Parameterized Kolmogorov–Smirnov Test for Normality

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I would recommend major revision. Please find my comments below:

Clarify in the abstract what is genuinely new in this work compared with your previous studies on Lilliefors type and Cramer von Mises tests, and state the main contributions in a short, explicit list.

Shorten the introduction by compressing the detailed literature overview around Table 1 and moving part of that material to supplementary information.

State clear research goals at the end of the introduction in a concise paragraph that tells the reader exactly what questions the paper answers and how the proposed tests are intended to be used in practice.

In teh section introducing the parameterized Kolmogorov Smirnov statistic, improve the explanation of the Bloom transformation and give more intuitive guidance on how the parameters control the sensitivity of the test.

Clarify in one place exactly what is new in this paper relative to Sulewski references 14 to 16 and other earlier works by the authors, for example by giving a short subsection that contrasts PKS with the previously proposed parameterized tests.

Justify the focus on sample sizes n equal to 10 and 20 in the simulation study and either extend the study to larger n or discuss explicitly how the results are expected to generalize to more typical sample sizes used in applied work.

Provide a clear description of how critical values for PKS are obtained for arbitrary sample sizes beyond those shown in Table 5, and indicate whether code or lookup tables will be made available as supplementary material for practitioners.

Give more intuition for the similarity measure M by adding a short explanation and perhaps one or two simple examples that show what values such as 0.5, 0.75 and 0.9 correspond to in terms of visible differences in the density function.

Improve the presentation of the groups of alternatives based on skewness and kurtosis in Table 2 by explaining more clearly what typical distributions belong to each group and how this grouping relates to practical modelling situations.

The manuscript is too long in its current form; reduce the length of the simulation results section by moving a substantial part of the detailed power tables, especially Tables 6 to 13, to suplementary information and replace them with summary plots and concise narrative conclusions.

In the description of the alternative distributions, highlight the most important families for applied users and give a short explanation of why each family is relevant rather than only listing parameter values.

In the simulation study, clarify the numerical settings such as the number of replications, random number generation method and convergence checks in a dedicated subsection so that the design is transparent and reproducible.

When discussing power results, add more interpretive statements that compare the performance of PKS variants with Shapiro wilk, Shapiro Francia and Anderson Darling in each main scenario instead of mainly reporting which test has the largest number in each row of the tables.

In the real data section, explain how the thirty examples were selected, what the variables represent and why they are suitable for evaluating normality tests rather than presenting them as a mostly anonymous list.

For the real data examples, add a more narrative discussion of a few representative cases where PKS leads to a different conclusion from Shapiro Wilk or Anderson Darling and explain the practical implications for researchers.

Improve the readability of Tables 15 to 17 by reducing the number of tests shown or by moving some of the less informative tests to supplementary material and focusing the main text on the most relevant variants.

In teh conclusions section, provide explicit recommendations for practitioners, for example which PKS parameter combinations are suggested for skew alternatives, for heavy tailed alternatives and for general use when there is no prior information.

Discuss more clearly the limitations of the proposed PKS tests, including the dependence on estimated parameters, the need for simulated critical values and potential computational cost compared with classical tests.

Check the notation throughout the paper for consistency, especially for the indices of groups A to H, for the naming of tests in Table 5 and for the use of symbols such as γ1, γ2 and δ, and standardize them in a notation subsection.

Imrove the English style and grammar in several places where sentences are hard to follow or unidiomatic, for example in some parts of the introduction and in the explanation of the grouping of alternatives, and consider a careful language edit.

Review the figures that display the similarity measure and the SKS measure and ensure that the captions clearly explain what each panel shows, how the axes are scaled and what conclusions the reader should draw from each figure.

Revisit the reference list to ensure that recent work on normality testing and goodness of fit tests is adequately cited and that the positioning of the PKS approach relative to state of the art methods is clearly explained.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors,

This paper presents a solid and technically sound contribution to the statistical literature on goodness-of-fit testing for normality. Your systematic extension of the parameterized testing framework to the Kolmogorov-Smirnov family, combined with rigorous Monte Carlo simulations, comprehensive classification of alternative distributions, and real data applications, represents a genuine advancement in this methodological area. Your work demonstrates the coherent research program you have been developing on parametrized goodness-of-fit testing, building systematically on your earlier contributions on modified Lilliefors and Cramér-von Mises tests.

The novel contribution consists of: (1) introducing the parameterized KS test PKS(α, β) through Bloom's transformation of the empirical distribution function, (2) expanding the EDF family with four new proposals strategically positioned to complete the parameter space, (3) classifying alternative distributions into nine groups based on skewness and excess kurtosis signs, and (4) conducting comprehensive power analysis across ten alternative distributions with appropriate emphasis on small sample sizes (n = 10, 20) relevant for experimental applications.

However, despite these merits, this reviewer considers it relevant to address some important issues below, providing recommendations for improvement.

Major Comments

1. Reproducibility and Computational Transparency

The manuscript would benefit substantially from enhanced reproducibility by:

• Providing exact computational algorithms: Offer detailed pseudocode or explicit step-by-step algorithms for computing the PKS(α, β) statistics, critical values, and power analysis. This would enable readers to verify results and implement the methods in alternative software environments.
• Confirming software implementation: Explicitly state whether an R package (or other software) implementing these variants is under development, or planned. Alternatively, provide the relevant code in supplementary materials. All practitioners who wish to apply your methods will benefit from this material.
• Documenting the simulation environment: I would suggest specifying the random seed(s) used for all Monte Carlo simulations, the computational environment, and approximate computational time for critical value generation and power calculations.

These additions would significantly enhance reader confidence in reproducing all values presented in Tables 5-13 and strengthen the scientific impact of your work.

2. Explanation of Anomalous Power Behavior

Tables 6-13 correctly identify cases where power fails to increase with sample size (SB and SU alternatives in Groups C, D, E) or fails to decrease as the similarity measure increases (P, PCM, LM, SB, SU, EECK alternatives in certain groups). While you acknowledge these observations, mechanistic explanations would strengthen the manuscript and guide practitioners:

• Why does power plateau or fail to increase monotonically with sample size for specific SB and SU alternatives?
• Are there theoretical properties of these distribution families (shape, modality, tail behavior) that explain this phenomenon?
• Should practitioners exercise caution when testing specifically against SB/SU alternatives in these parameter regions?

A brief discussion addressing these questions in Section 5 would clarify when the proposed PKS variants are most reliable.

Minor Comments

• Page 4, Figure 1: Consider adding a brief explanation in the caption regarding the significance of the β = α and β = -α + 1 lines to aid readers' interpretation.
• Page 5, Malachov inequality: A one-sentence intuitive explanation of why this constraint matters for alternative distribution selection would help readers unfamiliar with this concept.
• Page 6, last line. Replace “Figures 2-5” with “Figures 3-6”.

In summary, your paper makes genuine contributions to an important methodological area in goodness-of-fit testing. The theoretical development is rigorous, the simulation methodology is comprehensive, and the practical guidance for test selection is valuable. With the revisions outlined above—particularly enhanced reproducibility documentation and mechanistic explanation of power behavior anomalies—this manuscript would become a stronger and more impactful contribution to the statistical literature.

I am confident that careful attention to these suggestions will result in a significantly improved manuscript.

Thank you for your attention.

Best regards

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

1. My most serious comment involves the choice of the Bloom family (2) for study.  As specified by (2), this family allows the CDF approximation to be strictly less than 1 for values of x above the last order statistic, as long as beta<1, but the specification in the MS contains no ability for the CDF to be strictly > 0 for values of x below the first order statistic.  (Perhaps Bloom has a fuller discussion that allows this, but it does not appear in the present MS.)  In this respect, the family is inherently asymmetric.  I find this problematic, because it seems to me that if G is the best fit for the cdf of a variable X, then 1-G(-x) ought to be the best fit for the variable -X, but this isn't the case for this family.  This is important, because alpha=0, beta=1 are the most powerful methods for many situations.
2. A paragraph near the bottom of page 3 lays out objectives for the manuscript.  I think that these aims are too diffuse; the prime products of this paper seem to be a catalog of work on prior results on modifications of the empirical CDF in KS testing, and an extensive simulation study.  I suggest making aims more concise.
3. Some expositional issues:
   1. The formatting that puts tables filling almost all, but not all, of the page, interspersed with text, makes the text disjointed and difficult to absorb.
   2. On line 66, the third example of the Bloom family, is described as referring to the CM statistic.  I know what the authors intend here, but you have to have thought a lot about the relationship between the integral definition of the CM statistic and the formula via summation.  Description of this relationship to make the comment on this line give the right impression is outside the scope of this  paper.  If you want to talk about the CM statistic, I suggest referring to The Exact and Asymptotic Distributions of Cramér-von Mises Statistics on JSTOR
   3. I think that the indices in the outer maximum in (5) and (6) are incorrect.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Thanks to the authors for addressing my comments.