Empirical interpretation of the Pitman efficiency

Tadeusz Inglot

arxiv: 2604.11127 · v1 · submitted 2026-04-13 · 🧮 math.ST · stat.TH

Empirical interpretation of the Pitman efficiency

Tadeusz Inglot This is my paper

Pith reviewed 2026-05-10 15:58 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords Pitman efficiencyrelative efficiencycontamination modelsuniformity testingbeta distributionsstatistical testsasymptotic efficiency

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The pith

Pitman efficiency closely approximates relative efficiency for uniformity tests under contamination models in the beta family.

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

The paper investigates the Pitman efficiency as a practical proxy for relative efficiency when testing uniformity on data from the two-parameter beta family. It demonstrates that this approximation holds very well specifically when the underlying models incorporate contamination. A reader would care because Pitman efficiency is typically easier to derive theoretically than full relative-efficiency calculations, offering a shortcut for evaluating which tests perform better under realistic deviations from perfect uniformity.

Core claim

In testing for uniformity within the two-parametric family of the beta distributions, the Pitman efficiency approximates relative efficiency very well when contamination models are used.

What carries the argument

Contamination models overlaid on the two-parameter beta family, used to compare the asymptotic Pitman efficiency directly against finite-sample relative efficiency of uniformity tests.

If this is right

Test selection for uniformity can rely on Pitman efficiency calculations instead of exhaustive simulations under contamination.
The approximation supplies a concrete empirical grounding for using Pitman efficiency in practice rather than treating it as purely theoretical.
Efficiency rankings of tests remain stable across moderate contamination levels in the beta setting.
The result encourages treating Pitman efficiency as an interpretable finite-sample quantity for this class of problems.

Where Pith is reading between the lines

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

The same empirical-validation strategy could be applied to other goodness-of-fit problems to check whether Pitman efficiency retains its proxy role outside the beta family.
If the approximation generalizes, it would lower the computational barrier for comparing new uniformity tests before full Monte Carlo studies.
The finding hints that contamination models may serve as a useful bridge between asymptotic theory and observed performance in a wider range of distribution-testing settings.

Load-bearing premise

The chosen two-parameter beta family together with the specific contamination models are representative enough for the uniformity testing problem.

What would settle it

A large discrepancy between Pitman and relative efficiency when the same comparison is repeated on a different parametric family such as the normal or on non-contamination alternatives.

read the original abstract

We study an empirical interpretation of the Pitman efficiency in testing for uniformity in the two-parametric family of the beta distributions. We show that for contamination models the Pitman efficiency approximates relative efficiency very well.

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.

Desk Editor's Note private letter to a colleague

This paper runs a numerical check showing Pitman efficiency tracks relative efficiency closely for uniformity testing inside the two-parameter beta family under chosen contamination models, but the claim stays tightly scoped to that setting.

read the letter

The core contribution is a concrete empirical verification that Pitman efficiency approximates relative efficiency well when testing uniformity on beta distributions with contamination. It supplies specific numbers for that parametric family and those models, which adds a data point to the long-standing efficiency literature rather than a new general framework. That targeted check is the part worth noting; it is a legitimate, if incremental, exercise in seeing how the asymptotic concept behaves in finite samples for this case. The work is honest about staying empirical and does not overclaim a universal interpretation. The main limitation is the narrow base. Everything is confined to the beta family and the particular contamination constructions chosen. No checks appear against other local alternatives, different contamination mechanisms, or even nearby parametric families, so it is unclear whether the close approximation is robust or tied to these selections. Without sensitivity analysis or an argument for why the pattern should extend, the result functions more as a case study than a broad empirical interpretation of Pitman efficiency. The derivations and numerical evidence would need referee scrutiny to confirm the reported closeness is not sensitive to modeling choices. This is the sort of paper a specialist in efficiency theory or goodness-of-fit testing might want to see in detail, but it will not shift the wider field. A serious editor should send it to review so the calculations can be examined, even though the scope keeps the impact modest.

Referee Report

2 major / 2 minor

Summary. The manuscript studies an empirical interpretation of Pitman efficiency for testing uniformity on [0,1] within the two-parameter beta family. It claims that, under specific contamination models, the Pitman efficiency approximates the relative efficiency very well, based on numerical comparisons inside this parametric setting.

Significance. If the reported approximation is robust and not an artifact of the chosen family and contaminations, the work would supply a concrete, falsifiable link between an asymptotic efficiency measure and finite-sample relative performance under contamination. This could be useful for practitioners selecting tests for uniformity. The restriction to beta distributions and particular contamination constructions, however, confines the result to a case study unless broader representativeness is demonstrated.

major comments (2)

[Numerical results / empirical section] The central empirical claim (Pitman efficiency approximates relative efficiency 'very well' for contamination models) rests on the representativeness of the two-parameter beta family together with the specific contamination constructions employed. No sensitivity checks against other local or fixed alternatives (e.g., normal mixtures, logistic, or exponential contaminations) are reported, which is load-bearing for any broader 'empirical interpretation' of Pitman efficiency.
[Abstract and Section 3] The abstract and main text present the closeness as an observed fact rather than a quantity derived from the model; without explicit statements of the contamination mechanisms, the exact definition of relative efficiency used, and any post-selection of models or parameters, it is impossible to verify that the approximation does not depend on choices internal to the study.

minor comments (2)

[Notation and setup] Notation for the beta parameters and the contamination parameter should be introduced once and used consistently; currently the transition between population parameters and contaminated versions is not always explicit.
[Figures] Figure captions should state the exact sample sizes, number of Monte Carlo replications, and the precise definition of 'relative efficiency' plotted on the vertical axis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed and constructive feedback on our manuscript. We respond to each major comment below and indicate the revisions we plan to implement.

read point-by-point responses

Referee: The central empirical claim (Pitman efficiency approximates relative efficiency 'very well' for contamination models) rests on the representativeness of the two-parameter beta family together with the specific contamination constructions employed. No sensitivity checks against other local or fixed alternatives (e.g., normal mixtures, logistic, or exponential contaminations) are reported, which is load-bearing for any broader 'empirical interpretation' of Pitman efficiency.

Authors: We acknowledge that our study is confined to the beta family and the chosen contamination models, making it a specific case study rather than a general proof. This limitation is inherent to the empirical nature of the work. To strengthen the manuscript, we will revise the discussion section to explicitly frame the results as an empirical observation within this parametric family and note the need for future investigations with other distributions. Additionally, we will include a brief sensitivity analysis by reporting results for one additional contamination type, such as a mixture with a logistic distribution, to demonstrate that the approximation holds similarly. We believe this addresses the concern without expanding the scope beyond what is feasible. revision: partial
Referee: The abstract and main text present the closeness as an observed fact rather than a quantity derived from the model; without explicit statements of the contamination mechanisms, the exact definition of relative efficiency used, and any post-selection of models or parameters, it is impossible to verify that the approximation does not depend on choices internal to the study.

Authors: We agree that greater explicitness is needed. In the revised manuscript, we will update the abstract to state that the approximation is observed under the specified contamination models within the beta family. In Section 3, we will provide a detailed description of the contamination mechanisms (e.g., the specific ways the alternatives are constructed), the exact formula for the relative efficiency (ratio of sample sizes needed to achieve a given power), and confirm that all models and parameter values were selected based on prior literature and standard practices, with no post-hoc selection or data-driven choices. This will allow readers to fully verify and replicate the findings. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical comparison with no self-referential derivation

full rationale

The paper performs a numerical study of Pitman efficiency versus relative efficiency for uniformity tests inside the two-parameter beta family under chosen contamination models. The abstract and description frame the result as an observed approximation obtained from direct computation, not as a quantity derived from itself or forced by fitted parameters presented as predictions. No equations, self-citations, or ansatzes are indicated that would reduce the claimed approximation to the input data or models by construction. The representativeness of the beta family is treated as an explicit modeling choice rather than smuggled in via prior self-work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; the paper is presumed to rest on standard definitions of Pitman efficiency and relative efficiency from the statistical literature, plus the usual regularity conditions for beta distributions and contamination models. No explicit free parameters, new axioms, or invented entities are mentioned.

axioms (2)

standard math Standard asymptotic properties of Pitman efficiency and relative efficiency under local alternatives
Invoked implicitly when comparing the two efficiency measures.
domain assumption Beta distributions form a suitable two-parameter family for studying uniformity testing
The family is chosen as the setting for the uniformity tests.

pith-pipeline@v0.9.0 · 5299 in / 1100 out tokens · 48521 ms · 2026-05-10T15:58:12.783476+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Kallenberg, W. C. M. (1983). Intermediate efficiency, theory and examples,Ann. Statist., 111401-1420

work page 1983
[2]

Inglot, T. (1999). Generalized intermediate efficiency of goodness of fit tests,Math. Me- thods Statist.,8487-509

work page 1999
[3]

and Ćmiel, B., (2019), Intermediate efficiency in nonparametric testing problems with an application to some weighted statistics,ESAIM, Probability and Statistics,23697-738

Inglot, T., Ledwina, T. and Ćmiel, B., (2019), Intermediate efficiency in nonparametric testing problems with an application to some weighted statistics,ESAIM, Probability and Statistics,23697-738

work page 2019
[4]

Lehmann, E. L. and Romano, J. P. (2008),Testing Statistical Hypotheses, Springer Texts in Statistics, Springer, New York, 3rd ed

work page 2008
[5]

(1995),Asymptotic Efficiency of Nonparametric Tests, Cambridge University

Nikitin, Y. (1995),Asymptotic Efficiency of Nonparametric Tests, Cambridge University

work page 1995
[6]

Noether, G. E. (1955), On a theorem of Pitman,Ann. Math. Statist.,2664-68

work page 1955
[7]

(1980),Approximation Theorems of Mathematical Statistics, Wiley, New York

Serfling, R., J. (1980),Approximation Theorems of Mathematical Statistics, Wiley, New York. Appendix. Proof of Theorem. Fix 0< α < β <1, a sequencesn→0+ and denotet αn,vαnexact critical values of both compared tests. (i)First we proof thatN T (α,β,Pγ(sn))→∞. Letp 0(x),ps(x) denote denisties ofPγ0,Pγ(s)with respect toλandκn =κnT =N T (α,β,Pγ(sn)). Byp n0,p...

work page 1980

[1] [1]

Kallenberg, W. C. M. (1983). Intermediate efficiency, theory and examples,Ann. Statist., 111401-1420

work page 1983

[2] [2]

Inglot, T. (1999). Generalized intermediate efficiency of goodness of fit tests,Math. Me- thods Statist.,8487-509

work page 1999

[3] [3]

and Ćmiel, B., (2019), Intermediate efficiency in nonparametric testing problems with an application to some weighted statistics,ESAIM, Probability and Statistics,23697-738

Inglot, T., Ledwina, T. and Ćmiel, B., (2019), Intermediate efficiency in nonparametric testing problems with an application to some weighted statistics,ESAIM, Probability and Statistics,23697-738

work page 2019

[4] [4]

Lehmann, E. L. and Romano, J. P. (2008),Testing Statistical Hypotheses, Springer Texts in Statistics, Springer, New York, 3rd ed

work page 2008

[5] [5]

(1995),Asymptotic Efficiency of Nonparametric Tests, Cambridge University

Nikitin, Y. (1995),Asymptotic Efficiency of Nonparametric Tests, Cambridge University

work page 1995

[6] [6]

Noether, G. E. (1955), On a theorem of Pitman,Ann. Math. Statist.,2664-68

work page 1955

[7] [7]

(1980),Approximation Theorems of Mathematical Statistics, Wiley, New York

Serfling, R., J. (1980),Approximation Theorems of Mathematical Statistics, Wiley, New York. Appendix. Proof of Theorem. Fix 0< α < β <1, a sequencesn→0+ and denotet αn,vαnexact critical values of both compared tests. (i)First we proof thatN T (α,β,Pγ(sn))→∞. Letp 0(x),ps(x) denote denisties ofPγ0,Pγ(s)with respect toλandκn =κnT =N T (α,β,Pγ(sn)). Byp n0,p...

work page 1980