REVIEW 4 major objections 5 minor 19 references
On Rank Correlation Coefficients
T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read The paper introduces a weighted rank correlation coefficient and argues, via variance formulas and simulations, that it outperforms Pearson, Spearman, and Kendall when dependence is not close to linear.
desk verdict A genuinely new weighted rank correlation with a promising variance story, but the main variance proof does not close and the performance claim goes beyond the evidence. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The object carrying the argument is the weighted concordance sum $T_n = \sum_{i=2}^{n} \sum_{j=1}^{i-1} (n-i+j) I_{ji}$, where $I_{ji}$ is the indicator that $Y_{[j]} \le Y_{[i]}$ and $Y_{[j]}$ is the $Y$-concomitant of the $j$-th order statistic of the $X$-sample, that is, the $Y$ observation paired with $X_{(j)}$. The weight $n-i+j$ is largest when the two rank indices are close and decreases by one as the gap widens, translating the heuristic that neighboring order statistics are more strongly associated than distant ones. This weighting is what distinguishes $r_n$ from Kendall's $\tau_n$ and is what the paper credits for the improved variance.
What would settle it
Repeat the paper's variance comparison on a copula with strong tail dependence but non-linear overall structure, such as a Clayton or Gumbel copula at moderate Kendall tau. If the sample variance of $r_n$ is not below that of $\tau_n$ in that setting, the paper's claim that $r_n$ performs better whenever the relationship is not close to linear would be false.
Extended reading notes
Core claim
The central claim is that $r_n$ is the sample version of $r$ and that, for dependence structures that are not nearly linear, it estimates association more efficiently than the classical choices. The paper defines $r_n = 12 T_n/[n(n-1)(2n-1)] - 1$, where $T_n = \sum_{i=2}^{n}\sum_{j=1}^{i-1} (n-i+j) I_{ji}$ and $I_{ji}$ is the indicator that the $Y$-concomitant of the $j$-th $X$-order statistic does not exceed that of the $i$-th. Theorem 2.1 gives $E r_n = (1 - 3/(2n-1)) r - 3/(2n-1) + 12E(F(X,Y))/(2n-1)$, so $E r_n \to r$. Theorems 2.2 and 2.3 give asymptotic variance formulas for $r_n$ and Kendall's $\tau_n$ for any absolutely continuous $F$, and the Farlie-Gumbel-Morgenstern example shows $Var(r_n(t)) = (1/n)(1/4 - 7t^2/180) + O(1/n^2)$, while $Var(\tau_n(t)) = (1/n)(4/9 - 46t^2/2025) + O(1/n^2)$; the former is smaller for every $t$.
Load-bearing premise
The load-bearing premise is that association between two rank positions decays linearly with the distance between their indices, so the weights $n-i+j$ are the right ones; the paper justifies this only by a heuristic from the order-statistics literature and gives no theorem that this weighting is optimal or uniformly better for every non-linear dependence structure.
Editorial extensions
If this is right
- Outside the near-linear regime, $r_n$ becomes the preferred rank coefficient: lower variance at the same sample size means sharper estimates of association.
- The explicit asymptotic variance formulas allow $\tau_n$ and $r_n$ to be compared analytically for any absolutely continuous distribution, without running simulations.
- Because $r_n$ is a rank statistic, it inherits the outlier resistance of Spearman and Kendall coefficients and remains defined for heavy-tailed distributions where Pearson's coefficient does not exist.
- In the Farlie-Gumbel-Morgenstern family the ordering $Var(r_n) < Var(\tau_n)$ holds for every $t \in [-1,1]$, so the advantage is not limited to one parameter value.
Reading between the lines
- The paper leaves open whether the gain comes specifically from linear decay; one could test the family $(n-i+j)^\alpha$ with data-chosen $\alpha$ and use the same variance machinery to see whether further reductions are possible.
- If the efficiency gain carries over to tail-dependent copulas, $r_n$ could replace Kendall's tau in non-parametric independence tests, where lower variance typically translates into higher power.
- The simulations cover only three parametric families, so the 'not close to linear' rule is provisional until the comparison is repeated on heavy-tailed or asymmetric copulas.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a new rank correlation coefficient r_n, defined as a weighted sum of concordance indicators between concomitants of order statistics, with weights (n-i+j). It presents a formula for the expectation of r_n (Theorem 2.1) and asymptotic variance formulas for r_n and for Kendall's tau (Theorems 2.2 and 2.3). The performance of r_n is compared with Pearson, Spearman, and Kendall coefficients in three parametric examples (FGM copula, bivariate normal, bivariate Pareto) via simulation. The abstract claims that when the relationship between X and Y is not close to linear, r_n performs better than the other correlation coefficients, and the paper also claims analytic variance formulas that allow theoretical comparison.
Significance. If the variance formula in Theorem 2.2 is correct, it provides an explicit expression for the asymptotic variance of a new rank-based estimator under arbitrary absolutely continuous distributions, a non-trivial technical contribution. The FGM example gives a concrete parametric family where r_n has smaller asymptotic variance than Kendall's tau, which is interesting. However, the central performance claim is not established by theorem and is in fact contradicted by some of the paper's own simulation results. The value of the paper depends heavily on whether the proof of Theorem 2.2 can be corrected and on whether the empirical superiority claim can be qualified appropriately.
major comments (4)
- [Section 3, Proposition 3.2 and the proof of Theorem 2.2] The algebraic identity used to simplify the variance is incorrect as typeset. The correct expansion is E[(1+H)\bar F+(1-H)F]^2 = E[(F+\bar F)^2] + 2E[H(\bar F^2-F^2)] + E[H^2(\bar F-F)^2], but Proposition 3.2 and the line after equation (3.2) write E[H(\bar F-F)]^2 instead. In addition, the end of the proof of Proposition 3.2 states E(H^2(\bar F^2-F^2)), which differs from both forms. As written, the algebra leading to Theorem 2.2 does not follow, so the variance formula is not established. This is load-bearing for the paper's main technical claim and must be corrected and re-verified, including the FGM computation in Example 2.1.
- [Abstract, Conclusion, and Example 2.3] The central claim that r_n 'performs better than other correlation coefficients' when the relationship is not close to linear is contradicted by the paper's own Table 2.4: for the Pareto distribution at t=0.05, which is far from linear, the simulated variance of r_n is 2.976e-5, larger than both the Spearman variance 1.641e-6 and the Kendall variance 1.641e-5. The text itself notes this exception in the interval [0.0012,0.07]. The claim is also not characterized: no theorem or precisely defined class of distributions is given for which Var(r_n) ≤ Var(τ_n); the only analytic comparison is the FGM copula. The abstract and conclusion should be revised to state the supported, narrower finding, or a formal result covering a meaningful class of non-linear dependence structures should be supplied.
- [Section 2.2, Examples 2.2 and 2.3] The simulation evidence is limited to three parametric families with n=1000 and 1000 replications, and no standard errors or confidence intervals are reported for the estimated variances. In the normal example, the paper shows that r_n is the worst of the four coefficients at t=0.99 (Table 2.2), which is inside the paper's own set W where ρ_n is preferred; this is consistent with the 'close to linear' caveat, but the boundary of that region is not characterized for r_n. For the Pareto example, the ordering varies with t and in some regions r_n is outperformed by the other rank coefficients. The empirical claim should be accompanied by an analysis of simulation uncertainty and a discussion of how the ordering depends on the dependence structure.
- [Section 2, definition of r_n and weighting rationale] The estimator is built on the heuristic that association between distant order statistics is weaker than between neighboring ones, leading to linear weights (n-i+j). This heuristic is cited to Walsh (1969) but no theorem or simulation study addresses dependence structures with strong tail dependence, where distant order statistics can remain strongly associated (e.g., Clayton-type copulas). Since the claimed advantage of r_n rests on this weighting, the absence of any test with tail-dependent families leaves the rationale unsupported. Adding such an example, or at least a discussion of conditions under which the weighting is sensible, would materially strengthen the paper.
minor comments (5)
- [Section 3, proof of Proposition 3.1] Several index sums are typeset with missing plus signs, for example the expression for Σ_{m<k<j<i} reads 'I^(1)_3412;12,34 - I^(2)_3412;12,34 I^(3)_3412;12,34 - I^(4)_3412;12,34', and the Q2 formula contains 'I^(4)_1324;12,34 2I^(4)_3142;12,34 3I^(4)_1342;12,34' without separating plus symbols. These typographical errors make the proof very hard to follow and should be fixed.
- [Section 2.2, Example 2.2] The set W = (-1,-0.730072) ∪ (0.730072,1) is taken from Stepanov (2025b) and was established there for the comparison of ρ_n and τ_n; the paper should state explicitly that its extension to r_n is based on simulation and not on an asymptotic relative efficiency calculation.
- [Abstract and Theorem 2.3] The abstract says 'We also find analytically the values of Var(τ_n) and Var(r_n)', but Theorem 2.3 for Var(τ_n) is a restatement of a result from Stepanov (2025a). The paper should acknowledge this reuse more explicitly to avoid overstating novelty.
- [References] The reference list contains several formatting errors: one entry is a dangling 'Ann. Probab., 22, 126–144.' with no authors, 'correlatio ncoefficient' appears in the Hotelling entry, and 'Gulk, S.Van' should be cleaned up.
- [Tables 2.1–2.4] The notation S^2 is used without definition; please define it as the sample variance of the coefficient over the 1000 simulated replications.
Circularity Check
No significant circularity: r_n is independently defined, its asymptotic properties are derived from first principles, and the comparisons rest on explicit analytic and simulation evidence.
full rationale
The new coefficient r_n is defined directly as a weighted sum of concomitants, and its asymptotic expectation and variance are derived in Theorems 2.1, 2.2, and 2.3 from the joint density of order-statistic concomitants, not assumed from the target r. The identity r = (3τ − ρ_S)/2 is cited from Stepanov (2025b), but it is a definitional identity used only to express the limit; the convergence E r_n → r follows from the paper's own computation of E T_n. The comparisons with τ_n and ρ_{n,S} are supported by explicit variance formulas and by simulations on three parametric families, including the FGM copula, where Var(r_n) < Var(τ_n) is shown analytically for large n. The self-citations to Stepanov (2025a,b) supply background and technical lemmas, e.g., moment calculations for Kendall's tau, but those results are standard and externally checkable, and the central derivation of r_n does not reduce to them. The heuristic choice of weights (n − i + j) is a modeling assumption, not a fitted input, and no 'prediction' is forced by construction. An apparent typo in Proposition 3.2, where the squared H term in the statement does not match the proof's E[H^2(...)] term, is a correctness concern rather than a circularity concern.
Assumptions & free parameters
free parameters (1)
- linear weight function n - i + j
assumptions (4)
- standard math The joint densities of concomitants of order statistics, formulas (3.1), (3.3), (3.4)
- domain assumption The identity r = (3τ - ρ_S)/2
- domain assumption The variance of Kendall's tau, Var(τ_n), is 16(E[F+bar F]^2 - 4(EF)^2)/n + O(n^-2)
- domain assumption The heuristic that association between distant order statistics is weaker (Walsh 1969)
Cite this review
Pith. "Pith review of On Rank Correlation Coefficients." pith.science (2026). https://pith.science/paper/GFYYHY7Y
@misc{pith2026250606056,
author = {Pith},
title = {Pith review of: On Rank Correlation Coefficients},
year = {2026},
howpublished = {\url{https://pith.science/paper/GFYYHY7Y}},
note = {Machine review of arXiv:2506.06056}
}
abstract
In the present paper, we propose a new rank correlation coefficient $r_n$, which is a sample analogue of the theoretical correlation coefficient $r$, which, in turn, was proposed in the recent work of Stepanov (2025b). We discuss the properties of $r_n$ and compare $r_n$ with known rank Spearman $\rho_{S,n}$, Kendall $\tau_n$ and sample Pearson $\rho_n$ correlation coefficients. Simulation experiments show that when the relationship between $X$ and $Y$ is not close to linear, $r_n$ performs better than other correlation coefficients. We also find analytically the values of $Var(\tau_n)$ and $Var(r_n)$. This allows to estimate theoretically the asymptotic performance of $\tau_n$ and $r_n$.
Reference graph
Works this paper leans on
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Reviewed August 7, 2026 · model on record in the stance chip above.
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