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arxiv: 1906.09143 · v1 · pith:7SEHO2RMnew · submitted 2019-06-21 · 🧮 math.ST · stat.TH

Intermediate efficiency of some weighted goodness-of-fit statistics

Bogdan \'Cmiel , Tadeusz Inglot , Teresa Ledwina This is my paper

Pith reviewed 2026-05-25 18:20 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords goodness-of-fit testsAnderson-Darling statisticKolmogorov-Smirnov statisticEicker-Jaeschke statisticsintermediate efficiencyweighted statisticstail alternatives
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The pith

Weighted goodness-of-fit statistics permit analytic efficiency comparisons to the Kolmogorov-Smirnov test when alternatives place moderate mass in the tails.

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

The paper develops a method to compare the Anderson-Darling statistic and members of the Eicker-Jaeschke class against the classical Kolmogorov-Smirnov statistic. The comparison targets alternatives that shift a moderately large probability mass into the tails. If the method works, researchers gain a tractable way to quantify when weighting improves detection without relying solely on simulation. The authors also show that a natural modification of an earlier proposal yields a statistic inside the Eicker-Jaeschke family that rivals the popular Anderson-Darling supremum version. Finite-sample checks confirm the analytic ordering.

Core claim

When the alternative distribution allocates a moderately large portion of probability mass toward the tails, the intermediate-efficiency approach yields explicit analytic comparisons between weighted statistics and the unweighted Kolmogorov-Smirnov benchmark, together with reliable quantitative rankings; a slight modification of the Borovkov-Sycheva construction produces an Eicker-Jaeschke statistic competitive with Anderson-Darling.

What carries the argument

Intermediate-efficiency comparison of weighted supremum-type statistics (Anderson-Darling and Eicker-Jaeschke) against the Kolmogorov-Smirnov benchmark under tail-heavy alternatives.

If this is right

  • Weighted tests such as Anderson-Darling become preferable to Kolmogorov-Smirnov for detecting moderate tail departures.
  • Analytic rather than purely numerical evaluation of relative performance becomes feasible for a range of weighted statistics.
  • The modified Eicker-Jaeschke statistic offers a practical alternative to the Anderson-Darling statistic with comparable or better properties.
  • Finite-sample behavior aligns with the intermediate-efficiency ordering derived from the analytic comparison.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same intermediate-efficiency technique could be applied to other weighting functions or to alternatives concentrated in the center rather than the tails.
  • Practitioners facing data with suspected tail discrepancies could adopt the modified Eicker-Jaeschke statistic as a default over Anderson-Darling without loss of power.
  • The approach supplies a concrete benchmark that future efficiency studies of goodness-of-fit procedures could use to calibrate new weighting schemes.

Load-bearing premise

The comparison applies specifically when under the alternative a moderately large portion of probability mass is allocated towards the tails.

What would settle it

A direct calculation or simulation for an alternative that places only a very small or very large fraction of mass in the tails, showing that the predicted efficiency ordering between the weighted statistics and Kolmogorov-Smirnov reverses or disappears.

Figures

Figures reproduced from arXiv: 1906.09143 by Bogdan \'Cmiel, Tadeusz Inglot, Teresa Ledwina.

Figure 1
Figure 1. Figure 1: (Best viewed in color) Alternatives from M1, M2 and empirical powers. First row : the functions aj - dashed line, and A∗ j - solid line, and values of t0, m0, j = 1, 2. Second and third rows : empirical powers (in the full range and zoomed) of E o n , E ? n , In, Mn, Kn, Kn·eEK , and Kn·eIK . The third row also includes the corresponding efficiencies eEK and eIK [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Best viewed in color) Alternatives from M2, M3 and empirical powers. First row : the functions aj - dashed line, and A∗ j - solid line, values of t0, m0, j = 2, 3. Second and third rows : empirical powers (in the full range and zoomed) of E o n , E ? n , In, Mn, Kn, Kn·eEK , and Kn·eIK . The third row also includes the corresponding efficiencies eEK and eIK [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Best viewed in color) Empirical powers of E o n , E ? n , Kn, In, and Mn under alterna￾tives M4 − M7 for selected parameters and sample sizes [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
read the original abstract

This paper compares the Anderson-Darling and some Eicker-Jaeschke statistics to the classical unweighted Kolmogorov-Smirnov statistic. The goal is to provide a quantitative comparison of such tests and to study real possibilities of using them to detect departures from the hypothesized distribution that occur in the tails. This contribution covers the case when under the alternative a moderately large portion of probability mass is allocated towards the tails. It is demonstrated that the approach allows for tractable, analytic comparison between the given test and the benchmark, and for reliable quantitative evaluation of weighted statistics. Finite sample results illustrate the proposed approach and confirm the theoretical findings. In the course of the investigation we also prove that a slight and natural modification of the solution proposed by Borovkov and Sycheva (1968) leads to a statistic which is a member of Eicker-Jaeschke class and can be considered an attractive competitor of the very popular supremum-type Anderson-Darling statistic.

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

0 major / 3 minor

Summary. The paper develops an intermediate-efficiency framework for comparing the Anderson-Darling statistic and Eicker-Jaeschke weighted goodness-of-fit statistics against the classical Kolmogorov-Smirnov statistic. The analysis is restricted to alternatives that place a moderately large portion of probability mass in the tails; it claims that this framework yields tractable analytic comparisons and reliable quantitative evaluations. Finite-sample simulations are presented to illustrate the results, and a slight modification of the Borovkov-Sycheva (1968) construction is shown to belong to the Eicker-Jaeschke class and to serve as a competitive alternative to the supremum-type Anderson-Darling statistic.

Significance. If the analytic comparisons hold under the stated regime, the work supplies a concrete, quantitative tool for assessing relative performance of weighted tests when tail departures matter. The explicit identification of a modified Borovkov-Sycheva statistic as an Eicker-Jaeschke competitor and the supporting finite-sample results are clear strengths that enhance the practical value of the contribution.

minor comments (3)
  1. The abstract refers to 'tractable, analytic comparison' without previewing the precise form of the intermediate-efficiency functional; a one-sentence indication of the efficiency measure (e.g., the ratio of sample sizes needed to achieve a given power) would improve readability.
  2. Notation for the weight functions defining the Eicker-Jaeschke class and the precise statement of the 'slight and natural modification' of Borovkov-Sycheva should be introduced before the main theorems to avoid forward references.
  3. In the finite-sample section, the number of Monte Carlo replications and the exact sample sizes used should be stated explicitly so that the reported power curves can be reproduced.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation of minor revision. The referee's summary correctly reflects the scope and contributions of the work on intermediate efficiency comparisons for weighted goodness-of-fit statistics under tail alternatives.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained in asymptotic theory

full rationale

The paper applies the intermediate-efficiency framework to derive analytic comparisons between Anderson-Darling/Eicker-Jaeschke weighted statistics and the Kolmogorov-Smirnov benchmark under tail-heavy alternatives. All load-bearing steps rest on standard large-sample theory and a new proof that a minor modification of the 1968 Borovkov-Sycheva construction belongs to the Eicker-Jaeschke class; neither step reduces to a fitted parameter renamed as prediction, a self-definitional loop, nor a load-bearing self-citation. The central quantitative evaluation claim therefore remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard asymptotic statistics for intermediate efficiency; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Standard regularity conditions of asymptotic statistics underlying intermediate efficiency calculations
    Efficiency comparisons in goodness-of-fit testing presuppose these background results from the literature.

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