{"id":"e8c7e001-db73-43f6-94ff-14f11ef4da31","arxiv_id":"1908.08243","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Expectile-based skewness measures s2(alpha) and a stop-loss skewness function S_X(t) are introduced with asymptotic theory and simulations.","lead":"The authors introduce skewness measures based on expectiles, smoothed versions of quantiles, and prove that their empirical estimators are consistent and asymptotically normal. The measures are designed to avoid the tail instability of moment skewness and the inference problems of quantile skewness.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 11's displayed asymptotic variance misses a factor of 2 in the A(α)A(1−α)η(α,1−α) cross term; the delta-method quadratic form gives 2A(α)A(1−α)η, so interval (19) is mis-sized if the formula is faithful.","rationale":"The paper's central contribution has two pillars: a normalized expectile-based skewness functional with a symmetry characterization, and an asymptotic theory for its plug-in estimators. I checked the functional definitions and the ordering material: Proposition 4's bounds, the α→1/2 limit to 2F(μ)−1, and Theorem 9's crossing argument are internally coherent, although Theorem 9's proof is terse. The unresolved S3 property for s2 is explicitly labeled a conjecture, so it is not a hidden claim. The one place where a stated theorem seems numerically false is the variance formula in Theorem 11: a routine delta-method computation yields a factor of 2 on the expectile-expectile cross term that the displayed formula omits. Because Theorem 11 and equation (19) are what justify the confidence bands in Figure 2 and any inference using s2, this is load-bearing. The reader's uncertainty about the variance formula is on target, though the specific defect is different from the heavy-tail caveat identified in the reader's weakest_assumption. If the missing factor is confirmed, the paper should be accepted only after the formula is corrected; the rest of the central claims are not threatened.","tokens_in":17904,"tokens_out":22807,"duration_ms":218661,"concrete_test":"Symbolically form σ²=g^TΣg for a distribution with non-negligible η(α,1−α), for example Exp(1) with α=0.1, and compare the result with Theorem 11. Then simulate n=5000 with 10^5 replications and estimate the coverage probability of interval (19) using the paper's formula and the corrected coefficient 2A(α)A(1−α)η(α,1−α). If the displayed formula gives coverage materially below 0.95 while the corrected formula gives nominal coverage, the displayed formula is confirmed to be wrong.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Let L=e_X(α), U=e_X(1−α), D=U−L, c=1−2α, and write s2=(U+L−2μ)/(cD). The gradient of s2 is ∂s2/∂L=2(U−μ)/(cD²), ∂s2/∂U=2(μ−L)/(cD²), and ∂s2/∂μ=−2/(cD). For empirical expectiles, the asymptotic covariance of √n(ê(τ_i)−e(τ_i)) and √n(ê(τ_j)−e(τ_j)) is η(τ_i,τ_j)/(S(τ_i)S(τ_j)), where S(τ)=−τ−(1−2τ)F(e(τ)), so S(1/2)=−1/2. The contribution of the covariance between the two expectile estimators to σ² is 2·[2(U−μ)/(cD²)]·[2(μ−L)/(cD²)]·η(α,1−α)/(d_α d_u)=8A(α)A(1−α)η(α,1−α)/(c²D⁴). After factoring out 4/c², the bracket in Theorem 11 therefore needs the coefficient 2 in front of A(α)A(1−α)η(α,1−α); the displayed formula has coefficient 1. The mean-expectile cross terms appear correct. If the displayed formula is faithful, the confidence interval (19) and any coverage claims based on it are invalid. The issue is confined to Theorem 11 and is easy to fix, but it is load-bearing for the paper's asymptotic-inference contribution.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a family of expectile-based skewness measures, primarily s2(α) = (e_X(1−α)+e_X(α)−2μ) / ((1−2α)(e_X(1−α)−e_X(α))), and studies their theoretical and empirical properties. The authors show that these measures are normalized on (−1,1) with sharp bounds, characterize symmetry, are linked to Omega ratios and stop-loss transforms, and that a scale-invariant skewness function preserves the convex transformation order. They also derive plug-in estimators, prove strong consistency and asymptotic normality, provide an explicit asymptotic variance formula, and compare the new measures with quantile-based and moment-based skewness in simulations.","tokens_in":18202,"tokens_out":8237,"duration_ms":74083,"significance":"If the results are correct, the paper makes a useful contribution to the skewness-measurement literature: the expectile-based family avoids the heavy-tail sensitivity of the third central moment, is fully defined for finite-mean distributions, characterizes symmetry, and has convenient asymptotic inference without density estimation. The connection to Tajuddin's s3 as a limiting case, the sharp bounds in Proposition 4, and the ordering result in Theorem 9 are valuable. The paper also includes explicit proofs and a simulation study. The main mathematical claims are plausible and largely well supported, but the asymptotic variance in Theorem 11 contains an algebraic error that needs correction before the inference results can be used.","major_comments":[{"comment":"The displayed formula for σα^2 in Theorem 11 is missing a factor 2 in the covariance term between the two expectile estimators. Let L=e_X(α), U=e_X(1−α), D=U−L, c=1−2α, and S(τ)=τ+(1−2τ)F(e_X(τ)). The gradient of s2 with respect to (L,U,μ) is (2(U−μ)/(cD^2), 2(μ−L)/(cD^2), −2/(cD)). Using the notation A(α)=(U−μ)/S(α) and A(1−α)=(μ−L)/S(1−α), the covariance contribution between the two expectiles is 2·(2A(α)S(α)/(cD^2))·(2A(1−α)S(1−α)/(cD^2))·η(α,1−α)/(S(α)S(1−α)) = 8A(α)A(1−α)η(α,1−α)/(c^2D^4). After factoring out 4/c^2, the bracket must contain 2A(α)A(1−α)η(α,1−α), not A(α)A(1−α)η(α,1−α). As a consequence, the plug-in variance estimator and the confidence intervals (19) and (20) are mis-sized if the displayed formula is taken literally. The asymptotic normality itself remains valid; the fix is local, but it is load-bearing for the paper's inference contribution.","section":"Theorem 11 and Eq. (19)"}],"minor_comments":[{"comment":"The definition of p̂_t uses the true mean μ in the indicator 1_{μ−t<X_i≤μ+t}, while the displayed expression for σ̂_t^2 uses X_i−X̄ alongside p̂_t. For a genuine plug-in estimator, p̂_t should be defined with X̄ in place of μ.","section":"Section 5.1, after Theorem 13"},{"comment":"In Remark 10.2, the phrase 'the limiting measure s3(α)' is misleading because s3 does not depend on α; this should be simply s3.","section":"Section 3.1"},{"comment":"There is a typo: 'decrasing' should be 'decreasing'.","section":"Section 6.1"},{"comment":"In the reference to Bellini, Klar, and Müller (2018), the umlaut is missing ('Mller' should be 'Müller').","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The missing factor 2 in Theorem 11 is a straightforward algebraic correction, but it directly affects the stated confidence intervals and the plug-in variance estimator, so I recommend a major revision. The rest of the paper is technically sound in my reading. The reliance on Holzmann and Klar (2016), co-authored by one of the present authors, is prior published machinery and is acknowledged appropriately; the novel contribution is the application and the delta-method derivation, not the expectile asymptotics themselves."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuinely useful paper on expectile-based skewness measures, and the reader's ACCEPT-with-moderate-confidence is roughly right, but the stress-test note is correct. Theorem 11's displayed variance misses a factor of 2 on the A(α)A(1−α)η(α,1−α) cross term, so confidence interval (19) is mis-sized as written. That is a fixable, localized error, but it sits in the paper's main asymptotic-inference result.\n\nWhat is actually new: the s2(α) family, the stop-loss skewness function S_X(t), and the order ≤δ_μ. The convex-transform-order characterization (2) is a genuinely useful proof device. Proposition 4's sharp bounds are clean, and Theorem 9's ordering result appears correct and is the strongest positive claim in the paper. The paper is honest that the S3 property for s2 is only a conjecture, supported numerically; that is not a flaw. The connection to Omega ratios in Section 4 is neat and likely useful in finance.\n\nThe factor-of-two issue: let L=e_X(α), U=e_X(1−α), D=U−L, c=1−2α. The gradient of s2 is (2(U−μ)/(cD²), 2(μ−L)/(cD²), −2/(cD)). Using the standard expectile covariance η(τ_i,τ_j)/(S(τ_i)S(τ_j)), the S factors cancel against the A terms, and the expectile-expectile cross term contributes 8A(α)A(1−α)η(α,1−α)/(c²D⁴). After factoring out 4/c², the bracket in Theorem 11 needs 2A(α)A(1−α)η(...), not 1. The other terms match my computation. Because the proof of Theorem 11 is not given, a referee cannot see where the factor went; the authors should supply the delta-method computation and correct the statement.\n\nOther soft spots are minor. The CLT assumes EX²<∞ while one motivation is heavy tails; strong consistency under E|X|<∞ (Cor 12) is fine, but the asymptotic intervals do not cover infinite-variance cases. The simulation study is useful but no code or data are supplied, so I could not reproduce the figures. There are small notational slips in the confidence-level symbols around (20) and (22).\n\nWho it is for: statisticians working on distributional shape, expectile/quantile inference, and performance measures in finance. It deserves to go to referees. My recommendation: accept after minor revision, and require the variance formula to be corrected and proved.","headline":"Solid, useful paper on expectile-based skewness; the main asymptotic variance formula has a factor-of-two error that needs fixing before it can be used as stated.","tokens_in":18791,"tokens_out":8846,"would_cite":true,"duration_ms":78824,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G05","62G20","60E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Expectile ratios define a skewness measure that is bounded, exists for all finite-mean distributions, and is zero exactly for symmetric ones.","keywords":["expectiles","skewness measure","asymmetric least squares","Omega ratio","stop-loss transform","convex transform order","asymptotic normality","symmetry characterization"],"falsifier":"Draw many samples from a skewed distribution with finite mean but infinite variance, such as a Pareto distribution with shape between 1 and 2, compute $\\hat{s}_{2,n}(\\alpha)$ and its nominal 95% confidence intervals using $\\hat{\\sigma}_\\alpha$, and check whether the empirical coverage stays near 0.95; a large shortfall would show that the normal limit in Theorem 11 does not cover the motivating heavy-tailed case.","tokens_in":17643,"feed_emoji":"⚖️","tokens_out":9021,"duration_ms":73543,"temperature":0.7,"pith_summary":"Expectiles, the asymmetric least-squares analogues of quantiles, can be turned into a family of skewness measures that are normalized, always defined for finite-mean non-degenerate distributions, and zero exactly for symmetric ones. The paper proposes the family $s_2(\\alpha) = (e_X(1-\\alpha)+e_X(\\alpha)-2\\mu) / ((1-2\\alpha)(e_X(1-\\alpha)-e_X(\\alpha)))$ and shows that it characterizes symmetry, is bounded between $-1$ and $1$, and flattens to the limiting value $s_3 = 2F(\\mu)-1$ as $\\alpha\\to 1/2$. The plug-in estimator is strongly consistent whenever $E|X|<\\infty$ and asymptotically normal with an explicit variance under a finite second moment. A related scale-invariant skewness function, built from $\\Omega$ ratios and stop-loss transforms, is shown to respect the convex transformation order of distributions. This matters because the classical third-moment skewness is tail-dominated, hard to estimate for heavy-tailed data, and zero does not characterize symmetry.","feed_headline":"Expectile ratio measures skewness without relying on third moments","feed_subtitle":"The measure is always defined for finite-mean distributions and its estimator stays consistent for heavy-tailed data.","key_machinery":"The central object is the expectile skewness family $s_2(\\alpha)$, a normalized ratio of the expectile deviations $e_X(1-\\alpha)-\\mu$ and $\\mu-e_X(\\alpha)$. The expectile $e_X(\\alpha)$ is the minimizer of the asymmetric quadratic loss $\\alpha E[(X-t)_+^2]+(1-\\alpha)E[(X-t)_-^2]$, acting as a smoothed quantile and making the ratio sensitive to the whole distribution. The companion skewness function is $S_X(t)=\\frac{1}{t}\\{\\pi_X(\\mu+t)-\\pi_X(\\mu-t)\\}+1$, expressed through the stop-loss transform $\\pi_X(t)=E(X-t)_+$ and the $\\Omega$ ratio $\\Omega_X(t)=E(X-t)_+/E(X-t)_-$; its scale-invariant version $\\tilde{S}_X(t)=S_X(t\\delta_X)$ carries the convex-order comparison in Theorem 9.","core_discovery":"The paper's central claim is that the expectile function $e_X(\\alpha)$, defined by minimizing asymmetric quadratic loss, encodes skewness in a normalized ratio of upper and lower deviations from the mean. A distribution is symmetric exactly when $e_X(1-\\alpha)-\\mu = \\mu-e_X(\\alpha)$ for all $\\alpha\\in(0,1/2)$; the family $s_2(\\alpha)$ turns that condition into a bounded scalar between $-1$ and $1$. The paper proves that $s_2$ is location- and scale-invariant, derives its limiting value $s_3=2F(\\mu)-1$, and establishes that the plug-in estimator $\\hat{s}_{2,n}(\\alpha)$ is strongly consistent under $E|X|<\\infty$ and asymptotically normal with explicit variance under $EX^2<\\infty$ when the distribution has no point mass at the relevant expectiles. For the companion scale-invariant skewness function $\\tilde{S}_X(t)$, it proves monotonicity with respect to a skewness order that is weaker than the convex transformation order but stronger than a known weak skewness order.","pith_inferences":["If the paper's conjecture that $s_2$ itself satisfies van Zwet's convex-order property (S3) is confirmed, the family would inherit the strongest standard skewness-ordering guarantee rather than only its limiting case.","Because $s_2(\\alpha)$ is a smooth functional of the whole expectile curve, scanning a grid of $\\alpha$ values could detect skewness concentrated in particular regions of the distribution, a feature quartile-based measures miss.","The finite-variance requirement for asymptotic normality suggests a bootstrap or heavy-tailed central-limit extension would be needed to justify confidence intervals for very heavy-tailed data; the paper's own consistency result is weaker.","The connection to Omega ratios means the same construction could be imported directly into portfolio performance and risk measurement, where Omega ratios are already standard."],"forward_implications":["Skewness can be estimated and compared for heavy-tailed distributions with finite mean, where the classical moment measure is unstable or undefined.","Asymptotic confidence intervals for $s_2(\\alpha)$ follow from the delta method under a finite second moment, avoiding the density estimation needed for quantile-based skewness.","The limiting value $s_3=2F(\\mu)-1$ links the family to the sign-test statistic and gives a parameter-free representative of $s_2(\\alpha)$ near $\\alpha=1/2$.","The scale-invariant function $\\tilde{S}_X$ is monotone under the convex transformation order, so comparisons of skewness agree with a standard stochastic ordering of distributions."],"supporting_citations":[{"why":"defines expectiles as minimizers of an asymmetric quadratic loss and gives the identification condition used throughout the paper.","marker":"Newey and Powell (1986)"},{"why":"supplies the asymptotic normality of empirical expectiles on which the delta-method result for $\\hat{s}_{2,n}(\\alpha)$ relies.","marker":"Holzmann and Klar (2016)"},{"why":"introduced the convex transformation order and the skewness axioms S1-S3 used to evaluate the new measures.","marker":"van Zwet (1964)"},{"why":"established the convex-transform-order property for the quantile skewness measure $b_2$, the analogue of the new expectile measure.","marker":"Groeneveld and Meeden (1984)"},{"why":"provides the crossing conditions and skewness orderings that underpin Theorem 9.","marker":"Oja (1981)"},{"why":"defines the Omega ratio that connects expectiles to the equivalent skewness condition in terms of stop-loss transforms.","marker":"Keating and Shadwick (2002)"}],"fun_headline_variants":["Expectile ratio quantifies skewness, sidesteps tail issues","Bounded skewness via expectiles, no third moment needed","Expectile-based skewness: smooth, robust, simple","New skewness measure from expectiles beats moment and quantile","Expectile ratio: skewness that stays finite for heavy tails"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The asymptotic normality results require a finite second moment and no point mass exactly at the expectiles being used; when that fails, the practical confidence intervals are not justified, and only consistency under a finite mean remains.","fun_headline_variants_meta":{"raw":{"variants":["Expectile ratio quantifies skewness, sidesteps tail issues","Bounded skewness via expectiles, no third moment needed","Expectile-based skewness: smooth, robust, simple","New skewness measure from expectiles beats moment and quantile","Expectile ratio: skewness that stays finite for heavy tails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000343,"raw_usage":{"total_tokens":1875,"prompt_tokens":923,"completion_tokens":952,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":866}},"tokens_in":539,"tokens_out":952,"duration_ms":6339,"temperature":1.0,"reasoning_tokens":866,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:47:36.762122+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Draw many samples from a skewed distribution with finite mean but infinite variance, such as a Pareto distribution with shape between 1 and 2, compute $\\hat{s}_{2,n}(\\alpha)$ and its nominal 95% confidence intervals using $\\hat{\\sigma}_\\alpha$, and check whether the empirical coverage stays near 0.95; a large shortfall would show that the normal limit in Theorem 11 does not cover the motivating heavy-tailed case.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines expectiles as minimizers of an asymmetric quadratic loss and gives the identification condition used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the asymptotic normality of empirical expectiles on which the delta-method result for $\\hat{s}_{2,n}(\\alpha)$ relies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduced the convex transformation order and the skewness axioms S1-S3 used to evaluate the new measures."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"established the convex-transform-order property for the quantile skewness measure $b_2$, the analogue of the new expectile measure."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the crossing conditions and skewness orderings that underpin Theorem 9."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the Omega ratio that connects expectiles to the equivalent skewness condition in terms of stop-loss transforms."}],"review_version":1}