A Statistical Characterization of Median-Based Inequality Measures

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The work is excellent, but there is scope for improvement in the following areas:
1. Add a sentence to the summary that highlights the direct empirical applicability of the derived formulas, or the advantages they offer over alternative approaches (e.g. Gini and traditional quantiles).
2. It would be helpful to include a motivating example at the beginning, such as real data on inequality in Canada or the US, to engage the reader more quickly.
3. In terms of methodology, consider moving some of the more technical deductions to the appendices (e.g. the complete derivatives of variances) to make the main body more fluid. Adding a summary table of the main formulas could also improve readability.
4. A more analytical interpretation of the results could be included (e.g. the policy implications of changes in high incomes or the likely causes of changes in gender differences).
5. Include a section on practical and policy implications to make the article more accessible to policymakers and non-technical analysts.

Good Luck

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

1) The abstract currently reads more like a proposal outlining the paper’s aims rather than a summary of its main achievements. I encourage the authors to revise it so that it brings forward the methodological contributions and, more importantly, gives adequate weight to the empirical findings. In particular, the results showing a shrinking middle-income group and increasing upper-tail polarisation should be more clearly highlighted, as these are among the most compelling outcomes of the study.

2) The paper assumes that the income distribution is continuous, differentiable, and strictly increasing, which facilitates theoretical derivation. However, such conditions may not hold in many empirical datasets, especially those with grouped or discretised income values. I suggest the authors discuss the implications of this assumption more explicitly and consider how deviations from it might affect the robustness of the model’s inferences or applicability to real data.

3)The authors use the Comte and Genon-Catalot (2012) estimator to handle positive support in income data, which is methodologically appropriate. That said, the paper would benefit from a comparative evaluation against more standard kernel density estimators. A simulation or empirical analysis demonstrating this estimator’s robustness to rounding or top-coding would help justify its use and clarify the methodological choices for applied researchers.

4) Although the paper cites conventional inequality indices such as the Gini coefficient, it would be further strengthened by a more systematic comparison with these classical measures. Specifically, I recommend the authors clarify how their median-based framework captures intra- and inter-group distributional dynamics that are often missed by summary statistics. This would better establish the paper’s methodological contribution and its added empirical value.

5) While the statistical significance of the results is well demonstrated, the paper would benefit from a deeper economic interpretation. For instance, the finding that polarization is more pronounced among upper-income men raises interesting questions. Are structural shifts in industry employment, educational attainment, or labor market segmentation contributing to this trend? A brief discussion of these potential drivers would enhance the relevance and credibility of the findings.

6) The paper briefly suggests that statistical agencies publish median-based inequality indicators, which is a valuable recommendation. However, this point could be developed further. It would be useful for the authors to discuss how such indicators can improve the tracking of inequality trends, assist in communicating distributional changes to the public, and inform the design of more targeted redistributive policies. Doing so would help link the methodological contribution to real-world applications.

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Here are the comments on the paper "A Statistical Characterisation of Median-Based Inequality Measures" submitted to Econometrics. 

1. The literature review should be written in the past tense as the work has already been completed, similar to this paper's methods and empirical results.
2. The Introduction to the paper is plodding.  The authors should separate the introduction and the Review of the Literature sections.  Make it easier for the readers to follow the paper. 
3. In the derivations of this paper, all of the results from these derivations are asymptotic.  Consequently,  in small samples, normal approximation may be poor.
4. While common, real income data often have kinks due to tax brackets or the existence of wage floors.  Therefore, the differentiability of the CDF would undoubtedly be challenging.  How did the authors adjust the analysis for this?
5. While the authors carefully derived the analytical results, these derivations can be overwhelming for some readers without a guiding diagram or a schematic illustration. 
6. Also, the middle-income group estimates require joint asymptotics in the means, variances, and covariances.  Consequently, this would make such derivations in this case somewhat more complex?  
7. Regarding the data used in this study, the authors utilized data from 2000 to 2005.  Why did the authors use this time period as opposed to more recent years?  This referee thinks an explanation here would aid the reader when reading this paper.
8. This paper presents a rigorous applied econometric analysis with small standard errors, attributed to the large sample sizes used in this analysis.  Also, there is a great transparency of the different estimates with margins of errors as well as a good application of asymptotic theory to labor data.  Kudos to the authors for providing good empirical results based on their analytical derivations. 
9. This referee suspects that the rounding of the nearest $1000 may affect the kernel estimates.
10. There appears to be a lack of measures of inequality within each group or a failure to analyze group within-group inequality.  How could the authors adjust for this? 
11. The analytical framework focused on median-based cutoffs, which could understate dynamics within income deciles or percentiles. This is the referee's perception. Can the authors justify this?

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have responded fully to all of my previous comments. I am satisfied that my concerns have been addressed.

Reviewer 3 Report

Comments and Suggestions for Authors

Here are my comments on the paper "A Statistical Characterisation of Median-Based Inequality Measures," which was submitted to Econometrics. 

There are no additional comments on this paper.