{"id":"c73ba3e8-f8d9-4868-984c-cb4b00d089d9","arxiv_id":"2506.10370","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Method-of-moments SNR estimators for multivariate high-dimensional linear models are shown to be asymptotically normal, with plug-in standard errors for inference.","lead":"This paper extends method-of-moments estimation of signal-to-noise ratios from single-response linear models to multivariate responses, and gives formulas for confidence intervals in fixed-effects, random-effects, and heteroskedastic settings. If the theorems hold, researchers estimating heritability from many correlated phenotypes can quantify uncertainty instead of reporting a point estimate.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's CLT is mis-normalized: the displayed V is Cov(σ̂²,ρ̂²)=O(1/n), so n^{1/2}V^{-1/2}(θ̂-θ) diverges; the statement requires V to be n·Cov (O(1)), else the claimed normal convergence is false.","rationale":"The reader's weakest-assumption pick — the unproved total-variation bound in the proof of Theorem 3.1 — is legitimate and should be addressed. However, a more concrete and load-bearing problem appears in the same theorem's statement: the displayed V matrix has the wrong scale. Since Lemma 6.7 provides the covariance of (σ̂²,ρ̂²), which is O(1/n), writing n^{1/2}V^{-1/2}(θ̂-θ) yields a quantity with variance ~ n·I, so it cannot converge to N(0,I). This makes Theorem 3.1 false as written, regardless of the TV step. The fix is simple — replace V by nV in the theorem or change the normalization to V^{-1/2}(θ̂-θ) — and the numerical experiments apparently use the correct standard-error scale, so the empirical conclusions are probably unaffected. I therefore keep the reader's CONDITIONAL verdict, with the added condition that the scaling error in Theorem 3.1 be corrected and the proof of the TV bound supplied.","tokens_in":53441,"tokens_out":13356,"duration_ms":141353,"concrete_test":"Set q=1, B⊤B=ρ², and Σe=σ² in Theorem 3.1, and compute σ_r² from (3.1) using the displayed V. Compare with Dicker (2014), Theorem 1: the asymptotic variance of n^{1/2}(r̂²-r²) is O(1). If σ_r² from the paper is O(1/n), then σ_r=O(n^{-1/2}) and the CLT as stated cannot hold, so Theorem 3.1 needs V ← nV. Algebraically, verify that Var(n^{1/2}V^{-1/2}(θ̂-θ)) ≈ n·I, not I, when V is the covariance matrix from Lemma 6.7.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.1 states n^{1/2}V^{-1/2}(θ̂-θ) ⇒ N(0,I₂) with θ̂=(σ̂²,ρ̂²), but the displayed V coincides with the finite-sample covariance in Lemma 6.7, which is O(1/n) under Assumption 3.1. For example, V11 = 2/[q²(n+1)²n]{(n²+np)∥B⊤B∥²_F + 2pn tr(ΣeB⊤B) + (n²+np)tr(Σe²)} ≈ 2(1+τ)∥B⊤B∥²_F/(q²n) = O(1/n). Consequently V^{-1/2}=O(n^{1/2}), and n^{1/2}V^{-1/2}=O(n), while θ̂-θ=O_P(n^{-1/2}); the product diverges in probability. The theorem becomes correct only with V replaced by nV (or the n^{1/2} multiplier removed). This is a demonstrable scaling error in the central fixed-effects result, not merely a proof gap: any z-statistic built from the stated formula would be inflated by a factor of roughly √n. It also breaks the paper's claim that Theorem 3.1 recovers Dicker's univariate result: for q=1, the σ_r defined in (3.1) is O(n^{-1/2}), whereas Dicker's asymptotic variance for n^{1/2}(r̂²-r²) is O(1). The same issue does not affect Theorems 3.2 and 4.1, whose V entries are O(1) for fixed q, making Theorem 3.1 the outlier. The reader's total-variation concern is a real proof gap, but the normalization error is more immediate and falsifiable.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends method-of-moments SNR estimation to multivariate high-dimensional linear models under both fixed and random effects. It defines the SNR as r² = ρ²/(ρ²+σ²), proposes estimators (σ̂², ρ̂²) based on quadratic-form moment equations, and states asymptotic normality results with explicit covariance matrices in Theorem 3.1 (fixed effects), Theorem 3.2 (random effects), and Theorem 4.1 (heteroskedastic random effects). The paper also provides plug-in standard-error procedures, an operator-norm consistency result for covariance estimators under q = o(n) in Theorem 4.2, and extensive simulations for Gaussian and SNP-like designs, sparse and dense coefficients, and several noise heterogeneity models.","tokens_in":53760,"tokens_out":15614,"duration_ms":179552,"significance":"If the results hold, the paper provides a useful multivariate extension of Dicker's univariate SNR estimator, with explicit variance formulas that could support heritability-type inference on multiple correlated phenotypes. The random-effects and heteroskedastic parts are substantially worked out: the proofs use a published quadratic-form Gaussian approximation lemma, and the variance formulas are parameter-free in the sense that no fitted constants enter the derivation. The simulation study is broad and generally reports good coverage. However, the fixed-effects central limit theorem currently contains a demonstrable normalization error and an unsupported total-variation step, so the central fixed-effects claim needs substantial repair before the results can be relied upon.","major_comments":[{"comment":"The normalization in Theorem 3.1 is incorrect. The displayed V is the finite-sample covariance matrix of (σ̂², ρ̂²): from Lemma 6.7 and (6.4), V11 = (2 + o(1))/(q²(n+1)² n){(n²+np)∥BᵀB∥²_F + ...} = Θ(1/n), and similarly for the other entries. Hence V^{-1/2} = Θ(n^{1/2}), so n^{1/2}V^{-1/2}(θ̂-θ) = Θ(n) · O_P(n^{-1/2}) = O_P(n^{1/2}), which diverges. The subsequent statement n^{1/2}(r̂²-r²)/σ_r with σ_r² built from (3.1) has the same problem: σ_r = Θ(n^{-1/2}) makes the normalized statistic O_P(n^{1/2}). The theorem becomes correct only if either the n^{1/2} multipliers are removed (with V and σ_r² understood as finite-sample variances) or V is redefined as n times the displayed matrix, so that V = Θ(1) is the asymptotic covariance of n^{1/2}(θ̂-θ). This is a load-bearing error, not a typo: any implementation following the printed formula would produce z-statistics inflated by a factor of roughly n^{1/2}.","section":"Theorem 3.1; Eq. (3.1); Lemma 6.7"},{"comment":"The CLT for (σ̂², ρ̂²) is not proved. After deriving the covariance in (6.4), the proof states 'By an argument similar to that used in the proof of Theorem 1 in Dicker [2014] ... d_TV(h(σ̂², ρ̂²), w) = O(n^{-1/2})' and then asserts asymptotic normality. Dicker's theorem is for a univariate estimator; here the joint law of two quadratic-form statistics with Wishart cross-moments is needed. Lemma 6.6 provides second moments only, and moments do not imply convergence in distribution. The paper needs either a direct CLT argument (e.g., Stein/Lindeberg or the Lemma 6.4 approximation used in Section 6.3) or a precise reduction to a published multivariate result. As it stands, Theorem 3.1 lacks support exactly at the step that turns the covariance computation into a CLT.","section":"Section 6.2, proof of Theorem 3.1"},{"comment":"The proof of Theorems 3.2 and 4.1 requires the asymptotic covariance V to be nonsingular and bounded away from zero, but the invertibility condition is asserted rather than assumed. The text says 'Assumption 2.1 guarantees g2g4 - g3², g1g3 - g2² and g2 - g1² are lower bounded...', while Assumptions 2.1 and 2.4 do not exist in the paper; the assumptions are numbered 3.1–4.1. A precise non-degeneracy condition on the limiting spectral distribution H and τ, together with a proof (or a reference) that V_n is uniformly positive definite, must be added before V^{-1/2} is well-defined in the statement.","section":"Theorem 3.2 and Section 6.3, asymptotic distribution paragraph"},{"comment":"The paper does not establish consistency of the plug-in variance estimator in the fixed-effects setting. The matrix V in Theorem 3.1 depends on ∥BᵀB∥²_F, tr(ΣeBᵀB), and tr(Σe²). Remark 2 proposes to estimate these from ĉWb and bΣe in (2.2)–(2.3), but no theorem shows that these matrix estimators are consistent in the required norms for fixed q and p/n → τ. By contrast, the random-effects case has Theorem 4.2, but there is no fixed-effects analogue. Without such a result, the claimed confidence intervals in Tables 1–2 are not formally justified, though the simulations suggest the desired behavior.","section":"Remark 2 and Section 5.1"}],"minor_comments":[{"comment":"There are several typos: 'symmtric' in Theorem 3.2, 'entris' in Lemma 6.5, 'heterogerous' in the Introduction, and 'mutivariate' in Section 4.","section":"Throughout"},{"comment":"The empirical standard error is reported as 0.473 while the average estimated standard error is 4.45; the decimal is almost certainly misplaced (it should be 4.73). Please correct this entry.","section":"Table 4, t7 row (500, 1000)"},{"comment":"The notation d_TV(h(σ̂², ρ̂²)), w) is malformed. Also, h is introduced as a twice continuously differentiable function h : R → R, but it is applied to the bivariate vector (σ̂², ρ̂²); the notation should be clarified.","section":"Section 6.2"},{"comment":"The formula for E[(y_iᵀ y_i)² | X] is asserted without proof or reference. Since the estimator η̂ depends on this formula, a short derivation or citation would improve verifiability.","section":"Section 5.2.2, display preceding (5.1)"}],"recommendation":"major_revision","confidential_remarks":"The normalization slip in Theorem 3.1 is concrete and easily demonstrated; it is the kind of error that can be corrected in revision, and I would not reject on that basis alone. The deeper issue is that the fixed-effects CLT is asserted via an unproved analogy to Dicker (2014), and the plug-in variance consistency for fixed effects is not established. The random-effects part is in better shape and appears substantially correct. I recommend major revision and would want to see a corrected Theorem 3.1 with a real CLT proof, as well as a fixed-effects plug-in consistency result or an explicit caveat."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a careful look, but the fixed-effects theorem is not right as stated. Theorem 3.1 claims n^{1/2}V^{-1/2}(θ̂-θ) ⇒ N(0,I₂), but the displayed V is the O(1/n) finite-sample covariance. Then V^{-1/2} = O(n^{1/2}), so n^{1/2}V^{-1/2} = O(n), while θ̂-θ = O_P(n^{-1/2}). The product diverges. The fix is to use nV, or drop the n^{1/2} multiplier. This also breaks the claimed recovery of Dicker in the q=1 case: the σ_r in (3.1) is O(n^{-1/2}), so √n/σ_r diverges instead of giving a fixed normal limit. The random-effects Theorem 3.2 and heteroskedastic Theorem 4.1 have V entries that are O(1) for fixed q, so those are likely fine.\n\nThe paper isn't a wash. The multivariate extension of the method-of-moments estimator is a real problem, and Lemma 6.6, extending Wishart moment formulas to α≠β, is a genuine technical step. The random-effects analysis, including diverging q, and the heteroskedastic extension with the κ_tot term, are new and appear developed carefully. The simulations are extensive and mostly consistent with the claimed coverage; that is useful evidence.\n\nBeyond the normalization error, there are two smaller issues. The proof of Theorem 3.1 leans on an unproved total-variation analogy to Dicker (2014) at the end of Section 6.2; that gap needs to be filled or the theorem reduced to a conjecture. Also, the conditional moment equation in Section 2.2 has a sign typo (later corrected in Lemma 6.8), which will confuse readers.\n\nWho should read this: people working on high-dimensional SNR/heritability estimation, especially with multivariate responses. The random-effects and heteroskedastic sections are the stronger parts. But because the fixed-effects inference is the advertised central result and it is mis-normalized, the paper needs major revision before the results can be used. A serious referee should see it; the problem is worth solving and the authors are close on the random-effects side.","headline":"The multivariate fixed-effects SNR CLT has a normalization error; the random-effects theory is the stronger half.","tokens_in":54368,"tokens_out":2634,"would_cite":false,"duration_ms":28563,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H12","62J05","62E20","60B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the method-of-moments SNR estimator $\\hat r^2 = \\hat\\rho^2/(\\hat\\rho^2+\\hat\\sigma^2)$ is asymptotically normal for multivariate fixed-effects and random-effects high-dimensional linear models, with explicit variance…","keywords":["signal-to-noise ratio estimation","method of moments","multivariate linear model","heritability","high-dimensional asymptotics","random effects model","heteroskedastic noise","Wishart moments"],"falsifier":"Simulate the fixed-effects model with $q \\geq 2$ responses, Gaussian design with $p/n \\to \\tau$, and fixed $B$ and $\\Sigma_e$, then track the empirical distribution of $n^{1/2}(\\hat r^2 - r^2)/\\hat\\sigma_r$ as $n$ grows: if the distribution fails to approach the standard normal, or the finite-sample covariance of $(\\hat\\sigma^2, \\hat\\rho^2)$ does not approach the $V$ of Theorem 3.1, the unproved total-variation step is the point of failure. A sharper check computes the Stein-type bound on the total-variation distance for the bivariate statistic directly and tests whether the $O(n^{-1/2})$ decay asserted at the end of Section 6.2 actually holds.","tokens_in":53162,"feed_emoji":"🧬","tokens_out":14327,"duration_ms":138648,"temperature":0.7,"pith_summary":"This paper aims to extend method-of-moments estimation of the signal-to-noise ratio, $r^2 = \\rho^2/(\\rho^2+\\sigma^2)$, from univariate to multivariate high-dimensional linear models, and to give the extension a full asymptotic inference theory. Its central claim is that the plug-in estimator $\\hat r^2 = \\hat\\rho^2/(\\hat\\rho^2+\\hat\\sigma^2)$ is asymptotically normal at the $n^{1/2}$ rate, under fixed effects with Gaussian design, under random effects with general predictor covariance, and under heteroskedastic Gaussian noise, with an explicit variance in each case. If correct, the result converts multivariate SNR estimates, such as heritability of multiple correlated phenotypes in genomics, into fully quantified estimates with valid confidence intervals. The variance formulas make precise how the response covariance structure and, in the heteroskedastic case, the total degree of noise heterogeneity drive the estimator's uncertainty.","feed_headline":"Explicit variances make multivariate SNR intervals valid","feed_subtitle":"Method-of-moments SNR estimates stay asymptotically normal under fixed and random effects and heteroskedastic noise.","key_machinery":"The argument runs on the pair of method-of-moments estimators $(\\hat\\sigma^2, \\hat\\rho^2)$, each a linear combination of the two quadratic forms $\\mathrm{tr}(Y^\\top Y)$ and $\\mathrm{tr}(Y^\\top X X^\\top Y)$; the fixed-effects coefficients are constants, while the random-effects coefficients are rational functions of the sample spectral moments $\\hat g_k = \\frac{1}{p}\\mathrm{tr}(S_n^k)$ of the design, which converge to deterministic limits $g_k$ from random matrix theory. The proof has two load-bearing steps. First, Wishart cross-moment identities give the exact covariance of the two statistics: Lemma 6.6 extends the univariate moment results to the case $\\alpha \\neq \\beta$ using the invariant moment formula for the Wishart distribution, and this is what accounts for the response correlation structure in the variance $V$. Second, a total-variation normal approximation converts that covariance into the central limit theorem: for the fixed-effects case the paper asserts the same $O(n^{-1/2})$ total-variation bound proved in the univariate theory, and for the random-effects case a Stein-type bound conditional on the design controls the approximation. For heteroskedasticity, leave-one-out analysis of the design shows the variance depends on the noise only through the average $\\bar\\Sigma_e$ and the scalar $\\kappa_{\\mathrm{tot}} = \\sum_{k,l} \\kappa_{kl}$.","core_discovery":"This paper claims that the method-of-moments SNR estimator $\\hat r^2 = \\hat\\rho^2/(\\hat\\rho^2+\\hat\\sigma^2)$, formed from unbiased moment estimates of the signal variance $\\rho^2$ and noise variance $\\sigma^2$, is asymptotically normal in the multivariate linear model $Y = XB + E$. Theorem 3.1 gives $n^{1/2}(\\hat r^2 - r^2)/\\sigma_r \\Rightarrow N(0,1)$ under fixed effects, Theorem 3.2 under random effects, and Theorem 4.1 under heteroskedastic noise, with $\\sigma_r^2$ defined explicitly in each case from the displayed $2\\times2$ covariance matrix $V$ of $(\\hat\\sigma^2, \\hat\\rho^2)$. The $q=1$ case recovers the univariate estimator's known asymptotic behavior, and the fixed-effects theory imposes no sparsity on the coefficient matrix. In the heteroskedastic setting the noise covariance enters the variance only through its average $\\bar\\Sigma_e$ plus a single scalar $\\kappa_{\\mathrm{tot}}$ measuring total heterogeneity; the homogeneous case has $\\kappa_{\\mathrm{tot}}=0$ and reduces to Theorem 3.2. A further consistency result, Theorem 4.2, shows that plugging covariance estimates into the variance formulas is legitimate when $q = o(n)$.","pith_inferences":["A Wald test comparing SNR estimates across groups or across trait sets could be built directly from the explicit variances given here, although the paper does not discuss such tests.","The clean coverage seen under SNP-style non-Gaussian designs in the simulations suggests the Gaussian-design condition in Theorem 3.1 may be removable; a sub-Gaussian proof would plausibly preserve the same variance formula.","For fully general heteroskedasticity the paper estimates $\\kappa_{\\mathrm{tot}}$ only in structured models (scalar heterogeneity, subgroups); whether $\\kappa_{\\mathrm{tot}}$ is identifiable for arbitrary $\\Sigma_i$ marks the boundary between SNR estimation and noise-model estimation, a question the paper leaves open."],"forward_implications":["Plug-in standard errors from the explicit variance formulas give asymptotically valid confidence intervals for the multivariate SNR $r^2$, so heritability of multivariate phenotypes can be reported with uncertainty bounds.","The random-effects theory allows the number of responses $q$ to diverge, so the estimator scales to large phenotyping panels rather than a single trait.","Heteroskedastic noise, including heavy-tailed errors, Gaussian mixtures, and subgroup-structured noise, needs no new estimator: the same $\\hat r^2$ stays asymptotically normal, with the variance formula gaining only the $\\kappa_{\\mathrm{tot}}$ heterogeneity term.","The fixed-effects results hold without any sparsity assumption on the coefficient matrix $B$, covering both dense and sparse signal regimes."],"supporting_citations":[{"why":"The univariate method-of-moments SNR estimator and its central limit theorem; its Wishart moment identities (restated as Lemma 6.5) and its total-variation proof structure are the template that Theorem 3.1 extends.","marker":"Dicker [2014]"},{"why":"Supplies the invariant moment formula for the Wishart distribution, equation (B.2), from which Lemma 6.6 derives the multivariate cross-moment identities.","marker":"Letac and Massam [2004]"},{"why":"Provides the same invariant Wishart moment formula that Lemma 6.6 uses to compute moments with $\\alpha \\neq \\beta$.","marker":"Graczyk et al. [2005]"},{"why":"Supplies Lemma 6.1 (moments of quadratic forms) and Lemma 6.4 (a Stein-type bound for sub-Gaussian quadratic forms), which carry the random-effects CLT conditional on the design.","marker":"Dicker and Erdogdu [2017]"},{"why":"Convergence of the empirical spectral distribution of the design, the basis for the deterministic limits $g_k$ of the sample spectral moments.","marker":"Marčenko and Pastur [1967]"},{"why":"Limiting spectral distribution results supporting the convergence of $\\hat g_k$ to $g_k$ that the random-effects variance formulas rely on.","marker":"Silverstein and Choi [1995]"},{"why":"The Hanson-Wright inequality used in Lemma 6.10's concentration bounds, which prove the plug-in covariance estimators are operator-norm consistent.","marker":"Rudelson and Vershynin [2013]"},{"why":"Theorem 6.5, the spectral-norm concentration bound for sub-Gaussian sample covariances, used with leave-one-out analysis in the heteroskedastic proof.","marker":"Wainwright [2019]"}],"fun_headline_variants":["Multivariate SNR gets asymptotic normality with explicit variance","Method-of-moments SNR estimators go multivariate","Exact variances for multivariate SNR estimators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the fixed-effects central limit theorem (Theorem 3.1) assumes, without deriving it, that the two-dimensional statistic $(\\hat\\sigma^2, \\hat\\rho^2)$ satisfies the same $O(n^{-1/2})$ total-variation normal approximation that was proved for the univariate estimator, and this unproved step is what turns the computed covariance into the CLT.","fun_headline_variants_meta":{"raw":{"variants":["Multivariate SNR gets asymptotic normality with explicit variance","Method-of-moments SNR estimators go multivariate","Exact variances for multivariate SNR estimators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00023,"raw_usage":{"total_tokens":1482,"prompt_tokens":942,"completion_tokens":540,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":495}},"tokens_in":558,"tokens_out":540,"duration_ms":5432,"temperature":1.0,"reasoning_tokens":495,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:30:44.318220+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the fixed-effects model with $q \\geq 2$ responses, Gaussian design with $p/n \\to \\tau$, and fixed $B$ and $\\Sigma_e$, then track the empirical distribution of $n^{1/2}(\\hat r^2 - r^2)/\\hat\\sigma_r$ as $n$ grows: if the distribution fails to approach the standard normal, or the finite-sample covariance of $(\\hat\\sigma^2, \\hat\\rho^2)$ does not approach the $V$ of Theorem 3.1, the unproved total-variation step is the point of failure. A sharper check computes the Stein-type bound on the total-variation distance for the bivariate statistic directly and tests whether the $O(n^{-1/2})$ decay asserted at the end of Section 6.2 actually holds.","supporting_citations":[{"cited_title":"Variance estimation in high-dimensional linear models","cited_arxiv_id":null,"evidence_quote":"The univariate method-of-moments SNR estimator and its central limit theorem; its Wishart moment identities (restated as Lemma 6.5) and its total-variation proof structure are the template that Theorem 3.1 extends."},{"cited_title":"All Invariant Moments of the Wishart Distribution","cited_arxiv_id":null,"evidence_quote":"Supplies the invariant moment formula for the Wishart distribution, equation (B.2), from which Lemma 6.6 derives the multivariate cross-moment identities."},{"cited_title":"Flexible results for quadratic forms with applications to variance components estimation","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 6.1 (moments of quadratic forms) and Lemma 6.4 (a Stein-type bound for sub-Gaussian quadratic forms), which carry the random-effects CLT conditional on the design."},{"cited_title":"Distribution of eigenvalues for some sets of random matrices","cited_arxiv_id":null,"evidence_quote":"Convergence of the empirical spectral distribution of the design, the basis for the deterministic limits $g_k$ of the sample spectral moments."},{"cited_title":"Hanson-wright inequality and sub-gaussian concentration","cited_arxiv_id":null,"evidence_quote":"The Hanson-Wright inequality used in Lemma 6.10's concentration bounds, which prove the plug-in covariance estimators are operator-norm consistent."}],"review_version":1}