pith. sign in

arxiv: 2605.01606 · v1 · submitted 2026-05-02 · 📊 stat.ME · math.ST· stat.TH

L-Estimation of Population Quantiles Using Ranked Set Sampling

Mohammad Jafari Jozani , Ehsan Zamanzade , Reza Modarre This is my paper

Pith reviewed 2026-05-09 17:47 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH
keywords ranked set samplingquantile estimationL-estimatorsHarrell-Davis estimatorefficiency gainssimple random samplingpopulation quantiles
0
0 comments X

The pith

L-estimators under ranked set sampling provide more efficient quantile estimates than those from simple random sampling.

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

The paper develops methods to estimate population quantiles more efficiently when units can be ranked inexpensively but measured expensively. It extends two classic L-estimators to the ranked set sampling design, one direct and one using a decomposition into rank-based components for easier computation. Large-sample theory justifies the combined estimators, and simulations demonstrate consistent gains in efficiency across distributions and quantile levels, particularly for the Harrell-Davis variant at moderate and upper quantiles. An application to real biomedical data illustrates the practical value for estimating clinically relevant thresholds.

Core claim

We develop two families of RSS-based L-estimators for population quantiles that extend Stigler-type and Harrell--Davis estimators to the RSS framework. The first applies weighted-order-statistic estimation directly to the pooled ordered RSS sample. The second exploits a decomposition induced by the RSS design that constructs k pooled transformed-scale component estimators indexed by rank stratum and leads to a computationally scalable procedure. Large-sample results are derived for these component estimators under regularity conditions.

What carries the argument

The RSS design's induced decomposition into k pooled transformed-scale component estimators indexed by rank stratum, which allows scalable construction of combined L-estimators.

Load-bearing premise

The large-sample results for the component estimators hold under regularity conditions, and the ranked set sampling design induces a useful decomposition into rank strata.

What would settle it

Simulations or real data where the proposed RSS L-estimators show no efficiency improvement or even higher variance than standard empirical estimators under standard conditions would falsify the efficiency claims.

Figures

Figures reproduced from arXiv: 2605.01606 by Ehsan Zamanzade, Mohammad Jafari Jozani, Reza Modarre.

Figure 1
Figure 1. Figure 1: Relative efficiency (RE) of eight quantile estimators under simple random sampling view at source ↗
Figure 2
Figure 2. Figure 2: Relative efficiency (RE) of eight quantile estimators under SRS and RSS for ( view at source ↗
Figure 3
Figure 3. Figure 3: Relative efficiency (RE) of six quantile estimators under SRS and RSS for ( view at source ↗
Figure 4
Figure 4. Figure 4: Exploratory plots for the NHANES liver transient elastography illustration. The upper view at source ↗
Figure 5
Figure 5. Figure 5: Relative-efficiency curves for the NHANES RSS illustration with ( view at source ↗
read the original abstract

Quantile estimation is central when interest lies in thresholds or tail behavior rather than the mean. When exact measurement is costly but units can be ranked cheaply, ranked set sampling (RSS) provides an attractive alternative to simple random sampling (SRS). We develop two families of RSS-based L-estimators for population quantiles that extend Stigler-type and Harrell--Davis estimators to the RSS framework. The first applies weighted-order-statistic estimation directly to the pooled ordered RSS sample and serves primarily as an exact conceptual benchmark, since its computational burden increases rapidly with the set size. The second exploits a decomposition induced by the RSS design that constructs $k$ pooled transformed-scale component estimators indexed by rank stratum and leads to a computationally scalable procedure. We derive large-sample results for these component estimators under regularity conditions; these results provide a principled first-order motivation for the combined estimators employed in practice. Simulation results across several distributions, quantile levels, and ranking qualities show consistent efficiency gains over empirical quantile estimators under both SRS and RSS, with the RSS Harrell--Davis version performing especially well for moderate and upper quantiles. Beyond the simulation study, we demonstrate the practical relevance of the proposed estimators through an application to NHANES transient elastography data, highlighting their usefulness for estimating clinically meaningful quantiles in a biomedical setting

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

2 major / 2 minor

Summary. The paper develops two families of L-estimators for population quantiles under ranked set sampling (RSS): a direct weighted-order-statistic estimator applied to the pooled RSS sample (primarily as a conceptual benchmark) and a scalable version that exploits the RSS design to construct k pooled transformed-scale component estimators indexed by rank stratum. Large-sample consistency and asymptotic normality are derived for the component estimators under regularity conditions on the weight functions and density; these are presented as first-order motivation for the combined estimators. Simulations across distributions, quantile levels, and ranking qualities demonstrate efficiency gains relative to empirical quantile estimators from both SRS and RSS, with the RSS Harrell-Davis variant performing well for moderate and upper quantiles. An application to NHANES transient elastography data illustrates practical use for clinically relevant quantiles.

Significance. If the reported efficiency gains are robust, the work provides a useful extension of Stigler-type and Harrell-Davis L-estimators to the RSS setting, offering computationally feasible quantile estimation when ranking is inexpensive relative to measurement. The decomposition approach and NHANES application add practical value for biomedical and survey statistics contexts where tail or threshold estimation is important. The simulation design across multiple distributions and ranking qualities is a strength, as is the explicit acknowledgment that component asymptotics supply only motivation rather than a complete limiting theory for the combined estimator.

major comments (2)
  1. [Asymptotic results section] Asymptotic results section: Large-sample consistency and normality are established only for the k stratum-specific component estimators. The paper does not derive the joint asymptotic covariance structure of the components or apply a delta-method (or other) expansion to obtain the limiting distribution of the (possibly nonlinear) combination that yields the final RSS L-estimator. Because the efficiency gains reported in the simulation study are for this combined estimator, the component results supply motivation but leave the theoretical anchor for those gains incomplete.
  2. [Simulation study section] Simulation study section: The reported efficiency gains are presented without accompanying standard errors, confidence intervals, or details on the number of Monte Carlo replications, the precise mechanism used to induce ranking errors, or any data-exclusion rules. These omissions make it difficult to assess the statistical significance and robustness of the 'consistent efficiency gains' claim, particularly for the moderate and upper quantiles where the Harrell-Davis RSS version is highlighted.
minor comments (2)
  1. [Abstract] Abstract: The distinction between the direct pooled weighted-order-statistic estimator and the decomposed component-based estimator could be stated more explicitly to avoid potential reader confusion about which procedure is implemented in the simulations and application.
  2. [Methods section] Notation: The definition of the transformed-scale component estimators and the precise form of the weight functions (e.g., how they reduce to Stigler or Harrell-Davis in the SRS limit) would benefit from a single consolidated display equation early in the methods section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The comments highlight important areas for strengthening the theoretical and simulation components of the manuscript. We respond to each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Asymptotic results section] Asymptotic results section: Large-sample consistency and normality are established only for the k stratum-specific component estimators. The paper does not derive the joint asymptotic covariance structure of the components or apply a delta-method (or other) expansion to obtain the limiting distribution of the (possibly nonlinear) combination that yields the final RSS L-estimator. Because the efficiency gains reported in the simulation study are for this combined estimator, the component results supply motivation but leave the theoretical anchor for those gains incomplete.

    Authors: We agree that the manuscript provides asymptotic results only for the stratum-specific component estimators and does not derive the joint limiting distribution or apply a delta-method expansion to the final combined estimator. This was intentional, as the component results were presented explicitly as first-order motivation for the scalable procedure (see the abstract and Section 3). Deriving the full joint asymptotics would require a detailed characterization of the dependence structure across rank strata induced by the RSS design, which is technically involved and beyond the scope of the current contribution. In the revision we will expand the discussion at the end of the asymptotic results section to more clearly articulate this limitation, its implications for the simulation evidence, and the conditions under which the component-wise consistency still supports practical use of the combined estimator. We will also include a brief outline of how a delta-method argument could proceed under an approximate-independence approximation. This is a partial revision. revision: partial

  2. Referee: [Simulation study section] Simulation study section: The reported efficiency gains are presented without accompanying standard errors, confidence intervals, or details on the number of Monte Carlo replications, the precise mechanism used to induce ranking errors, or any data-exclusion rules. These omissions make it difficult to assess the statistical significance and robustness of the 'consistent efficiency gains' claim, particularly for the moderate and upper quantiles where the Harrell-Davis RSS version is highlighted.

    Authors: We appreciate this observation and agree that the simulation section would benefit from greater transparency. The Monte Carlo experiment used 5,000 replications for each combination of distribution, quantile level, set size, and ranking quality. Ranking errors were generated using the probabilistic ranking model of Dell and Clutter (1972) with error probabilities calibrated to produce the three ranking-quality regimes described in Section 4.1; no observations were excluded. In the revised manuscript we will report Monte Carlo standard errors for all efficiency ratios in the tables and figures, add the replication count and ranking-error mechanism to the text of Section 4, and explicitly state that no data-exclusion rules were applied. These additions will allow readers to evaluate the statistical significance and robustness of the reported gains. revision: yes

Circularity Check

0 steps flagged

No significant circularity; estimators extend prior L-estimators via RSS decomposition with independent asymptotics

full rationale

The paper constructs two families of RSS L-estimators by direct extension of Stigler-type and Harrell-Davis forms to the ranked set sampling design, using the induced decomposition into k stratum-specific pooled components. Large-sample consistency and normality are derived for the components under stated regularity conditions on weights and density; these supply first-order motivation for the combined estimator but do not reduce it to a fitted input or self-referential definition. Simulations and the NHANES application serve as separate empirical checks. No self-citation is load-bearing for the central construction, no ansatz is smuggled, and no result is renamed or forced by prior author work. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the estimators extend existing L-estimation and RSS concepts without additional postulates or fitted constants identified.

pith-pipeline@v0.9.0 · 5536 in / 1209 out tokens · 54049 ms · 2026-05-09T17:47:33.655338+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

31 extracted references · 23 canonical work pages

  1. [1]

    and Balakrishnan, N

    Arnold, Barry C. and Balakrishnan, N. and Nagaraja, H. N. , title =. 2008 , series =. doi:10.1137/1.9780898719062 , url =

    work page doi:10.1137/1.9780898719062 2008

  2. [2]

    ASTIN Bulletin , year =

    Brazauskas, Vytaras and Serfling, Robert , title =. ASTIN Bulletin , year =. doi:10.2143/AST.33.2.503698 , url =

    work page doi:10.2143/ast.33.2.503698

  3. [3]

    Statistica Sinica , year =

    Chen, Zehua , title =. Statistica Sinica , year =

  4. [4]

    Journal of Statistical Planning and Inference , volume=

    On ranked-set sample quantiles and their applications , author=. Journal of Statistical Planning and Inference , volume=. 2000 , publisher=

    2000

  5. [5]

    , title =

    Chen, Zehua and Bai, Zhidong and Sinha, Bimal K. , title =. 2004 , publisher =. doi:10.1007/978-0-387-21664-5 , url =

    work page doi:10.1007/978-0-387-21664-5 2004

  6. [6]

    and Lockhart, W.L

    Evans, M.S. and Lockhart, W.L. and Doetzel, L. and Low, G. and Muir ,D. and Kidd, K. and Stephens, G. and Delaronde J. , title =. Science of the Total Environment , year =. doi:10.1016/j.scitotenv.2004.12.086 , url =

    work page doi:10.1016/j.scitotenv.2004.12.086 2004

  7. [7]

    , title =

    Giorgi, Giovanni M. , title =. Handbook of Income Inequality Measurement , series =. 1999 , publisher =. doi:10.1007/978-94-011-4413-1_9 , url =

    work page doi:10.1007/978-94-011-4413-1_9 1999

  8. [8]

    and Davis, C

    Harrell, Frank E. and Davis, C. E. , title =. Biometrika , year =. doi:10.1093/biomet/69.3.635 , url =

    work page doi:10.1093/biomet/69.3.635

  9. [9]

    Statistical Methods in Medical Research , year =

    Hatefi, Armin and Jafari Jozani, Mohammad , title =. Statistical Methods in Medical Research , year =. doi:10.1177/0962280215601458 , url =

    work page doi:10.1177/0962280215601458

  10. [10]

    Human Health Risk Assessment of Mercury in Fish and Health Benefits of Fish Consumption , year =

  11. [11]

    and Leslie, William D

    Jafari Jozani, Mohammad and Ayilara, Olawale F. and Leslie, William D. , title =. Statistics in Medicine , year =. doi:10.1002/sim.7655 , url =

    work page doi:10.1002/sim.7655

  12. [12]

    , title =

    Kvam, Paul H. , title =. Journal of Agricultural, Biological, and Environmental Statistics , year =. doi:10.1198/1085711032156 , url =

    work page doi:10.1198/1085711032156

  13. [13]

    Statistics in Medicine , year =

    Loewen, Neve and Jafari Jozani, Mohammad , title =. Statistics in Medicine , year =. doi:10.1002/sim.70362 , url =

    work page doi:10.1002/sim.70362

  14. [14]

    Communications in Statistics---Simulation and Computation , year =

    Mahdizadeh, Mahdi and Arghami, Naser Reza , title =. Communications in Statistics---Simulation and Computation , year =. doi:10.1080/03610918.2011.624236 , url =

    work page doi:10.1080/03610918.2011.624236 2011

  15. [15]

    Statistical gua rantees for the EM algorithm: From population to sample-based analysis

    Mason, David M. and Shorack, Galen R. , title =. The Annals of Probability , year =. doi:10.1214/aop/1176989529 , url =

    work page doi:10.1214/aop/1176989529

  16. [16]

    McIntyre, G. A. , title =. Australian Journal of Agricultural Research , year =. doi:10.1071/AR9520385 , url =

    work page doi:10.1071/ar9520385

  17. [17]

    and Gastwirth, Joseph L

    Modarres, Reza and Nayak, Tapan K. and Gastwirth, Joseph L. , title =. Computational Statistics & Data Analysis , year =. doi:10.1016/S0167-9473(01)00094-9 , url =

    work page doi:10.1016/s0167-9473(01)00094-9

  18. [18]

    , title =

    Nourmohammadi, Mohammad and Jafari Jozani, Mohammad and Johnson, Brad C. , title =. Computational Statistics & Data Analysis , year =. doi:10.1016/j.csda.2013.11.020 , url =

    work page doi:10.1016/j.csda.2013.11.020 2013

  19. [19]

    , title =

    Presnell, Brett and Bohn, Lora L. , title =. Journal of Nonparametric Statistics , year =. doi:10.1080/10485259908832756 , url =

    work page doi:10.1080/10485259908832756

  20. [20]

    and Al-Sagheer, Omar A

    Samawi, Hani M. and Al-Sagheer, Omar A. M. , title =. Biometrical Journal , year =. doi:10.1002/1521-4036(200106)43:3<357::AID-BIMJ357>3.0.CO;2-Q , url =

    work page doi:10.1002/1521-4036(200106)43:3

  21. [21]

    , month = nov, year =

    Serfling, Robert J. , title =. 1980 , publisher =. doi:10.1002/9780470316481 , url =

    work page doi:10.1002/9780470316481 1980

  22. [22]

    Fish growth rates modulate mercury concentrations in walleye (

    Simoneau, Michel and Lucotte, Marc and Garceau, Sylvain and Lalibert. Fish growth rates modulate mercury concentrations in walleye (. Environmental Research , year =. doi:10.1016/j.envres.2004.05.007 , url =

    work page doi:10.1016/j.envres.2004.05.007 2004

  23. [23]

    Minimax test and neyman-pearson lemma for capacities

    Stigler, Stephen M. , title =. The Annals of Statistics , year =. doi:10.1214/aos/1176342756 , url =

    work page doi:10.1214/aos/1176342756

  24. [24]

    Insurance: Mathematics and Economics , year =

    Wang, Shaun , title =. Insurance: Mathematics and Economics , year =. doi:10.1016/0167-6687(95)00010-P , url =

    work page doi:10.1016/0167-6687(95)00010-p

  25. [25]

    NHANES August 2021--August 2023 , year =

    2021

  26. [26]

    August 2021--August 2023 Examination Data , year =

    2021

  27. [27]

    2024 , howpublished =

    Liver Ultrasound Transient Elastography (. 2024 , howpublished =

    2024

  28. [28]

    2024 , howpublished =

    Body Measures (. 2024 , howpublished =

    2024

  29. [29]

    2024 , howpublished =

    Demographic Variables and Sample Weights (. 2024 , howpublished =

    2024

  30. [30]

    and Petroff, D

    Karlas, T. and Petroff, D. and Sasso, M. and Fan, J.-G. and Mi, Y.-Q. and de L. Individual patient data meta-analysis of controlled attenuation parameter (. Journal of Hepatology , year =. doi:10.1016/j.jhep.2016.12.022 , url =

    work page doi:10.1016/j.jhep.2016.12.022 2016

  31. [31]

    and Sasso, Magali and Allison, Matthew and Tsochatzis, Emmanuel and Anstee, Quentin M

    Eddowes, Peter J. and Sasso, Magali and Allison, Matthew and Tsochatzis, Emmanuel and Anstee, Quentin M. and Sheridan, David and Guha, Indra N. and Cobbold, James F. L. and Deeks, Jonathan J. and Paradis, Valerie and Bedossa, Pierre and Newsome, Philip N. , title =. Gastroenterology , year =. doi:10.1053/j.gastro.2019.01.042 , url =

    work page doi:10.1053/j.gastro.2019.01.042 2019