A new class of tests for convex-ordered families based on expected order statistics

Mohammed Es-Salih Benjrada; Tommaso Lando

REVIEW 2 major objections 4 minor 40 references

A new class of tests for convex-ordered families based on expected order statistics

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A general class of nonparametric tests uses expected order statistics to decide whether an unknown distribution lies in a convex-ordered family relative to a known reference, with unbiasedness, monotone power, and consistency for heavy…

desk verdict Solid new tests for convex-ordered families with an honest heavy-tail extension; the finite-sample proof of monotone power has a real but repairable gap that needs a referee's attention. read the letter →

arxiv 2501.14075 v2 pith:MLFYK2MA submitted 2025-01-23 math.ST stat.TH

Tommaso Lando , Mohammed Es-Salih Benjrada This is my paper · ORCID

classification math.STstat.TH MSC 62G1062G2062G30

keywords convextransformorderexpectedstatisticsL-estimatornonparametrictesthazardrateheavytailsstochasticinfinitemean

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a general nonparametric test for whether an unknown distribution $F$ belongs to a family defined by the convex transform order with respect to a known reference distribution $G$, such as increasing hazard rate, increasing or decreasing odds rate, and decreasing reversed hazard rate families. The guiding observation is that if $G^{-1}\circ F$ is convex or concave, then the probability that $X$ falls below a certain expected order statistic is bounded by a known number $\pi^G_{j:m}$; the paper turns this into a test by estimating the expected order statistics with L-estimators and comparing the empirical non-exceedance probabilities to the bounds. The authors prove the resulting tests are unbiased, consistent, and have power that increases as $F$ moves away from $G$ in the convex order, and they show the consistency survives even when the mean is infinite by proving a strong law for the L-estimators under domain-of-attraction conditions. If the claims are correct, the method supplies the first tests for several recently studied families, works for any absolutely continuous $G$, and does not restrict the supports of $F$ and $G$.

What carries the argument

The engine is the identity $\pi^G_{j:m} = G(E(G^{-1}(B_{j:m})))$, where $B_{j:m}$ is a $\beta$$(j, m-j+1)$ random variable, together with the Jensen bound of Proposition 1 relating $P(X \leq \mu_{j:m})$ to this quantity. The expected order statistics $\mu_{j:m}(F)$ are estimated by L-estimators, weighted linear combinations of sample order statistics, and Theorem 1 gives an almost-sure strong law for these estimators when $E|X|$ is infinite, under maximum-domain-of-attraction and regular-variation conditions. The finite-sample theory rests on Lemma 1, which uses the linearly interpolated empirical quantile composition to show that $F \le_c H$ implies $\widetilde F_n(\mu_{j:m}(F_n)) \le_{\mathrm{st}} \widetilde H_n(\mu_{j:m}(H_n))$; this stochastic monotonicity is what carries the unbiasedness and monotone-power results for every sample size.

What would settle it

Simulate from a pair $F \le_c H$ (for example exponential versus Weibull with shape 1.5) and check every sample path for whether $\widetilde H_n^{-1}\circ\widetilde F_n$ is convex between consecutive order statistics; a single sample with a downward kink disproves the Lemma 1 premise, and Monte Carlo can then test whether rejection probabilities under $F$ and $H$ violate the claimed monotonicity.

Watch

Extended reading notes

Core claim

The central claim is that, for any absolutely continuous reference distribution $G$, the test statistics $T^{G+}_{m,p}$ and $T^{G-}_{m,p}$ --- the positive and negative parts of the gap between the bounds $\pi^G_{j:m}$ and the linearly interpolated empirical CDF evaluated at estimated expected order statistics --- solve the problem of testing $H_0: F$ belongs to the location-scale family of $G$ against the alternatives that $F$ is strictly convex- or concave-ordered with respect to $G$. Proposition 1 provides the load-bearing bound: $F \in \mathcal{F}^{cx}_G$ implies $P(X \leq \mu_{j:m}) \leq \pi^G_{j:m}$, and $F \in \mathcal{F}^{cv}_G$ implies the reverse inequality. The paper proves that these tests are unbiased and have monotone power at every fixed sample size, and that they are consistent both in the finite-mean case and, through a constrained version that discards non-converging ranks, for heavy-tailed distributions with infinite or undefined mean. It also reports simulations indicating that the tests work for the IHR/DHR, IOR/DOR, and DRHR/IRHR families, including detection of non-monotone hazard rates where the normalized-spacings test can be misleading.

Load-bearing premise

The finite-sample results depend on the claim that linearly interpolating the empirical distribution preserves the convexity of the transformed quantile function in Lemma 1; if a sample can make the smoothed quantile composition non-convex where the true composition is convex, unbiasedness and monotone power no longer follow.

Editorial extensions

If this is right

The same testing recipe applies to every absolutely continuous reference $G$, including uniform, exponential, negative exponential, log-logistic, Frechet, and Cauchy, with no support restrictions.
Families such as IOR, DOR, and DRHR, for which the paper says no comparable tests currently exist, become testable with guaranteed size control and consistency.
The constrained version of the test remains consistent when the mean is infinite, so a Cauchy or other very heavy-tailed $F$ can still be tested using ranks near the median, such as $\hat\mu_{2:3}$.
For the exponential reference, running the IHR and DHR versions together can detect a non-monotone hazard rate: the proposed tests reject in favour of both alternatives, while normalized-spacings tests may incorrectly report a single monotone direction.
The tests complement goodness-of-fit procedures: rejecting $H_0$ while a goodness-of-fit test accepts the shape class provides evidence that $F$ lies in the convex-ordered family outside the location-scale family.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension, not developed in the paper, is to automate the choice of safe ranks by estimating the tail index from the sample (for example with a Hill-type estimator) and then selecting $m$ and $\ell$ data-adaptively rather than from a prior lower bound on $\alpha$.
The same bounds could support confidence statements about shape rather than only tests: the distance between $\pi^G_{j:m}$ and $F(\mu_{j:m})$ is a population measure of how far $F$ is from the location-scale family, and its L-estimator could be used to construct an effect-size summary.
Because the tests are location- and scale-invariant and do not restrict support, they could be applied iteratively to narrow down the shape of a distribution, as the paper does for river-flow data; this suggests a model-selection workflow where families are accepted or rejected one by one.
The strictness of Jensen's inequality in the consistency proof means the tests distinguish the location-scale family from the rest of the convex-ordered family; they do not by themselves certify membership in the shape class, so a combined use with existing goodness-of-fit tests is the intended reading.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: this is a solid, genuinely new testing method for convex-ordered families (IOR, DOR, DRHR, IHR/DHR), and it fills a real gap—most families in this class had no tests at all. The central idea is simple and sound: if G^{-1}∘F is convex, Jensen gives bounds on P(X ≤ μj:m) in terms of the reference distribution G, and the paper builds a test by comparing empirical non-exceedance probabilities at estimated expected order statistics against those bounds. The heavy-tail extension (a strong law for L-estimators when the mean is infinite, under max-domain-of-attraction conditions) is a genuine contribution, and the simulations are honest: they openly report that Proschan–Pyke has higher power under genuinely monotone hazard rates, while their test handles non-monotone hazard rates and new families. Code is on GitHub, and the real-data example is a nice use case. The citation pattern is mostly the authors' own prior work—the key bound is from their JAP paper—but it is a published, parameter-free Jensen inequality, and the testing machinery built on it is new, so the self-citation does not bother me.

The soft spots, in proportion. First, the proof of Theorem 2, which carries the finite-sample unbiasedness and monotone-power claims, is not valid as written. It establishes componentwise stochastic inequalities via Lemma 1 and then invokes Theorem 1.A.3 of Shaked–Shantikumar, which requires the coordinates within each vector to be independent. Here the coordinates are all functions of the same empirical CDF. The gap is real and repairable: the coupling inside the proof of Lemma 1 (F_n = H_n∘H^{-1}∘F) gives the inequality for all j simultaneously on the same probability space, so the joint stochastic order follows in one line. But the paper does not present that argument, and a referee should insist on it. Second, the interpolation step in Lemma 1 is terse but actually fine—piecewise linear interpolation of points on a convex function is convex by the chord-slope property—so I would not worry about that one. Third, the constrained heavy-tail test needs the user to know or estimate the tail index α to choose ℓ; the paper says so in Section 5.3, but the abstract's "broadly applicable" claim is a bit strong. Fourth, there is a sign-flipped exponent in a displayed integral in the proof of Theorem 1; the stated convergence condition is correct, so this is a typo, not a mathematical error.

Net: the central argument holds up, the proof gap is genuine but clearly fixable, and the paper is honest about its limitations. It deserves a serious referee. I would send it to review and ask for a corrected proof of Theorem 2, the typo fixed, and a short paragraph on how to choose ℓ when α is unknown.

Referee Report

2 major / 4 minor

Summary. This paper proposes a class of nonparametric tests for convex-ordered families F^cx_G and F^cv_G, with null hypothesis H^G_0: F belongs to the location-scale family of a known absolutely continuous reference distribution G. The test statistics are L_p-norms of the positive or negative parts of the vector pi^G_{j:m} - \tilde F_n(\hat\mu_{j:m}(F_n)), where pi^G_{j:m} = G(E(G^{-1}(B_{j:m}))) and \hat\mu_{j:m} is an L-estimator of the expected order statistic. The paper proves an almost-sure limit theorem for these L-estimators under extreme-value domain-of-attraction conditions, including cases where the mean is infinite, and claims finite-sample unbiasedness and monotone power through Lemma 1 and Theorem 2, plus consistency through Propositions 5 and 6. Simulations compare the method with the Proschan-Pyke test and apply it to river flow data.

Significance. If the finite-sample claims are properly established, this is a useful contribution: it provides a unified testing method for several shape-constrained families, including IOR, DOR, DRHR and IRHR, for which direct tests are largely unavailable, and it does so without support restrictions and with robustness to infinite means. Theorem 1, the strong law for L-estimators under heavy tails, is a valuable standalone result, and the paper is transparent about code and simulations. The central inequality Proposition 1 is taken from the authors' prior published work, but it is a Jensen-type bound rather than a fitted quantity, so I do not see a circularity problem. The main caveat is that the proof of the finite-sample stochastic-order result is incomplete as written, so the advertised unbiasedness and monotone power properties are not yet logically guaranteed by the text.

major comments (2)

[Section 4.1, proof of Theorem 2] The proof passes from componentwise stochastic inequalities \tilde U_j ≤_st \tilde Z_j, obtained from Lemma 1, to stochastic ordering of the increasing function ψ_p(z) = ||z||_p by invoking Theorem 1.A.3 of Shaked and Shanthikumar. That theorem requires the coordinates within each vector to be independent. Here the coordinates are all functions of the same empirical CDF H_n (or F_n), so they are dependent, and componentwise stochastic dominance does not imply multivariate stochastic dominance for dependent coordinates. This is a load-bearing gap because Corollary 2 and the advertised finite-sample unbiasedness and monotone power rest on Theorem 2. The gap appears repairable: the proof of Lemma 1 constructs a coupling F_n = H_n ∘ H^{-1} ∘ F for which the inequalities hold pointwise for all j simultaneously, and applying ψ_p to both sides of that pointwise vector inequality would establish the desired stochastic order directly. The repair should be written into the proof.
[Section 4.1, Theorem 2(2)] The second displayed inequality in Theorem 2 has the wrong direction. Under F ≤_c H, Lemma 1 gives \tilde F_n(μ_{j:m}(F_n)) ≤ \tilde H_n(μ_{j:m}(H_n)). Since x ↦ (π^G_{j:m} - x)_- is decreasing, the componentwise inequality reverses for the statistic T^{G-}_{m,p}. The paragraph after Lemma 1 also states that under F ≤_c H the rejection probability for H^{G}_{1-} is smaller under F than under H, and Corollary 2(2) requires P(T^{G-}_{m,p}(F_n) ≥ c^-_{α,n}) ≤ P(T^{G-}_{m,p}(H_n) ≥ c^-_{α,n}) when F ≤_c H. Therefore part 2 of Theorem 2 should read ≤, not ≥, or should be reformulated with the order reversed. As written, the theorem contradicts Corollary 2(2) when H = G and F ≤_c G.

minor comments (4)

[Section 4.1, proof of Lemma 1] The statement that \tilde H_n^{-1} ∘ \tilde F_n 'is a convex function, interpolating H_n^{-1} ∘ F_n' needs one explicit justification: this composition is the piecewise-linear interpolant of the convex function H^{-1} ∘ F at the increasing knots x_i, and a piecewise-linear interpolant of a convex function with increasing knots is convex because its successive slopes are nondecreasing. Adding this sentence would make the proof easier to follow.
[Section 4.2, proof of Theorem 1] Several displays in the proof of Theorem 1 have lost their superscript formatting. In particular, the expression X_{n-k+1:n} n^{-1-(m-j)} k^{m-j} and the conditions E(h,j,m) and Q(h,j,m) are hard to parse; the exponents should be typeset explicitly so the reader can verify the convergence conditions.
[Section 5.4 and Section 6] The references to 'Table 2b-(a)' and 'Table 2b-(b)' are confusing; the table should be given distinct sublabels or separate tables. There are also small typos, such as 'well-kown' in Section 2.1 and 'Kologorov-Smirnov' in Section 6, which should be corrected.
[Section 4.2] The constrained test statistic T^{G+}_{m,ℓ,p} is introduced with notation that is not fully defined before first use; please state explicitly that ℓ is the number of selected indices j_k and that the indices are chosen so that the corresponding L-estimators converge under Theorem 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the central inequality is a parameter-free Jensen bound derived in the text, and the test construction does not fit or reuse its own output. A minor self-citation is not load-bearing; a separate proof gap in Theorem 2 is a correctness concern, not circularity.

full rationale

The paper's central claim rests on Proposition 1: if G^{-1}∘F is convex then P(X ≤ μ_{j:m}) ≤ G(E(G^{-1}(B_{j:m}))). The paper itself derives this from Jensen's inequality in Section 2.2 ('Jensen's inequality implies that EX_{j:m} ≤ F^{-1}∘G(E(G^{-1}∘F(X_{j:m}))) = F^{-1}∘G(E(G^{-1}(B_{j:m})))'), then cites Arab et al. (2025) only as a summary of the property. This is a parameter-free bound whose assumptions do not include the test's conclusions, so the self-citation is not load-bearing. The test statistics T^{G+}_{m,p} and T^{G-}_{m,p} compare empirical non-exceedance probabilities with these fixed bounds; no parameter is fitted to data and then renamed as a prediction, and the consistency proofs (Propositions 5 and 6) use standard Glivenko-Cantelli and L-estimator convergence arguments rather than assuming the claimed result. Lemma 1 and Theorem 2 attempt a stochastic-monotonicity proof of unbiasedness and monotone power; here the text contains a genuine correctness gap: the proof of Theorem 2 invokes Theorem 1.A.3 of Shaked and Shantikumar (2007) while stating only componentwise stochastic order, omitting that theorem's independence condition, which the dependent coordinates eU_j and eZ_j (all functions of the same empirical CDF) do not satisfy. This is repairable via the coupling in Lemma 1, but as written it is a missing-support issue, not a circular equation. Similarly, the finite-sample properties of the constrained statistics T^{G+}_{m,ℓ,p} are asserted without proof ('It is easy to check...'), another omitted-support issue. No step in the derivation reduces a claimed prediction to an input by construction, so the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities or fitted constants. The test family has tuning parameters (m, p, ℓ) and relies on standard regularity assumptions plus tail-domain conditions from extreme value theory. The main external ingredient is Mason's strong law for L-estimators.

free parameters (3)

m (number of order statistics in test statistic) = user-selected; simulations suggest 10-20% of n for IHR/IOR, larger for heavy-tailed DOR
Tuning parameter of the test family; affects power and the set of j available under Theorem 1.
p (norm order) = p=1 chosen after simulations (Table 1)
Lp norm parameter; p=1 recommended from simulation power comparisons, not derived from theory.
ℓ (number of order statistics used in constrained heavy-tailed test) = user-selected, e.g., ℓ=m-2 for IOR; for DOR m=25,ℓ=5 etc.
Subset of order statistics for which L-estimator convergence is guaranteed; depends on tail index α.

assumptions (5)

domain assumption G is absolutely continuous with known density g
Used to define generalized hazard rate and πG_j:m; if G has atoms the convex transform framework changes.
standard math F is absolutely continuous or has a density a.e. under the alternative
Convexity of G^{-1}∘F implies F has a density a.e., cited from Marshall and Olkin, Proposition C.2.
standard math Jensen's inequality applied to convex/concave transforms to obtain Proposition 1 bounds
Core bound on P(X ≤ μj:m) from Arab et al. (2025).
standard math Mason (1982) strong laws for linear functions of order statistics
Provides a.s. convergence of L-estimators under tail conditions in the proof of Theorem 1.
domain assumption F ∈ D+(Φα) or D-(Φβ) domain of attraction conditions for heavy-tailed consistency
Assumed for consistency in the infinite-mean case; these are tail regularity assumptions that can be checked or estimated.

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Cite this review

Pith. "Pith review of A new class of tests for convex-ordered families based on expected order statistics." pith.science (2026). https://pith.science/paper/MLFYK2MA

@misc{pith2026250114075,
  author       = {Pith},
  title        = {Pith review of: A new class of tests for convex-ordered families based on expected order statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MLFYK2MA}},
  note         = {Machine review of arXiv:2501.14075}
}

abstract

Consider a pair of cumulative distribution functions $F$ and $G$, where $F$ is unknown and $G$ is a known reference distribution. Given a sample from $F$, we propose tests to detect the convexity or the concavity of $G^{-1}\circ F$ versus equality in distribution (up to location and scale transformations). This framework encompasses well-known cases, including increasing hazard rate distributions, as well as some other relevant families that have garnered attention more recently, for which no tests are currently available. We introduce test statistics based on the estimated probability that the random variable of interest does not exceed a given expected order statistic, which, in turn, is estimated via L-estimation. The tests are unbiased, consistent, and exhibit monotone power with respect to the convex transform order. To ensure consistency, we show that our L-estimators satisfy a strong law of large numbers, even when the mean is not finite, thereby making the tests suitable for heavy-tailed distributions. Unlike other approaches, these tests are broadly applicable, regardless of the choice of $G$ and without support restrictions. The performance of the method under various conditions is demonstrated via simulations, and its applicability is illustrated through a concrete example.

Figures

Figures reproduced from arXiv: 2501.14075 by the authors.

**Figure 2.** Simulated power of the proposed IOR tests under [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗

**Figure 3.** Simulated power of the proposed DOR tests under [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Simulated power of the proposed IHR tests compared to that of P&P under [PITH_FULL_IMAGE:figures/full_fig_p031_4.png]

**Figure 5.** Figure 5: Simulated power of the proposed tests evaluated for different combinations of [PITH_FULL_IMAGE:figures/full_fig_p032_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

, year 2013

author Anis, M. , year 2013 . title A family of tests for exponentiality against ifr alternatives . journal Journal of Statistical Planning and Inference volume 143 , pages 1409--1415

work page 2013

[2] [2]

, author Basu, K

author Anis, M. , author Basu, K. , year 2014 . title Tests for exponentiality against NBUE alternatives: a M onte C arlo comparison . journal Journal of Statistical Computation and Simulation volume 84 , pages 231--247

work page 2014

[3] [3]

, author Lando, T

author Arab, I. , author Lando, T. , author Oliveira, P. , year 2025 . title Inequalities and bounds for expected order statistics from convex ordered families . journal Journal of Applied Probability , pages 1--19

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