximation; the paper leaves this unexplored.
  • Because the Assumption 3(v) exponential-moment bound is verified for Han's estimator only under subgaussian designs and smooth conditional densities, one might expect the normal approximation to break down even earlier for heavy-tailed designs; this is not tested in the paper.
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Editorial analysis

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

Referee Report

4 major / 4 minor

Summary. This paper develops asymptotic theory for M-estimators whose objective functions are U-processes, possibly discontinuous, in an increasing-dimension setting where both the data dimension m_n and parameter dimension p_n grow with n. The main results are: a maximal inequality for degenerate U-processes in increasing dimensions (Theorem 2.1); consistency under ν_n/n → 0 (Theorem 2.2); a (ν_n∨p_n)/n rate of convergence (Theorem 2.3); a Bahadur-type bound and normal approximation under stronger scaling (Theorem 2.4); and consistency of numerical-derivative covariance estimators with step-size calibration (Theorem 2.6). The general results are applied to four rank estimators: Han's maximum rank correlation, Cavanagh–Sherman, Khan–Tamer, and Abrevaya–Shin, with corollaries giving p_n/n^{1/2} estimation rates, Bahadur-type bounds of order log(n/p_n^2)p_n^{3/2}/n^{5/4}, and normal approximation under log(n/p_n^2)p_n^{3/2}/n^{1/4}=o(1). Simulations for Han's MRC illustrate coverage deterioration with p_n.

Significance. The paper addresses a genuine gap: existing increasing-dimension M-estimation theory excludes discontinuous U-process objectives, and rank estimators have only been analyzed for fixed p. The new maximal inequality for degenerate U-processes (Theorem 2.1) and the careful tracking of ν_n, p_n, and m_n are valuable technical contributions, and the conclusion that normal inference requires a much stronger scaling condition than estimation is economically important. The extensive proofs for the general M-estimator and for Han's MRC are detailed and represent serious work. However, the advertised Bahadur-type rates are not supported by the proof as written (missing square root; see major comments), and the minimax optimality claim is not backed by a lower bound. With those corrected, the framework would be a solid contribution to the nonparametric and semiparametric econometrics literature.

major comments (4)
  1. [§A.3.5, Theorem 2.4(i)] The displayed Bahadur bound is not implied by the proof. After (A.28) one has 0 ≤ −(1/2)(t̂_n − t*_n)^T V(t̂_n − t*_n) ≤ 2nφε, and with Assumption 3(ii) this yields ‖t̂_n − t*_n‖ = O_P(√(nφε)), so ‖θ̂_n − θ0 + V^{-1}P_n∇_1τ(·;θ0)‖_2 = O_P(√φε), not O_P(φε). The rate displayed in Theorem 2.4(i) and the rates in Corollaries 3.1(iii), 3.2(iii), 3.3(iii), and 3.4(iii) must be replaced by square-root versions; for Corollary 3.1(iii), for example, the proof supports O_P((log(n/p_n^2))^{1/2}p_n^{3/4}/n^{5/8} + p_n^{5/4}/n^{3/4}) under the stated scaling, not the displayed log(n/p_n^2)p_n^{3/2}/n^{5/4}. The asymptotic-normality scaling condition log(n/p_n^2)p_n^{3/2}/n^{1/4}=o(1) is unchanged because it is equivalent to √φε = o(1/√n), but the quantitative claims in the abstract and corollaries need revision.
  2. [Abstract and §1.1] The abstract and Section 1.1 call (p_n/n)^{1/2} the 'minimax optimal' rate, but no minimax lower bound is proved for the increasing-dimension problems studied. The reference (Yu, 1997) supplies lower bounds only in fixed-dimensional settings; the triangular-array framework with changing parameter spaces requires a new lower-bound argument. As it stands, Theorems 2.3 and Corollaries 3.1–3.4 only establish upper bounds, so the optimality claim should either be proved or softened.
  3. [Appendix A.4.1, Lemmas A.7–A.9] Lemmas A.7–A.9, which bound sup_{θ∈B(θ0,r)} E{h^C(·,·;θ)}^2, E{h^K(·,·;θ)}^2, and E{h^A(·,·;θ)}^2, are omitted with the statement that their proofs are similar to Lemma A.6. These lemmas are load-bearing for Corollaries 3.2–3.4, and the similarity is not immediate: the censored-duration objective includes R_i and V_i, and the Abrevaya–Shin objective includes a kernel K_b(W_i−W_j) with bandwidth, which changes the differentiation and moment arguments. The proofs should be supplied or at least the differences from Lemma A.6 detailed.
  4. [§3.2–§3.4, Assumption 3(v)] Assumption 3(v), the exponential moment condition on the smoothed Hessian, is verified only for Han's MRC in Theorem 3.1 under Conditions 1–3. For the Cavanagh–Sherman, Khan–Tamer, and Abrevaya–Shin estimators, no primitive conditions are given under which Assumption 3(v) (or Assumption 3(iii)) holds; the corresponding corollaries therefore rely on an unverified high-level condition. The authors should state sufficient design and smoothness conditions for these estimators or explicitly flag Assumption 3 as a high-level condition that must be checked case by case.
minor comments (4)
  1. [§A.3.5] The quantity φε is used in (A.26) before it is defined in (A.27); reorder the display or define φε earlier.
  2. [§1.4] The notation 'P− →' in Section 1.4 appears garbled; standard notations for convergence in probability should be used.
  3. [Corollary 3.4(iii)] In Corollary 3.4(iii), the term n^{-δJ} should be made explicit as O_P(n^{-δJ}), and the dependence of the constant on J and on the kernel K(·) should be stated.
  4. [Section 4] Tables 1–3 report coverage probabilities for three projection directions, and Figures 1–3 are not explicitly cross-referenced to those directions in the text; a sentence stating which figure corresponds to which projection would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained and no prediction reduces to a fitted input or self-citation chain.

full rationale

The paper's central results are derived under explicit assumptions (Assumptions 1–6, Conditions 1–3) rather than fitted to the data, and the main theorems are proved from external empirical-process tools (Nolan and Pollard 1987, Kosorok 2007, Spokoiny 2012a/b, 2013, Sherman 1993). The only self-citation is Han et al. (2017), mentioned in the concluding remarks as a direction for future smoothing work ('if no further smoothing (cf. Han et al. (2017)) is made'); it is not used to establish consistency, rates, Bahadur bounds, or normality, so it is not load-bearing. The claimed sqrt(p/n) estimation rate follows from Theorem 2.3 after bounding the VC dimension of the rank-estimator function classes, and the Bahadur-type bound in Corollary 3.1(iii) is obtained by plugging the moment bound from Lemma A.6 into Theorem 2.4, with no parameter renamed as a prediction. The asymptotic covariance estimator consistency is also proved from the assumed scaling of the numerical-derivative step size rather than chosen to match the conclusions. The reader-submitted concern about a missing square root in Theorem 2.4's displayed rate is a potential correctness or proof-error issue, not circularity, because the theorem's conclusion does not reproduce an input by construction. Accordingly, no specific circular step can be exhibited, and the score is 0.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new entities. Its technical core rests on external standard results (Hoeffding, Nolan-Pollard, Kosorok, Spokoiny) and on explicit modeling assumptions (identifiability, smoothness, exponential moments, VC structure). The only user-chosen quantities are the numerical derivative step size and the kernel bandwidth in one example.

free parameters (2)
  • step size epsilon_n in numerical derivative covariance estimation = epsilon_n ~ (p_n/n)^{1/6} (recommended)
    Chosen to satisfy epsilon_n sqrt(p_n) = o(1) and epsilon_n^{-2} p_n / sqrt(n) = o(1). It is a tuning parameter, not fitted to data, but its rate is essential for consistency of the covariance estimator.
  • bandwidth b in Abrevaya-Shin estimator = b = c n^{-delta} with 1/J < delta < 1/5
    Kernel bandwidth in the partially linear index model; the order J kernel and the rate condition for b are required for the bias to vanish. It is chosen by the user, not fitted to data.
assumptions (7)
  • standard math Hoeffding decomposition of the U-process objective into a smooth empirical process plus a degenerate U-process
    Invoked in Section 1.3 and used throughout the proofs; a classical result from Hoeffding (1948).
  • standard math Nolan and Pollard (1987) maximal inequality for degenerate U-processes
    The basis for the new Theorem 2.1; the paper modifies the argument to handle increasing dimensions.
  • standard math Covering number bounds for VC-subgraph classes (Theorem 9.3 in Kosorok 2007)
    Used to bound entropy and moment terms in Lemmas A.2, A.3, and the proof of Theorem 2.1.
  • standard math Spokoiny's finite-sample theory for differentiable M-estimators with increasing dimension (Spokoiny 2012a, 2013)
    The smoothed estimator theta_tilde_n is analyzed by invoking Spokoiny's bracketing results in the proof of Theorem 2.3 and Theorem 2.4.
  • domain assumption Assumption 1: identifiability with a uniform gap xi0 between the maximum and the boundary
    Standard identification condition for M-estimators, stated in Section 2.2; needed for consistency.
  • domain assumption Assumption 3: local strong convexity, smoothness of E tau, and subgaussian exponential moment condition on the Hessian of zeta
    Central assumption for the Bahadur bound and normal approximation; for the examples it is verified under primitive conditions, but it restricts the design and error distribution.
  • domain assumption The function class F is a uniformly bounded VC-subgraph class with VC dimension nu_n
    Required by the empirical process theory used throughout; the rank estimators are shown to satisfy this with nu_n of order p_n.

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Pith. "Pith review of On rank estimators in increasing dimensions." pith.science (2026). https://pith.science/paper/LVOWM5JZ

@misc{pith2026190805255,
  author       = {Pith},
  title        = {Pith review of: On rank estimators in increasing dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LVOWM5JZ}},
  note         = {Machine review of arXiv:1908.05255}
}
abstract

The family of rank estimators, including Han's maximum rank correlation (Han, 1987) as a notable example, has been widely exploited in studying regression problems. For these estimators, although the linear index is introduced for alleviating the impact of dimensionality, the effect of large dimension on inference is rarely studied. This paper fills this gap via studying the statistical properties of a larger family of M-estimators, whose objective functions are formulated as U-processes and may be discontinuous in increasing dimension set-up where the number of parameters, $p_{n}$, in the model is allowed to increase with the sample size, $n$. First, we find that often in estimation, as $p_{n}/n\rightarrow 0$, $(p_{n}/n)^{1/2}$ rate of convergence is obtainable. Second, we establish Bahadur-type bounds and study the validity of normal approximation, which we find often requires a much stronger scaling requirement than $p_{n}^{2}/n\rightarrow 0.$ Third, we state conditions under which the numerical derivative estimator of asymptotic covariance matrix is consistent, and show that the step size in implementing the covariance estimator has to be adjusted with respect to $p_{n}$. All theoretical results are further backed up by simulation studies.

Figures

Figures reproduced from arXiv: 1908.05255 by the authors.

Figure 1
Figure 1. Plots of the kernel density estimates of the normalized estimates (blue) v.s. N(0, 1) (red) under the first projection direction (n = 100, 200, 400 from top to bottom). p = 1 p = 2 p = 3 p = 4 -6 -4 -2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized projection N(0,1) -6 -4 -2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized projection N(0,1) -6 -4 -2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9… view at source ↗
Figure 2
Figure 2. Plots of the kernel density estimates of the normalized estimates (blue) v.s. N(0, 1) (red) under the second projection direction (n = 100, 200, 400 from top to bottom). 51 [PITH_FULL_IMAGE:figures/full_fig_p051_2.png] view at source ↗
Figure 3
Figure 3. Plots of the kernel density estimates of the normalized estimates (blue) v.s. N(0, 1) (red) under the third projection direction (n = 100, 200, 400 from top to bottom) [PITH_FULL_IMAGE:figures/full_fig_p052_3.png] view at source ↗

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