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observed data consist of ( Y, T, X ), and the full data consist of (Y0, Y1, T, X, U ) with latent variables ( Y0, Y1, U) in the case of MSM. The observed outcome Yis determined from ( Y0, Y1) through the consistency assumption. The unmeasured variable U can be arbitrary in the sense that no knowledge is assumed, except... | https://arxiv.org/abs/2504.08301v1 |
the preceding perspective, MSM constraint (1b) posits a bound on the conditional asso- ciation between UandTin the odds ratio scale as in (2), but leaves the association between U and ( Y0, Y1) arbitrary. This may often lead to wide population sharp bounds under MSMs. To address this limitation when inferring µ1, we po... | https://arxiv.org/abs/2504.08301v1 |
to derive implementable representations of the sharp bounds µ1− eMSMandµ1+ eMSM. By symmetry, we mainly discuss the sharp upper bound. To start, for any joint distribution Qallowed in (6), we note that the mean EQ(Y1) can be expressed as follows: EQ(Y1) =E TY+ (1−T)EQ(Y1|T= 0, X) , (8a) EQ(Y1|T= 0, X) =Z ηQ(X, u)λQ(X,... | https://arxiv.org/abs/2504.08301v1 |
in (4b), except for the factor ψ1+(X). More- over, the sharp bounds in Proposition 1 can be achieved by a joint distribution Qas in Corollary 1, with Y0omitted for simplicity. See Figure 2 for an illustration. Corollary 1 (Worst-case unmeasured confounding) .A joint distribution Qon(Y0, Y1, T, X, U ) which achieves the... | https://arxiv.org/abs/2504.08301v1 |
arbitrary, then ν1+ eMSM(X) reduces to E(Y|T= 1, X), which identifies ν1(X) under ignobility. If ∆ 1(X) = ∆ 2(X) = 0 but (Λ 1(X),Λ2(X)) may be arbitrary, then ν1+ eMSM(X) also reduces to E(Y|T= 1, X). These “degenerate” cases are consistent with our earlier discussions in Section 3.1. Second, Proposition 1 demonstrates... | https://arxiv.org/abs/2504.08301v1 |
confounder Uacts as a selection indicator deciding how Y1 is related to Yin the two segments of the distribution PY(·|T= 1, X). Fourth, by the definition of q∗ 1,τ(X) as a minimizer of E{ρτ(Y, q)|T= 1, X}over q∈R, the formula (12) shows that conditionally on X, ν1+ eMSM(X) = inf q∈Rν1+ eMSM(X;q), where ν1+ eMSM(X;q) is... | https://arxiv.org/abs/2504.08301v1 |
loss E{ρτ(Y, q∗ 1,τ)|T= 1, X}, when evaluating the sharp upper bound of µ1. Taking δ1+(X) =δ2+(X)≡δwould lead to the specification [−∆1,U(X),∆2,U(X)] = [−δ/τ, δ/ (1−τ)]·E{ρτ(Y, q∗ 1,τ)|T= 1, X}, (15) where δ∈[0,1] is a constant. The sharp upper bound ν1+ eMSM(X) would become ν1+ eMSM,U(X) =E(Y|T= 1, X) + (Λ 2−Λ1)δE{ρτ(... | https://arxiv.org/abs/2504.08301v1 |
treated as 1. If the conditional distribution of Yis symmetric around its median, then (22) is trivial because the ratio of the two optimized quantile losses in (22) is 1. For an asymmetric conditional distribution, inequality (22) appears to be new. See Supplement Section VI.4 for a proof. Second, inequality (22) also... | https://arxiv.org/abs/2504.08301v1 |
important aspects of uncertainty to infer about µ1as discussed in Tan (2024, 2025). In short, our approach treats the sensitivity parameters (Λ 1,Λ2) and δdata- independently and then evaluate the sensitivity bounds which are data-dependent. 15 The preceding discussion leaves open the question of how to decide plausibl... | https://arxiv.org/abs/2504.08301v1 |
and m∗ −(X;q) = E{˜Y−(q)|T= 1, X}respectively, with the transformed response ˜Y−(q) =Y−(Λ2−Λ1)δρ1−τ(Y, q). Similar results to Proposition 4 also hold. 4 Sample estimation For sensitivity analysis using eMSMs, the population bounds developed in Section 3 can be esti- mated, with associated confidence intervals, from sam... | https://arxiv.org/abs/2504.08301v1 |
correctly specified, then the relaxed bound µ1+ eMSM(hT¯βWQ,+) reduces to the sharp bound µ1+ eMSM. (ii) ˆµ1+ CALadmits the asymptotic expansion ˆµ1+ CAL= ˆµ1+(¯γCAL,¯βWQ,+,¯αWL,+) +op(n−1/2), (31) regardless of misspecification of model (25), (26) or (27). The expansion (31) enables the construction of valid Wald conf... | https://arxiv.org/abs/2504.08301v1 |
Y. We consider binary Yand focus on the conditional bounds given Xas in the main results of DV. For simplicity, all probability expressions are implicitly conditional on X. See Supplement Section IV for DV unconditional bounds and sample estimation. With binary outcome, the eMSM sharp bounds of µ1=E(Y1) in Propositions... | https://arxiv.org/abs/2504.08301v1 |
ORRP(T= 1) + P(T= 0) min {B(Λ2,Θ),1/p1} P(T= 1)/B(Λ2,Θ) + P(T= 0). Removing respectively the argument 1 /p0and 1 /p1in the min operators in (42) leads back to the DV bound (39). However, due to the separate use of (37a) or (37b), the improved DV bounds (42) 21 are not sharp in the DV model defined by (37a), (37b), (38a... | https://arxiv.org/abs/2504.08301v1 |
the proof in Supplement Section VI.6. Second, the sharp bounds of µ1in Proposition 5 (ii)–(iii) are derived by accounting for the constraints (37a), (37b), and (38b) jointly, and hence are tighter than the DV bounds in (40) or (41). See Figure 3 for a numerical demonstration. As a special case, setting Λ 1= 0 or Λ 2=∞ ... | https://arxiv.org/abs/2504.08301v1 |
This situation differs from that of Section 3.3, where MSM with recommended specification (19) achieves the sharp upper and lower bounds from two MSMs respectively with specifications (15) and (17) for the same δ. Finally, the relationship (43) and (44) can be used in a converse direction to relate a specified eMSM to ... | https://arxiv.org/abs/2504.08301v1 |
the intensive care units of five medical centers. For each patient, the data consist of treatment status T(= 1 if RHC was used within 24 hours of admission and 0 otherwise), binary outcome Y(30-day survival status), and a list of 75 covariates Xspecified by medical specialists in critical care. For sensitivity analysis... | https://arxiv.org/abs/2504.08301v1 |
eMSM at Λ = 1 are the same for all δ). The MSM sensitivity intervals include 0 at all Λ ∈ {1.2,1.5,2}as in Tan (2024). The eMSM intervals are below 0 at Λ = 1 .2 and δ∈ {0.2,0.5}and (Λ = 1 .5, δ= 0.2), almost below 0 at (Λ = 2 , δ= 0.2) with an upper interval point of 0.0030 (Supplement Table S7), but include 0 at the ... | https://arxiv.org/abs/2504.08301v1 |
constant. We consider sensitivity parameters δ∈ {0.2,0.5,0.8,1}, same as in Section 6.1, and Λ ∈ {1,10,20,30,50}, which are selected for presentation after obtaining results for a denser sequence from 1 to 50. Figure 5 shows the point bounds and 90% confidence intervals on ATE using working models with main effects and... | https://arxiv.org/abs/2504.08301v1 |
outcome sensitivity constraint (5) controls the U-Y1association by bounding the deviation of η∗(X, U) from the reference value, E(Y|T= 1, X), under unconfoundedness. For completeness, we point out two alternative sensitivity models with outcome sensitivity constraints that are based on η∗(X, U). The first outcome sensi... | https://arxiv.org/abs/2504.08301v1 |
estimates of continuous- valued interventions. NeurIPS , 35:13892–13907. 31 Jin, Y., Ren, Z., and Zhou, Z. (2022). Sensitivity analysis under the f-sensitivity models: a distributional robustness perspective. arXiv:2203.04373 . Koenker, R. and Bassett, Jr., G. (1978). Regression quantiles. Econometrica , 46:33–50. Koen... | https://arxiv.org/abs/2504.08301v1 |
) Fix Λ1(X)andΛ2(X). (i) A distribution of (Y0, Y1, T, X, U )that satisfies (1)also satisfies (S2). (ii) For a distribution of (Y0, Y1, T, X )that satisfies (S2), there exists an unmeasured variable Uthat satisfies (1a), i.e., (Y0, Y1)⊥T|X, U, and the distribution of (Y0, Y1, T, X, U )satisfies (1b). (iii) A distributi... | https://arxiv.org/abs/2504.08301v1 |
equivalent, there is a subtle differ- ence between their formulations. While MSM (S2) characterizes unmeasured confounding via the density ratio of ( Y0, Y1) between the treatment groups, MSM (1) does so via the density ratio of a responsible unmeasured confounder Uas in the treatment sensitivity constraint (1b). The i... | https://arxiv.org/abs/2504.08301v1 |
) is constant in U, then E(Y1|T= 1, X, U ) is constant in U. □ Conditional on X, Lemma S2 (i) and (ii) show that if Uis not associated with TorY1in either treatment group, then unconfoundedness holds in Y1. Lemma S2 (iii) is variation of result (ii), dealing with unconfoundedness in means only and corresponding to the ... | https://arxiv.org/abs/2504.08301v1 |
to those from the two partial eMSMs, which is proved in Section VI.3. Moreover, the proof shows that the sharp lower bound of µ0and the sharp upper bound of µ1under respective partial eMSMs (or the sharp upper bound of µ0and the sharp lower bound of µ1) are attained simultaneously by a single distribution Qallowed in C... | https://arxiv.org/abs/2504.08301v1 |
E(Y1|X)/E(Y0|X). With B(x, y) =xy/(x+y−1),the DV bounds of µ1(X) and µ0(X) are µ1+ DV(X) =p1(X){P(T= 1|X) +P(T= 0|X)B(Λ2(X),Θ(X))}, µ1− DV(X) =p1(X) P(T= 1|X) +P(T= 0|X) B(1/Λ1(X),Θ(X)) , µ0+ DV(X) =p0(X){P(T= 0|X) +P(T= 1|X)B(1/Λ1(X),Θ(X))}, µ0− DV(X) =p0(X) P(T= 0|X) +P(T= 1|X) B(Λ2(X),Θ(X)) . 7 The main result o... | https://arxiv.org/abs/2504.08301v1 |
dMSM(∆) are no tighter than those under individual eMSM with ∆ 1(X) + ∆ 2(X)≤∆(X), and can be derived from those under eMSM with proper choice of ∆ 1(X) and ∆ 2(X). Specifically, the sharp bounds of µ1under dMSM(∆) are µ1+ dMSM(∆) = E E(Y|T= 1, X) + (1 −T){Λ2(X)−Λ1(X)}× min τ(X){1−τ(X)}∆(X), E{ρτ(Y, q∗ 1,τ)|T= 1, X}... | https://arxiv.org/abs/2504.08301v1 |
c∈(0,1], then ηQ(u) satisfies E Y−c−1ρc(Y, q∗ c) ≤ηQ(u)≤E Y+c−1ρ1−c Y, q∗ 1−c , (S13) where q∗ cis the c-quantile of PY, the observed distribution of Y, and ρc(Y, q) is the check function. Let (Ω ,F, Q) be the probability space on which YandUare defined. It follows from the definition of conditional expectation tha... | https://arxiv.org/abs/2504.08301v1 |
λ(nonnegative) and ηofU|T= 1 are subject to the following constraints λ(U)∈[Λ1,Λ2], (S19) η(U)−E(Y|T= 1)∈[−∆1,∆2], (S20) Z λ(u)dQU(u|T= 1) = 1 , (S21) Z η(u)dQU(u|T= 1) = E(Y|T= 1). (S22) Fix any ( QU(·|T= 1), η) allowed in (S18), it follows from Property 2.2 of Francis and Wright (1969) (related to Proposition 2 of Do... | https://arxiv.org/abs/2504.08301v1 |
bound and lower bound for ν1 under the partial eMSM specified (1) and (5). min ν1+ MSM, ν1+ e =E(Y|T= 1)+ 15 (Λ2−Λ1) min τ∆1,(1−τ)∆2, E{ρτ(Y, q∗ 1,τ)|T= 1} , (S27) max ν1− MSM, ν1− e =E(Y|T= 1)− (Λ2−Λ1) min (1−τ)∆1, τ∆2, E{ρ1−τ(Y, q∗ 1,1−τ)|T= 1} .(S28) Similarly, (S29) and (S30) below are respectively valid uppe... | https://arxiv.org/abs/2504.08301v1 |
(1a), i.e., ( Y0, Y1)⊥T|U, for claim (ii). Next, we derive relevant quantities to prove claims (ii) and (iii). From (S31), (S32) and (S33a), QU(1|T= 1) = PU(1|T= 1) = p1,+(1−τ+τψ1+) +p1,−{1−τ−(1−τ)ψ1+}+ p1,∼ 1−τ−(1−τ)ψ1++1−τ−p1,+ p1,∼ψ1+ = 1−τ+{p1,+τ−p1,−(1−τ)−p1,∼(1−τ) + 1−τ−p1,+}ψ1+ 17 = 1−τ, (S34) QU(1|T= 0) = PU(... | https://arxiv.org/abs/2504.08301v1 |
consequently Proposition 1. Proposition 2 follows from the proof for ν0 Qby flipping the label of T, which leads to λQbeing replaced with its reciprocal and (Λ 1,Λ2) replaced with (Λ−1 2,Λ−1 1). Corollary 1 follows from the construction of Q. Note that the construction of Qdoes not seem to be unique when ψ1+orψ0−is str... | https://arxiv.org/abs/2504.08301v1 |
E{φ1+(π, q, m +)}=µ1+ eMSM(q). Minimization at true quantile The minimization of µ1+ eMSM(q) at q∗ 1,τfollows from that of the conditional quantile loss E{ρτ(Y, q)|T= 1, X}atq∗ 1,τ(X). See Koenker and Bassett (1978) for details. VI.6 Proof of Proposition 5 We state a stronger result that includes the simultaneous sharp... | https://arxiv.org/abs/2504.08301v1 |
By Proposition S1 (i), µ1+ DV(Θ) = max ∆∈D(Θ)µ1+ eMSM(∆) = max (∆1,∆2):θ1(∆1,∆2)≤Θµ1+ eMSM(∆1,∆2), (S51) µ0− DV(Θ) = min ∆∈D(Θ)µ0− eMSM(∆) = min (∆′ 1,∆′ 2):θ0(∆′ 1,∆′ 2)≤Θµ0− eMSM(∆′ 1,∆′ 2), (S52) where the second equality in (S51) and that in (S52) follow from our discussion in Section III, that C-eMSM shares the sa... | https://arxiv.org/abs/2504.08301v1 |
the collection of eMSM(∆ 1,∆2) with ∆ 1(X) + ∆ 2(X)≤∆(X), and hence µ1+ dMSM(∆) = max ∆1+∆2≤∆µ1+ eMSM(∆1,∆2). For any full data distribution Q∈eMSM(∆ 1,∆2) with ∆ 1(X) + ∆ 2(X)≤∆(X), max uηQ(X, u)−min uηQ(X, u) 25 = max uηQ(X, u)−E(Y|T= 1, X)−n min uηQ(X, u)−E(Y|T= 1, X)o = ∆ 2(X) + ∆ 1(X)≤∆(X), which shows thatS ∆1+∆2... | https://arxiv.org/abs/2504.08301v1 |
(1−pr)dQU′(u) if u∈ {u:η′(U′) =qη′,τ}, z=−1 0, ifu∈ {u:η′(U′) =qη′,τ}, z= 0, 27 and dQU′ Z(U′=u, Z=z) = dQU′(u) if u /∈ {u:η′(U′) =qη′,τ}, z= 0 0 if u /∈ {u:η′(U′) =qη′,τ}, z̸= 0. We define λ′ Zandη′ ZforU′ Zas follows: λ′ Z(U′=u, Z=z) = λ′(u) if z= 0 Λ2 ifz= 1 Λ1 ifz=−1, andη′ Z(U′=u, Z=z) =η′(... | https://arxiv.org/abs/2504.08301v1 |
δ0.2 0.5 0.8 10.4 0.6 0.8 1.0N Auntreated( µ0 ) δ0.2 0.5 0.8 10.4 0.6 0.8 1.0N A (d) Λ = 2 Figure S1: Point bounds (x) and 90% confidence intervals (–) on 30-day survival probabilities in the RHC study, similarly as Figure 4. 29 treated( µ1 ) δ0.2 0.5 0.8 10.4 0.5 0.6 0.7 0.8 0.9 1.0N Auntreated( µ0 ) δ0.2 0.5 0.8 10.4... | https://arxiv.org/abs/2504.08301v1 |
0.0 0.1CRR δ0.2 0.5 0.8 10.7 0.8 0.9 1.0 1.1 1.2(a) Λ = 1 treated( µ1 ) δ0.2 0.5 0.8 10.50 0.55 0.60 0.65 0.70 0.75untreated( µ0 ) δ0.2 0.5 0.8 10.50 0.55 0.60 0.65 0.70 0.75ATE δ0.2 0.5 0.8 1−0.2 −0.1 0.0 0.1CRR δ0.2 0.5 0.8 10.7 0.8 0.9 1.0 1.1 1.2 (b) Λ = 1 .2 treated( µ1 ) δ0.2 0.5 0.8 10.50 0.55 0.60 0.65 0.70 0.7... | https://arxiv.org/abs/2504.08301v1 |
0.915 0.020 0.915 0.020 0.915 0.020 MSM lin 0.916 0.021 0.916 0.021 0.916 0.021 0.916 0.021 eMSM logit 0.915 0.020 0.915 0.020 0.915 0.020 0.915 0.020 eMSM lin 0.916 0.021 0.916 0.021 0.916 0.021 0.916 0.021 Upper bounds and SEs for µ1/µ0 MSM logit 0.915 0.020 0.915 0.020 0.915 0.020 0.915 0.020 MSM lin 0.916 0.021 0.9... | https://arxiv.org/abs/2504.08301v1 |
under MSM and eMSM with Λ = 1 .2, using working models with only main effects in the RHC study. δ= 0.2 δ= 0.5 δ= 0.8 δ= 1 Bounds SEs Bounds SEs Bounds SEs Bounds SEs Lower bounds and SEs for µ1 MSM logit 0.598 0.013 0.598 0.013 0.598 0.013 0.598 0.013 MSM lin 0.601 0.014 0.601 0.014 0.601 0.014 0.601 0.014 eMSM logit 0... | https://arxiv.org/abs/2504.08301v1 |
0.574 0.011 0.565 0.011 0.560 0.011 Upper bounds and SEs for µ1 DV Θ+/2 0.625 0.012 0.651 0.013 0.685 0.014 0.703 0.014 DV Θ+0.650 0.013 0.676 0.013 0.694 0.014 0.703 0.014 DV 3Θ+/2 0.666 0.013 0.684 0.014 0.697 0.014 0.703 0.014 Lower bounds and SEs for µ0 DV Θ+/2 0.690 0.008 0.673 0.008 0.655 0.007 0.646 0.007 DV Θ+0... | https://arxiv.org/abs/2504.08301v1 |
0.007 MSM lin 0.730 0.007 0.730 0.007 0.730 0.007 0.730 0.007 eMSM logit 0.700 0.008 0.711 0.008 0.723 0.007 0.730 0.007 eMSM lin 0.700 0.008 0.711 0.008 0.723 0.007 0.730 0.007 Lower bounds and SEs for µ1−µ0 MSM logit -0.180 0.016 -0.180 0.016 -0.180 0.016 -0.180 0.016 MSM lin -0.175 0.016 -0.175 0.016 -0.175 0.016 -0... | https://arxiv.org/abs/2504.08301v1 |
0.907 0.019 0.777 0.016 0.659 0.014 0.605 0.012 DV Θ+0.790 0.017 0.691 0.014 0.632 0.013 0.605 0.012 DV 3Θ+/2 0.728 0.015 0.662 0.014 0.623 0.013 0.605 0.012 Upper bounds and SEs for µ1/µ0 DV Θ+/2 0.907 0.021 1.058 0.024 1.246 0.030 1.360 0.033 DV Θ+1.040 0.024 1.190 0.028 1.300 0.031 1.360 0.033 DV 3Θ+/2 1.128 0.026 1... | https://arxiv.org/abs/2504.08301v1 |
0.027 1.208 0.027 1.208 0.027 eMSM logit 0.971 0.021 1.061 0.022 1.159 0.024 1.229 0.026 eMSM lin 0.969 0.021 1.053 0.023 1.144 0.025 1.208 0.027 Note : MSM logit and eMSM logit denote CAL under MSM and eMSM using logistic outcome mean regression. MSM lin and eMSM lin denote CAL under MSM and eMSM using linear outcome ... | https://arxiv.org/abs/2504.08301v1 |
using working models with main effects and interactions in the RHC study. δ= 0.2 δ= 0.5 δ= 0.8 δ= 1 Bounds SEs Bounds SEs Bounds SEs Bounds SEs Lower bounds and SEs for µ1 MSM logit 0.636 0.011 0.636 0.011 0.636 0.011 0.636 0.011 MSM lin 0.636 0.011 0.636 0.011 0.636 0.011 0.636 0.011 eMSM logit 0.636 0.011 0.636 0.011... | https://arxiv.org/abs/2504.08301v1 |
eMSM lin 0.629 0.011 0.618 0.011 0.607 0.011 0.601 0.012 Upper bounds and SEs for µ1 MSM logit 0.667 0.010 0.667 0.010 0.667 0.010 0.667 0.010 MSM lin 0.666 0.010 0.666 0.010 0.666 0.010 0.666 0.010 eMSM logit 0.642 0.011 0.652 0.010 0.661 0.010 0.667 0.010 eMSM lin 0.642 0.011 0.651 0.010 0.660 0.010 0.666 0.010 Lower... | https://arxiv.org/abs/2504.08301v1 |
0.641 0.009 0.641 0.009 0.641 0.009 0.641 0.009 MSM lin 0.642 0.009 0.642 0.009 0.642 0.009 0.642 0.009 eMSM logit 0.678 0.008 0.664 0.008 0.650 0.009 0.641 0.009 eMSM lin 0.678 0.008 0.664 0.008 0.651 0.009 0.642 0.009 Upper bounds and SEs for µ0 MSM logit 0.729 0.007 0.729 0.007 0.729 0.007 0.729 0.007 MSM lin 0.728 ... | https://arxiv.org/abs/2504.08301v1 |
logit 0.700 0.008 0.718 0.007 0.737 0.007 0.749 0.006 eMSM lin 0.699 0.008 0.718 0.007 0.736 0.007 0.749 0.006 Lower bounds and SEs for µ1−µ0 MSM logit -0.260 0.014 -0.260 0.014 -0.260 0.014 -0.260 0.014 MSM lin -0.259 0.014 -0.259 0.014 -0.259 0.014 -0.259 0.014 eMSM logit -0.093 0.013 -0.156 0.013 -0.218 0.013 -0.260... | https://arxiv.org/abs/2504.08301v1 |
S13–S17 show numerical estimates and standard errors using working models with only main effects and Tables S18–S22 show numerical estimates and standard errors using working models with main effects and interactions. Comparing Figures 5 and S6, we see notable differences in point bounds between using working models wi... | https://arxiv.org/abs/2504.08301v1 |
0.5 0.8 10.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 (e) Λ = 50 Figure S6: Point bounds (x) and 90% confidence intervals (–) on mean blood mercury and ATE in the NHANES study, similarly as Figure 5, but using working models with main effects only. 47 Table S14: Point bounds and SEs for CAL under MSM and eMSM with Λ = 10, using wor... | https://arxiv.org/abs/2504.08301v1 |
0.5 δ= 0.8 δ= 1 Bounds SEs Bounds SEs Bounds SEs Bounds SEs Lower bounds and SEs for µ1 MSM lin -0.458 0.050 -0.458 0.050 -0.458 0.050 -0.458 0.050 eMSM lin 0.447 0.090 0.108 0.068 -0.232 0.052 -0.458 0.050 Upper bounds and SEs for µ1 MSM lin 1.916 0.064 1.916 0.064 1.916 0.064 1.916 0.064 eMSM lin 0.921 0.091 1.294 0.... | https://arxiv.org/abs/2504.08301v1 |
0.038 -1.097 0.038 -1.097 0.038 eMSM lin -1.097 0.038 -1.097 0.038 -1.097 0.038 -1.097 0.038 Upper bounds and SEs for µ0 MSM lin -1.097 0.038 -1.097 0.038 -1.097 0.038 -1.097 0.038 eMSM lin -1.097 0.038 -1.097 0.038 -1.097 0.038 -1.097 0.038 Lower bounds and SEs for µ1−µ0 MSM lin 1.862 0.082 1.862 0.082 1.862 0.082 1.8... | https://arxiv.org/abs/2504.08301v1 |
3.709 0.126 3.709 0.126 3.709 0.126 3.709 0.126 eMSM lin 2.242 0.082 2.784 0.090 3.347 0.111 3.709 0.126 Note : MSM lin and eMSM lin denote RCAL under MSM and eMSM using linear outcome mean regression. Table S21: Point bounds and SEs for RCAL under MSM and eMSM with Λ = 30, using working models with main effects and in... | https://arxiv.org/abs/2504.08301v1 |
High-dimensional Clustering and Signal Recovery under Block Signals Wu Su Center for Big Data Research, Peking University, Beijing, China and Yumou Qiu School of Mathematical Sciences, Peking University, Beijing, China Abstract This paper studies computationally efficient methods and their minimax optimal- ity for high... | https://arxiv.org/abs/2504.08332v1 |
spectral clus- tering has gained more attention in high-dimensional data analysis recently. L¨ offler et al. (2021) established the optimal convergence rate of spectral clustering in Gaussian mixture models with isotropic covariance matrix. Abbe et al. (2022); Zhang and Zhou (2024) devel- oped the leave-one-out analysi... | https://arxiv.org/abs/2504.08332v1 |
and filter out useful features from clustered data based on pairwise U-statistics. The proposed CFA method prevents signal cancellation from unknown cluster labels and avoids estimating unknown variances. It works under non-Gaussian distributions and is more general than IF- PCA designed for the Gaussian distribution w... | https://arxiv.org/abs/2504.08332v1 |
sample ofp-dimensional independent observations of size n, where Xi= (Xi1, . . . , X ip)T. Here, the dimension can be much larger than the sample size. Following the high-dimensional clustering model proposed in Jin et al. (2017), we assume the cluster label of each obser- vation is independent and identically distribu... | https://arxiv.org/abs/2504.08332v1 |
between blocks satisfies d0/(max 1≤g≤m|I0,g|)→ ∞ . Assumption 3 There exists µ0>0andc0≥1such that c−1 0µ0≤ |µj| ≤c0µ0for any j∈S(µ), and sign(µj) =egforj∈I0,gandeg=±1. 7 Assumption 1 assumes the data are sub-Gaussian distributed, which is commonly made in high-dimensional literature (Bickel and Levina, 2008; Cai et al.... | https://arxiv.org/abs/2504.08332v1 |
m≪p p/b. The minimax results in Section 5 show that consistent clustering requires stronger signals than signal recovery for sparse block signals, and vice versa for dense block signals. This indicates that we should conduct pre- clustering signal recovery first and cluster the observations based on the selected featur... | https://arxiv.org/abs/2504.08332v1 |
a window ˜ c2, where Ep({j, . . . , j + ˜c1},˜c2) ={j−˜c2, . . . , j + ˜c1+ ˜c2} ∩ { 1, . . . , p }, (3.2) and ˜c1,˜c2>0. For I1∈Jp(h1) and a window size h2>0, we consider its cross products with blocks that are at least h2apart. Let ˜I1= argmax I2∈Jp(h1), I2∩E(I1,h2)=∅|W(I1, I2)|and W0(I1) =W(I1,˜I1) (3.3) be the larg... | https://arxiv.org/abs/2504.08332v1 |
signal-to-noise condition in (3.5) is for block feature selection, which ensures the expectation√nµ(I1)µ(I2) of the cross-products√nWi(I1, I2) diverges to infinity if I1andI2are the true signal blocks. The second signal-to-noise condition on min {(n/m)1/2,1}∥µ∥2in (3.5) is much weaker than the classical condition min {... | https://arxiv.org/abs/2504.08332v1 |
call this method as moving average PCA (MA-PCA) for clustering. In the SM, we demonstrate that the signal-to-noise ratio in Y(ma)Y(ma)Tincreases with h3when h3≲b, but decreases with further increases inh3when b=o(h3). The following theorem establishes the consistency of ˆℓmatoℓup to a sign flip. Theorem 3 For the clust... | https://arxiv.org/abs/2504.08332v1 |
midentified blocks from Algorithm 2. The following theorem shows that the proposed post-clustering signal recovery procedure can 16 consistently identify all true signal blocks. Theorem 4 Under the model (2.1), Assumptions 1-3, max{∥Σ∥2,∥Σ−1∥2} ≤C,√ nbµ 0> c√logpfor a constant c >0, and additionally µj=τgforj∈I0,gandg=... | https://arxiv.org/abs/2504.08332v1 |
for the estimated signal set ˆSfrom a signal recovery procedure, we measure its performance using the normalized expected Hamming distance, defined as Hamm sig(ˆS;µ, ϖ) = (pε)−1E ˆS∆S(µ) , (5.3) where S(µ) is the true signal set of µ,ε=|S(µ)|/p=s/pis the proportion of signals out ofpvariables, and A∆B= (A\B)∪(B\A) deno... | https://arxiv.org/abs/2504.08332v1 |
ηsig θ,α(β)and (2) any polynomial-time signal recovery procedure ˆSifr >˜ηsig θ,α(β). Theorems 5 and 6 characterize both the statistical and computational minimax lower bounds for clustering and signal recovery under block signals. They demonstrate that when r > ηclu θ,α(β) and r > ηsig θ,α(β), no statistical algorithm... | https://arxiv.org/abs/2504.08332v1 |
diagram, nor study the minimax results for signal recovery. Therefore, their results don’t show the statistical-computational gaps in the phase transi- tion diagram, and cannot compare the difficulty of clustering and signal recovery across different sparsity regimes from a minimax view. LetˆScfa=∪ˆI∈ˆIcfaˆIandˆSma=∪ˆI... | https://arxiv.org/abs/2504.08332v1 |
block of I(t)defined in (3.2). Using W0(I), the CFA method in Algorithm 1 can be extended to tensor-valued data to identify the tensor blocks with nonzero means. Once the estimated blocks ˆI1, . . . , ˆIˆmare obtained from Algorithm 1, we construct the aggregated data matrix for those selected blocks as ˆY(cfa)= (ˆY(cf... | https://arxiv.org/abs/2504.08332v1 |
sizes along the two coordinates were randomly drawn from a uniform distribution over [ Lmin, Lmax]. After the location of the first block, denoted as I0,1, was selected, we removed the region E(I0,1, d0) that expands I0,1byd0from the candidate set of grids for selecting the next block. This guarantees the randomly sele... | https://arxiv.org/abs/2504.08332v1 |
IF-PCA were quite similar and better than MA-PCA and spectral clustering under the sparse signal setting. The performance of k-means was ranked in the middle for dense signals, but was the worst for sparse signals. Meanwhile, for signal recovery, CFA-PCA and IF-PCA were comparable and much better than the other three m... | https://arxiv.org/abs/2504.08332v1 |
the strong signals over northern Siberia. The means of the two clusters exhibited a completely opposite pattern in the Canadian region and northern Siberia. However, these two regions, collectively known as the boreal forests, share very similar climate patterns. This shows the results by k-means might be unreasonable.... | https://arxiv.org/abs/2504.08332v1 |
, 105(491):1156– 1166. 30 Jin, J. (2015). Fast community detection by score. The Annals of Statistics , 43(1):57–89. Jin, J., Ke, Z. T., and Wang, W. (2017). Phase transitions for high dimensional clustering and related problems. Ann. Statist. , 45(5):2151–2189. Jin, J. and Wang, W. (2016). Influential features PCA for... | https://arxiv.org/abs/2504.08332v1 |
Regime) signal recovery error 0.5 1 1.5 signal strength00.51 0.5 1 1.5 signal strength00.51Figure 2: Simulation results under the non-block signal settings. The clustering errors (upper panels) and signal recovery errors (lower panels) of CFA-PCA, MA-PCA, IF-PCA, spectral clustering and k-means under different signal s... | https://arxiv.org/abs/2504.08332v1 |
Standardization of Weighted Ranking Correlation Coefficients Pierangelo Lombardo Abstract. A relevant problem in statistics is defining the correlation of two rankings of a list of items. Kendall’s τand Spearman’s ρare two well es- tablished correlation coefficients, characterized by a symmetric form that en- sures zer... | https://arxiv.org/abs/2504.08428v1 |
to define weighted correlation coefficients able to capture the greater importance of top ranks. In fact, in many contexts, when we compare two rankings, the rel- evance of rank mismatches depends on the positions in- volved, and top ranks are typically considered more im- portant; two prominent examples are content-ba... | https://arxiv.org/abs/2504.08428v1 |
all items. This definition contains both Spear- man’s ρand Kendall’s τ. Indeed, with aij=aj−aiandbij=bj−bi, where {ai}and{bi}are the rankings to be compared, we obtain the Spearman ρ[23] (2) ρ=P i,j(ai−aj)(bi−bj)qP i,jai−aj)2P i,j(bi−bj)2. Introducing the quantities ¯a=P jajand¯b=P jbj, Eq. 2 can also be reformulated a... | https://arxiv.org/abs/2504.08428v1 |
for large n, the multiplicative scheme can- not discriminate different rankings when they differ only by the exchange of a top rank aiand a low rank bi, as the factor f(bi)makes the quantity winegligible. For this reason, the additive scheme is more suitable for several use cases where such discrimination is relevant [... | https://arxiv.org/abs/2504.08428v1 |
for Standard Ranking Correla- tion Coefficients. For both the standard Spearman ρand Kendall τcoefficients, the expected value in Eq. 19 can be straightforwardly calculated due to the symmetry of their form in the permutation formalism. In fact, if we fo- cus on Eq. 16, we can notice that, for any permutation π, we can... | https://arxiv.org/abs/2504.08428v1 |
the rankings Γandg(Γ), i.e., given any pair of permutations π1andπ2, ifΓ(π1)>Γ(π2), then g(Γ(π1))> g(Γ(π2)). Note that, since the original coeffi- cient Γis already constrained in the [−1,1]domain, con- sistency condition 1 is a consequence of conditions 2 and 4. We search for a piecewise polynomial function in the dom... | https://arxiv.org/abs/2504.08428v1 |
this case, the monotonic requirement (consistency condition 4) determines a set of inequalities for g0. For several combinations of coefficient type, weighting scheme, weighting function, and ranking lengths, these constraints are compatible with g0= 0, which is therefore a fairly natural choice, as it implies g(¯Γ) = ... | https://arxiv.org/abs/2504.08428v1 |
¯Γbased on numerical sampling. • Use polynomial regression to fit numerical sam- ples, obtaining an estimate of ¯Γas a function of the ranking length nin a wider domain.Algorithm 1 Setting g0 1:Input: ¯Γ,V,Vℓ 2:A= 3: B = 4: C = 5:D=C−2(1−¯Γ) 2(1−¯Γ)−B 6:E=−C B 7:F=2(1+ ¯Γ)−C 2(1+ ¯Γ)+B 8:blow 1=−1andbup 1= 1 ▷Bound i 9... | https://arxiv.org/abs/2504.08428v1 |
ranking lengths na, i.e., Strain={na= [qa]}, where ais an integer with amin≤a≤amaxand[x]repre- sents the rounding of xto the closest integer. In this work, we chose the following values. •q= 1.3, so that we have approximately 9 points na before an increase of a factor 10. •amin= 9, corresponding to namin= 11 , as the e... | https://arxiv.org/abs/2504.08428v1 |
the MSE for several poly- nomial degrees Dγ, as shown in Figure 2, considering the ratio of the MSEs associated with subsequent de- grees. We allow increasing the degree to Dγonly if the ratio MSE( Dγ)/MSE( Dγ−1)is below 0.8or if MSE( Dγ+1)/MSE( Dγ−1)is below 0.5, where the last condition was introduced to reduce the c... | https://arxiv.org/abs/2504.08428v1 |
be minimized is the standard sum of squared residuals, i.e., we do not consider different weights for the residuals. • The logarithmic transformation x(n) = 1/log(n)is used only for the multiplicative weighting scheme applied to the weighting function 1/(n+n0)2, as thex(n) = 1/ntransformation seems acceptable in all ot... | https://arxiv.org/abs/2504.08428v1 |
Eqs. 23. 10 TABLE 2 Results for Spearman coefficient with additive weighting scheme. f(i) 1/i 1/(i+n0)2 n0 - 0 1 2 ¯Γ3 -0.0351391 -0.12458 -0.0542795 -0.0302762 ¯Γ4 -0.055027 -0.1860625 -0.0908478 -0.0542042 ¯Γ5 -0.0699207 -0.2311256 -0.1214359 -0.0758427 ¯Γ6 -0.0820016 -0.2667351 -0.1480683 -0.0958261 ¯Γ7 -0.0921957 -... | https://arxiv.org/abs/2504.08428v1 |
40000 is_logvℓFalse True True True w0 0.004903 -0.013756 -0.00869 -0.005547 w1 2.397079 0.304449 0.120127 0.041578 w2 -68.24124 -0.746029 0.157677 0.52811 w3 1006.321 1.043273 - -0.477712 w4 -5002.647 - - - been implemented through the pseudo-random number generator included in the numpy.random Python library, using an... | https://arxiv.org/abs/2504.08428v1 |
is_logvFalse True True True z0 0.036587 0.025408 0.074257 0.115488 z1 2.836327 0.583835 0.368796 0.154672 z2 -36.09014 - - - z3 218.8208 - - - Vℓ3 0.2068706 0.2443723 0.215398 0.202704 Vℓ4 0.1345549 0.1595142 0.1437786 0.1350048 Vℓ5 0.1052886 0.1236619 0.1146279 0.1073594 Vℓ6 0.087606 0.1032788 0.0974049 0.0906875 Vℓ7 ... | https://arxiv.org/abs/2504.08428v1 |
tic matching. In European semantic web symposium 61–75. Springer. [9] H AUKE , J. and K OSSOWSKI , T. (2011). Comparison of values of Pearson’s and Spearman’s correlation coefficients on the same sets of data. Quaestiones geographicae 3087–93. [10] I MAN , R. L. and C ONOVER , W. (1987). A measure of top–down correlati... | https://arxiv.org/abs/2504.08428v1 |
arXiv:2504.08435v1 [math.ST] 11 Apr 2025High-dimensional Gaussian approximations for robust mean s Anders Bredahl Kock∗ University of Oxford Department of Economics 10 Manor Rd, Oxford OX1 3UQ anders.kock@economics.ox.ac.ukDavid Preinerstorfer WU Vienna University of Economics and Business Institute for Statistics and ... | https://arxiv.org/abs/2504.08435v1 |
limsupn→∞d nm/2−1+ξ>0, then limsupn→∞ρn= 1. Conversely, it is a simple consequence of Theorem 3.2 in Chernozhukov et al. (2023a), cf. Theorem 2.2 in Kock and Preinerstorfer (2024), thatρn→0 uniformly over a large class of distributions with bounded mth moments if there exists a ξ∈(0,∞) such that limn→∞d nm/2−1−ξ= 0. He... | https://arxiv.org/abs/2504.08435v1 |
a small exponent for mclose to two. An important advantage over the trimmed mean st udied inResende (2024) is that one does not need to know the number of moments mthat theXipossess in order to implement our winsorized and trimmed mea ns — they “adapt” tom. Furthermore, as m→ ∞, we allow dto grow almost as fast as exp(... | https://arxiv.org/abs/2504.08435v1 |
with entries possessing a continuous cdf is more restrictive. Liu and Lopes (2024) impose (the stronger) existence of a bounded density of the X1,j/σ2,jwith respect to the Lebesgue measure, whereas Re- sende(2024) avoids this assumption.1Finally, all results remain valid absent adversarial contamination, i.e., for ηn= ... | https://arxiv.org/abs/2504.08435v1 |
dependence of which on c, viaεn, is suppressed) in the form of an upper bound on ρn,W:= sup t∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingleP/parenleftbigg max j=1,...,dSn,W,j≤t/parenrightbigg −P/parenleftbigg max j=1,...,dZj≤t/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle,whereZ∼Nd(... | https://arxiv.org/abs/2504.08435v1 |
n 2.1 in the previous reference to imply that P/parenleftBig max j=1,...,dZn,j≤t/parenrightBig in the previous display can be replaced by P/parenleftBig max j=1,...,dZj≤t/parenrightBig . Finally, we show that for j∈ {1,3}one has that max j=1,...,d|In,j,l|are sufficiently small for a Gaussian anti-concentration inequality... | https://arxiv.org/abs/2504.08435v1 |
satisfied with m >2. Further- more, assume that Y1,jY1,kpossesses a continuous cdf for all 1≤j < k≤d. Ifε′ n∈(0,1/2), withε′ nas in(9), then for a constant C=C(b2,c,m), P/parenleftBigg max 1≤j,k≤d/vextendsingle/vextendsingle˜Σn,j,k−Σj,k/vextendsingle/vextendsingle> C/bracketleftbigg η1−2 mn+/parenleftBiglog(dn) n/parenr... | https://arxiv.org/abs/2504.08435v1 |
an estimate of σ2,jto bring the variables on the same scale. We now describe how the Gaussian and bootst rap approximations established so far remain valid upon normalization by the di agonal elements ˜ σn,j=˜Σ1/2 n,j,j of the robust estimator ˜Σnof Σ, cf. ( 10). Writing D= diag(σ2,1,...,σ 2,d) and Σ 0= D−1ΣD−1for the ... | https://arxiv.org/abs/2504.08435v1 |
2 in Resende (2024) where the amount of trimming depends on m. Thus, the same discussion as that following Theorem 2.1also applies to the trimmed mean that we study in ( 15). The following theorem is the trimmed mean analogue to Theore m3.2on bootstrap approximation to the distribution of winsorized means. Theorem 5.2.... | https://arxiv.org/abs/2504.08435v1 |
J. Li, A. Moitra, and A. S tewart (2019): “Robust estimators in high-dimensions without the computa tional intractability,” SIAM Journal on Computing , 48, 742–864. Diakonikolas, I. and D. Kane (2023):Algorithmic high-dimensional robust statistics , Cambridge University Press. Fang, X. and Y. Koike (2021): “High-dimens... | https://arxiv.org/abs/2504.08435v1 |
and bootstrap a pproximations for trimmed sample means,” arXiv preprint arXiv:2410.22085 . Rockafellar, R. T. (1997):Convex Analysis , Princeton University Press. Zhang, D. and W. B. Wu (2017): “Gaussian approximation for high dimensional time series,”Annals of Statistics , 45, 1895–1919. 19 A A useful decomposition We... | https://arxiv.org/abs/2504.08435v1 |
equality, that /vextendsingle/vextendsingleET1,jT2,k/vextendsingle/vextendsingle ≤E/parenleftbig |X1,j−µj||µk+σm/(1−εn−c−1εn)1/m−X1,k|1/parenleftbig X1,k<αk/parenrightbig/parenrightbig ≤E/parenleftbig |X1,j−µj||X1,k−µk|+|X1,j−µj|σm/(1−εn−c−1εn)1/m/parenrightbig 1/parenleftbig X1,k<αk/parenrightbig ≤/parenleftBig E/pare... | https://arxiv.org/abs/2504.08435v1 |
mn=o/parenleftbig n2m−4 5m−2η1−2 mn/parenrightbig and n2m−4 5m−2η1−2 mn→0⇐⇒n2m2−4m (5m−2)(m−2)ηn→0. The latter convergence is satisfied since√nη1−1 m→0 by assumption, which is equivalent tonm 2(m−1)ηn→0, andn2m2−4m (5m−2)(m−2)ηn≤nm 2(m−1)ηn→0 form >2. Next, log2(d)/bracketleftBiglog(d) n/bracketrightBig1−2 m =log(d)3−2 ... | https://arxiv.org/abs/2504.08435v1 |
1/2 implies the existence of a constant C1depending only on b1,b2,candmsuch that sup t∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingleP/parenleftbig Yn≤t/parenrightbig −P/parenleftbigg max j=1,...,dZj≤t−√n(In,1+In,3)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C1/parenleftbigglog5−2... | https://arxiv.org/abs/2504.08435v1 |
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