Finite population inference for skewness measures

Leo Pasquazzi

REVIEW 3 major objections 5 minor 20 references

Finite population inference for skewness measures

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper derives asymptotic normality for plug-in estimators of Bowley's skewness and the Groeneveld–Meeden index in finite population sampling.

desk verdict Honest, useful extension of plug-in inference to two quantile-based skewness measures, but the asymptotic theory is explicitly conjectural and the density estimator bias is unquantified; worth refereeing, not worth treating as settled. read the letter →

arxiv 2411.18549 v3 pith:KQPU3AW5 submitted 2024-11-27 stat.ME

Leo Pasquazzi This is my paper · ORCID

classification stat.ME MSC 62D0562G0562G20

keywords skewnessBowley'sindexGroeneveld–MeedenfinitepopulationinferencefunctionaldeltamethodcalibrationestimatorHájekSen–Yates–Grundyvariance

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

This paper tries to establish that two classic skewness measures—Bowley's $b_2(r)$ and the Groeneveld–Meeden $b_3$—can be estimated from a finite population sample with a variance estimator that makes normal confidence intervals valid. For plug-in estimators built from the Hájek cdf estimator and from a Deville–Särndal calibration cdf estimator, the paper derives asymptotic variance formulas by a von Mises (functional delta) expansion and proposes Horvitz–Thompson and Sen–Yates–Grundy type variance estimators. The paper's central claim is asymptotic normality of the standardized estimator, $(\widehat{b} - b)/V \to N(0,1)$, under broad conditions, plus the assertion that the proposed variance estimators approximate $V$. Simulations under simple and stratified random sampling show coverage rates at or above nominal levels for $b_2(0.75)$ and $b_3$, while the variance estimators retain a noticeable bias that the paper traces to plug-in density estimation.

What carries the argument

The machinery is the von Mises expansion of the plug-in estimator, also called the functional delta method. For $b_3$ the paper builds a one-parameter path $F_\lambda = F + \lambda(\widehat{F} - F)$ and differentiates the functional to obtain the influence weight $g(t)$; the derivative exists only under a smoothness condition, so the paper replaces it with assumption B1 that the expansion holds with a negligible remainder even for the discrete Hájek and calibration cdf estimators. The variance calculation then reduces to the variance of the weighted sample sum $\sum_{i\in s} d_i g(y_i)$, and the proposed variance estimators are the standard Horvitz–Thompson and Sen–Yates–Grundy forms applied to those weighted sums. A plug-in density estimator for $f(\nu_r)$ completes the construction, and the paper notes that this density estimator is the suspected source of the variance estimators' bias.

What would settle it

Generate a finite population with a known skewness value and a design with strongly unequal inclusion probabilities, then compute $\widehat{b}$ and the variance estimate over many samples and check the empirical distribution of $(\widehat{b} - b)/\widehat{V}$. If the 95% interval covers in well under 90% of samples, the central claim is false for that setting. A more direct check is to compute the remainder $R_N$ from the von Mises expansion for increasing $N$ and see whether it vanishes relative to the integral term.

Watch

Extended reading notes

Core claim

The claimed discovery is that quantile-based skewness measures, despite being nonlinear functionals of a step-function cdf estimator, admit the same linearization that makes survey inference work for smooth functionals. Specifically, the paper asserts that $\widehat{b}_\bullet - b_\bullet = \int g(t)\,d[\widehat{F}(t) - F(t)] + R_N$ with $R_N$ asymptotically negligible, where $\bullet$ is $2$ or $3$ and $g$ is the explicit influence-function-type weight given in formulas (5) and (8). Under that expansion plus a central limit theorem for the integral, the standardized estimator converges to $N(0,1)$ with asymptotic variance $V_N^2 = N^{-2}\operatorname{var}(\sum_{i\in s} d_i g(y_i))$, where $d_i$ are Hájek or calibration weights. The paper then provides Horvitz–Thompson and Sen–Yates–Grundy estimators for this variance. The simulation evidence is presented as supporting the claim: confidence intervals for $b_2(0.75)$ and $b_3$ tend to cover at or above the nominal rate, even though the variance estimators themselves show sizable relative bias.

Load-bearing premise

The load-bearing premise is that the von Mises expansion holds with a negligible remainder and the leading linear term obeys a central limit theorem for the discrete Hájek and calibration cdf estimators; the paper states it does not prove sufficient conditions. If that assumption fails, the normal confidence intervals are not justified.

Editorial extensions

If this is right

Normal confidence intervals for $b_2(r)$ and $b_3$ become available for fixed-size survey designs, with the Sen–Yates–Grundy variance estimator as the default.
The calibration cdf estimator offers little improvement over the Hájek estimator for skewness in the simulations, in contrast with the large gains it delivers for the mean; auxiliary information does less work for these functionals.
Variance estimates inherit bias from plug-in density estimation, and the relative bias need not shrink from $n=40$ to $n=80$; users should treat interval lengths as approximate at moderate sample sizes.
For the mean, the same designs show undercoverage, while the skewness intervals overcover; inference for these skewness measures is not a carbon copy of mean inference.

Reading between the lines

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

If the high-level assumption holds across a wider class of designs and populations, the same linearization recipe could be carried over to other quantile-based shape measures, such as kurtosis or tail-weight indices, as long as their influence functions are available.
The bias pattern in the variance estimators suggests that a bootstrap-calibrated version of the interval, or the variance-stabilizing transformation the paper mentions, would likely deliver more accurate coverage at small sample sizes.
A quick way to stress-test the central claim is to check whether the remainder $R_N$ in the von Mises expansion is actually negligible under designs with highly unequal inclusion probabilities; simulation studies varying the spread of the $\pi_i$ would settle this.
Since the paper leaves sufficient conditions for B1 and B2 open, bridging that gap with uniform quantile-process results would turn a heuristic argument into a theorem.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the paper does something useful and is refreshingly honest about what it doesn't prove. It gives plug-in estimators and variance estimators for Bowley's b2 and Groeneveld-Meeden's b3 under finite population sampling, with the functional delta method as the engine. The influence functions were already in Groeneveld (1991), so the novelty is the finite-population variance machinery and the simulation evaluation, not the discovery of a new method. That's fine; it fills a real gap for survey practitioners who want standard errors for these descriptive measures.

What's good: the derivation is clear, the variance estimators for the Hajek and calibration cdf estimators are carefully laid out, and the simulation study is honest. The author reports coverage rates, tail errors, relative bias and stability of the variance estimators, and they note when the variance estimator's relative bias is large (e.g., 1.36 in Table 3) and that it doesn't always shrink with n. That kind of transparency is worth a lot.

The soft spots are real but not hidden. The main theorem rests on Assumptions B1 and B2 in Appendix A, which are the von Mises expansion with negligible remainder and a CLT for the integral term. The paper explicitly says it does not investigate sufficient conditions and only conjectures they hold. That's a load-bearing gap: without a proof, the entire normal-CI claim is heuristic. The density estimator in (3) is acknowledged to carry unanalyzed bias, and the large relative biases in the variance estimator are consistent with that bias converging slowly. The simulations cover only two populations and two designs, so they're suggestive, not decisive.

None of this kills the paper. The author says the right things about what is and isn't established, and the empirical evidence is consistent with the claims. But a referee should push for either a proof of B1/B2 under finite-population asymptotics (there are tools in Conti-Marella, Han-Wellner, Dey-Chaudhuri) or a careful analysis of the density estimator's bias, and ideally code/data release.

Who should read it: survey methodologists who need variance estimates for skewness measures, and anyone applying plug-in inference to distribution-based parameters. It deserves a serious referee, though the outcome should be conditional on the theoretical gaps being addressed or substantially reduced.

Referee Report

3 major / 5 minor

Summary. The paper develops plug-in estimators for two quantile-based skewness measures, Bowley's b2(r) and the Groeneveld-Meeden b3, in finite population sampling. Estimators are formed by inserting the Hajek or a Deville-Sarndal calibration cdf estimator into the definitions of the skewness measures. The paper derives asymptotic variance formulae by a functional delta method argument, proposes Horvitz-Thompson and Sen-Yates-Grundy variance estimators, and evaluates the resulting normal confidence intervals in a simulation study with two synthetic populations, two sampling designs, and two sample sizes. The central claim is that, under broad but unproved conditions, the standardized plug-in estimators are asymptotically normal with variance given by the proposed formulae, so that the proposed intervals have nominal coverage asymptotically.

Significance. If the central asymptotic claim holds, the paper makes a useful contribution by extending quantile-based skewness inference from i.i.d. settings to design-based finite population sampling and by providing explicit influence-function-type variance formulae for two cdf estimators. The paper is commendably honest: it labels the functional delta method argument as heuristic, explicitly states Assumptions B1 and B2 as conjectures rather than theorems, and openly notes that the density estimator bias is not formally analyzed. The simulation study, though limited, provides encouraging evidence that coverage rates are often conservative and that variance estimator stability is comparable to that reported in earlier survey-sampling work. The main limitation is that the entire normal-CI procedure rests on unproved high-level assumptions, so the paper currently supplies methodology and evidence but not a rigorous foundation for its headline inferential claim.

major comments (3)

[Appendix A, Assumptions B1 and B2] The paper's central claim that (bb - b)/V_N converges in distribution to N(0,1) is stated to follow from Assumptions B1 and B2, but these assumptions are only conjectured: the text says 'we do not investigate sufficient conditions under which B1 and B2 hold' and merely cites Conti and Marella (2015), Han and Wellner (2021), and Dey and Chaudhuri (2024). This is load-bearing because Condition A, under which the von Mises expansion is formally derived, requires continuous and positive densities for F(t) and Fhat(t), while the Hajek and calibration cdf estimators are step functions. The remainder R_N in equation (4) contains second-order terms at estimated quantiles divided by density factors, and controlling it for step-function cdf estimators requires a uniform Bahadur-type representation and rate conditions on the density estimator, none of which are stated or proved. Without a proof or at least a precise statement of sufficient conditions, the variance formula in equation (6) is not established as the asymptotic variance of bb, and the claimed validity of normal confidence intervals is unsupported.
[Section 2, equation (3), and Appendix B] The proposed variance estimators depend on the density estimator fhat(nu_r) defined in equation (3), and the paper acknowledges that this estimator carries bias whose formal analysis is outside the scope. The simulation results show that this bias is not negligible: in Table 3 the relative bias of the variance estimator for bb2,Ha(0.75) is 1.363 at n=40 and 0.370 at n=80, and several other entries exceed 0.30 at n=80 (Tables 3, 5, 7). These numbers are consistent with the paper's own conjecture that the bias decreases slowly, but they also mean that the variance estimators can severely overestimate the MSE, which is relevant to the practical claim that normal intervals are valid. The paper should either provide a first-order analysis of the density estimator bias and its effect on V-hat, or propose a bias-corrected density estimator, or clearly restrict the validity claim to settings where the bias is asymptotically negligible and show that the simulations support that restriction.
[Section 3 and Appendix B] The simulation study uses only two populations, both generated from the same model with normal errors and lognormal X, two sampling designs (SRS and stratified SRS), and two sample sizes (n=40,80) from a population of N=800. This is too narrow to substantiate the statement in the introduction that normal confidence intervals work 'under broad conditions.' In particular, the behavior of quantile-based skewness estimators and of the Woodruff-based density estimator may depend strongly on the local density at the relevant quantiles and on the design, and the simulations do not explore heavy-tailed distributions, unequal-probability designs without stratification, or populations where the target quantiles lie in regions of low density. At minimum, the conclusions should be tempered to describe the evidence as preliminary, and the simulation design should be expanded or justified as representative of the settings for which the theory is intended.

minor comments (5)

[Appendix A, paragraph after equation (4)] There is a garbled sentence in the derivation of the von Mises derivative: 'In order to make sure that However, partial_nu_lambda/partial_lambda ...' appears to be an incomplete edit. This should be corrected.
[Section 2, equation (3)] The notation fhat(nu_r) is introduced with square brackets that are easy to confuse with a floor or indicator function; a clearer notation such as widehat{f}(hat{nu}_r) would improve readability.
[Section 2, equation (2)] The claim that the calibration equations (2) are 'always solvable' should be accompanied by a reference or a brief condition on the support of X and the sample design, since solvability of exponential calibration equations is not completely unconditional.
[Appendix B] The tables report only the Sen-Yates-Grundy variance estimators; the paper mentions Horvitz-Thompson-type estimators as alternatives, but does not report their performance. A sentence noting that SYG was chosen because of fixed-size designs is present, but it would be helpful to state explicitly that the HT versions were not evaluated.
[Introduction] The paper cites Groeneveld (1991) for influence functions of b2(r) and b3 but does not explain how those influence functions relate to the g-functions in Appendix A beyond a one-sentence remark; a short display of the connection would make the paper more self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: variance formulae are derived by linearization and estimated by standard plug-in methods; the asymptotic normality claim is explicitly conditional on unproved assumptions B1-B2, a stated limitation rather than a circular reduction.

full rationale

The derivation chain is not circular. The estimators are plug-ins of the Hajek and calibration cdf estimators into the quantile-based definitions of b2(r) and b3, and the asymptotic variance V_N^2 is obtained by differentiating the functional along the linear path F_lambda; the g functions in eqs. (5) and (8) are derived from those derivatives, not fitted. Equation (6) expresses V_N^2 as the variance of the linearized estimator sum d_i g(y_i), which is a standard influence-function calculation. The variance estimators then use the usual Horvitz-Thompson and Sen-Yates-Grundy forms with plug-in estimates of g, including the Woodruff-type density estimator in eq. (3); plugging estimated nuisance parameters into an influence-function variance formula is standard plug-in estimation rather than a renamed prediction. The only load-bearing theoretical step is that Appendix A states, rather than proves, Assumptions B1 and B2: the von Mises expansion with a negligible remainder and the CLT for the main integral term. The paper is transparent about this, saying 'we do not investigate sufficient conditions' and labeling the argument a conjecture. The CLT conclusion then follows from B1 and B2 in a one-line deduction, which is a missing proof or stated limitation, not a circular reduction: B1 and B2 are not fitted values, not renamed predictions, and not supported by a load-bearing self-citation. No such self-citation is present. Therefore the paper's variance formulae have independent content and the circularity score is zero.

Assumptions & free parameters 1 free parameters · 2 assumptions · 0 invented entities

The paper introduces no new entities or global parameters. The central claim depends on unproven high-level asymptotic assumptions (B1, B2) and a sample-fitted density estimate with unanalyzed bias.

free parameters (1)

density estimate hat f(nu_r) = per-sample, eq. (3)
Central variance estimators depend on this density plug-in, which is fitted to the sample by matching CI lengths; its bias is acknowledged but not analyzed.

assumptions (2)

ad hoc to paper Assumptions B1 and B2 (von Mises expansion with negligible remainder and CLT for the integral)
Stated without proof in Appendix A; the paper says sufficient conditions are not investigated. Central to asymptotic normality.
domain assumption F has continuous positive density at the quantiles of interest
Assumption A in Appendix A; needed for derivatives of quantile and absolute deviation functionals to exist.

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Cite this review

Pith. "Pith review of Finite population inference for skewness measures." pith.science (2026). https://pith.science/paper/KQPU3AW5

@misc{pith2026241118549,
  author       = {Pith},
  title        = {Pith review of: Finite population inference for skewness measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KQPU3AW5}},
  note         = {Machine review of arXiv:2411.18549}
}

abstract

In this article we consider Bowley's skewness measure and the Groeneveld-Meeden $b_{3}$ index in the context of finite population sampling. We employ the functional delta method to obtain asymptotic variance formulae for plug-in estimators and propose corresponding variance estimators. We then consider plug-in estimators based on the H\'{a}jek cdf-estimator and on a Deville-S\"arndal type calibration estimator and test the performance of normal confidence intervals.

Figures

Figures reproduced from arXiv: 2411.18549 by the authors.

**Figure 1.** Population from the model with γ = 0 (upper part) and γ = 1 (lower part). Different symbols (circle, triangle, plus sign) are used to show strata membership in the stratified sampling designs. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

Bowley, A. L. (1901). Elements of statistics . Number viii, 330 p. P. S. King, London

work page 1901
[2]

Bowley, A. L. (1920). Elements of statistics . P.S. King & Son, Ltd. ; C. Scribner's Sons, London, New York, 4th ed edition

work page 1920
[3]

Chambers, R. L. and Dunstan, R. (1986). Estimating distribution functions from survey data. Biometrika , 73(3):597--604

work page 1986
[4]

Conti, P. L. and Marella, D. (2015). Inference for quantiles of a finite population: Asymptotic versus resampling results. Scandinavian Journal of Statistics , 42(2):545--561

work page 2015
[5]

and Särndal, C.-E

Deville, J.-C. and Särndal, C.-E. (1992). Calibration estimators in survey sampling. Journal of the American Statistical Association , 87(418):376--382

work page 1992
[6]

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Dey, A. and Chaudhuri, P. (2024). Quantile processes and their applications in finite populations

work page 2024
[7]

Francisco, C. A. and Fuller, W. A. (1986). Estimation of the distribution function with a complex survey. Proc. Sec. Survey Res. Methods, Amer. Statist. Association, Washington D.C. , pages 37--45

work page 1986
[8]

Groeneveld, R. A. (1991). An influence function approach to describing the skewness of a distribution. The American Statistician , 45(2):97--102

work page 1991

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[9]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Bowley, A. L. (1901). Elements of statistics . Number viii, 330 p. P. S. King, London

work page 1901

[2] [2]

Bowley, A. L. (1920). Elements of statistics . P.S. King & Son, Ltd. ; C. Scribner's Sons, London, New York, 4th ed edition

work page 1920

[3] [3]

Chambers, R. L. and Dunstan, R. (1986). Estimating distribution functions from survey data. Biometrika , 73(3):597--604

work page 1986

[4] [4]

Conti, P. L. and Marella, D. (2015). Inference for quantiles of a finite population: Asymptotic versus resampling results. Scandinavian Journal of Statistics , 42(2):545--561

work page 2015

[5] [5]

and Särndal, C.-E

Deville, J.-C. and Särndal, C.-E. (1992). Calibration estimators in survey sampling. Journal of the American Statistical Association , 87(418):376--382

work page 1992

[6] [6]

and Chaudhuri, P

Dey, A. and Chaudhuri, P. (2024). Quantile processes and their applications in finite populations

work page 2024

[7] [7]

Francisco, C. A. and Fuller, W. A. (1986). Estimation of the distribution function with a complex survey. Proc. Sec. Survey Res. Methods, Amer. Statist. Association, Washington D.C. , pages 37--45

work page 1986

[8] [8]

Groeneveld, R. A. (1991). An influence function approach to describing the skewness of a distribution. The American Statistician , 45(2):97--102

work page 1991