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10%. We carry out both one-sided and two-sided tests. For simplicity, we choose the same type of hypothesis for both outcomes.9 To implement our approach, we must obtain an estimate ˆVof the variance-covariance matrix of the estimated treatment effects. We accomplish this by estimating the fixed- effect regression (6)s... | https://arxiv.org/abs/2503.22369v1 |
specification with multiple treatments, outcomes, and subgroups. In addition to the four subgroups from Section 8 and the outcomes from Section 8, they define three treatments corresponding to the randomized matching ratios 1:1, 1:2, and 1:3. This gives rise to 48 treatment effects corresponding to the unique combinati... | https://arxiv.org/abs/2503.22369v1 |
selection. 7 Table A.2: Inference after multiple hypothesis testing for Table 5 in List, Shaikh, and Xu (2019) based on two-sided testing and the BY2001 or the RW2005 procedures. For conditional inference, the point estimate is the conditionally median-unbiased estimator, and the 90% conditional confidence interval is ... | https://arxiv.org/abs/2503.22369v1 |
given incl match red county in red state 3:1 vs. control0.40 (0.18, 0.61)0.35 (0.03, 0.60)0.51 (0.26, 0.98)0.34 (0.03, 0.50)0.39 (0.23, 0.51) Dollars given not incl match blue county in blue state 1:1 vs. control-0.00 (-0.06, 0.06)- - - - Dollars given not incl match blue county in blue state 2:1 vs. control0.01 (-0.06... | https://arxiv.org/abs/2503.22369v1 |
Wolf (2005, RW2005). B.1. More step-down rules The table below contains a list of some well-known step-down rules. αj Step-down rule Controls α mBonferroni correction FWER 1−(1−α)1/m ˇSid´ ak correction ( ˇSid´ ak 1967) FWER α m+1−jBonferroni-Holm correction (Holm 1979) FWER 1−(1−α)1 m+1−jˇSid´ ak-Holm correction FWER ... | https://arxiv.org/abs/2503.22369v1 |
σ−1 n(i) =σ−1 n+1(i) for all i={1, . . . ,|Sc|} \ { j, j+ 1}. Moreover, suppose that under σn, the neighbors are ordered in descending order, i.e., xh> x h′. Ifσnsatisfies the sufficient condition, so does 13 σn+1. To prove this, we have to verify the two conditions xh<¯xσn+1(h)and xh′<¯xσn+1(h′). These conditions are ... | https://arxiv.org/abs/2503.22369v1 |
impose all required restrictions on h /∈S, and we only have to check the restrictions imposed by the selection event for h∈S. To find intersections of{xz,h(xs)}h∈S, we can use the Bentley-Ottmann algorithm (Bentley and Ottmann 1979) that finds all intersections in O(˜nIlog|S|) time, where ˜nInow the number of intersect... | https://arxiv.org/abs/2503.22369v1 |
and similar to the modifications described above. We split the real line into intervals Ias described in Section 5 of the main text and define σ∗ I∈ E({1, . . . , m }) as the permutation that orders the elements of xz(xs) in descending order for xs∈I. We then iterate over all Iand find subsets [ ℓ(I), u(I)] ofIthat sat... | https://arxiv.org/abs/2503.22369v1 |
j= 1, . . . , k and xσ−1(j)≥¯xk+1for all j=k+ 1, . . . , m. Proof The result follows by taking the union over the conditions in Lemma B.1 for all S′⊇S. 19 E. Mathematical proofs of results in the main text Proof (Proof of Lemma 1) The only if part follows by taking σ=˜σsuch that ˜σ−1(j) = ( >, j) for j= 1, . . . ,|S|. ... | https://arxiv.org/abs/2503.22369v1 |
Idefined by the intersection points. This joint value is computed by σ∗ I=σ∗(¯x). To check xs∈ X s(z, S) we can therefore replace σ∗(xs) by σ∗ Iin(7). For h∈Swe have to check xh=xh(xs) = Ω h,sxs+zh≥¯xσ∗ I,h. If Ω h,s>0, this is equivalent to xs≥¯xσ∗ I,h−zh Ωh,s. If Ω h,s<0, this is equivalent to xs≤¯xσ∗ I,h−zh Ωh,s. Th... | https://arxiv.org/abs/2503.22369v1 |
We now argue that, provided ϵis small enough, hypothesis r∗must be downward sloping, i.e., C(s) r∗<0 and hence xr∗(xs) is decreasing in xs. Letσ∗ −(x) denote the limit of σ∗(x) as we approach xfrom the left, i.e., the limit of a sequence σ∗(xn) with xn→with xn∈(−∞, x). Let σ+(x) denote the correspondingly defined limit... | https://arxiv.org/abs/2503.22369v1 |
F(x;µ, σ)be the cumulative distribution function of a Gaussian random variable X∼N(µ, σ2)conditional on X∈S¯k k=1[lk, uk], where −∞ ≤ l1< u 1< l2< u 2<···< u ¯k≤ ∞ . It holds that, for any given x∈S¯k k=1(lk, uk)andσ >0, 1.F(x;µ, σ)is strictly decreasing in µ; 2.limµ→∞F(x;µ, σ) = 0 andlimµ→−∞ F(x;µ, σ) = 1 . The condit... | https://arxiv.org/abs/2503.22369v1 |
that Pµn,Ωn(ˆS=S) is bounded away from zero, it suffices to show that there do not exist subsequences on which Pµn,Ωn ˆS=S∧Fs Xs,|Z(s), Xs−cα−ϵ, S <1−α/2 ≥κ (14) and Pµn,Ωn ˆS=S∧Fs Xs|Z(s), Xs−cα−ϵ, S >1−α/2 ≥κ. (15) Assume that there exists a subsequence on which either (14)or(15)holds. We pass to this subsequ... | https://arxiv.org/abs/2503.22369v1 |
0, either inequality (17) is satisfied for all ξ∈Ror Y0(0,˜z, ν,Ω) =∅. For h∈Swith µh= Φ−1(νh) =∞, inequality (16)is satisfied for all ξ∈R. For h /∈Swith µh= Φ−1(νh) =−∞, inequality (17)is satisfied for all ξ∈R. Since ˜ xs∈ Y 0(0,˜z, ν,Ω) implies that Y0(0,˜z, ν,Ω) is non-empty, we have Yc(˜xs,˜z, ν,Ω) =R. Along with t... | https://arxiv.org/abs/2503.22369v1 |
We now establish the convergence of different quantities along appropriately chosen subsequences. We first consider the sequence of threshold function ¯xn. The domain Dof¯xn is the space of all subsets of {1, . . . , m }, which has cardinality 2m. Therefore, we can think of¯xnas a finite-dimensional vector that is cont... | https://arxiv.org/abs/2503.22369v1 |
ξ∈R: (1 + η)|ξ| −bn−min σ,h¯x∗ σ,h+η≥0 ≤P ξ∈n ξ∈R: (1 + η)|ξ| −bn−min nσ, h¯x∗ σ,h+η≥0o +ϵ/4≤ϵ/2. Fornlarge enough such that Pn(E1,η)≤ϵ/4 and Pn(E2,bn)≤ϵ/4, we have Pn(ˆS=S)≤ϵ. Since ϵis an arbitrarily small positive number, this contradicts Pn(ˆS=S)≥κ. Similarly, we can arrive at a contradiction if we assume that ... | https://arxiv.org/abs/2503.22369v1 |
is Fs(Ξ| Z, θs, S) =Fs(Ξ| Z, θs,¯x∗, v∗ s,Ω∗, S) =g∅,ℓ(Ω∗, µ∗,Z,¯x∗,Ξ,0). The sandwich Ys,H bΩ, µn,cMsdiag−1/2(ˆv)(ˆθ−θn),ˆ¯x ⊆Ys bΩ, µn,cMsdiag−1/2(ˆv)(ˆθ−θn),ˆ¯x ⊆Ys,H bΩ, µn,cMsdiag−1/2(ˆv)(ˆθ−θn),ˆ¯x ∪An, bounds gH,ℓ bΩ, µn,cMsdiag−1/2(ˆv)(ˆθ−θn),ˆ¯x,ˆXs−ˆv−1/2 sθn,s, an ≤g∅,ℓ bΩ, µn,cMsdiag−1/2(ˆv)(ˆθ−θn)... | https://arxiv.org/abs/2503.22369v1 |
The inclusions, A∩Sn⊆(A∩S)∪(Sn∩Sc) A∩S⊆(A∩Sn)∪(S∩Sc n) and the union bound for probabilities imply |qn|=|Pn(A∩Sn)−Pn(A∩S)| PnSn≤Pn(Sn△S) PnSn→0. Next, we show pn→QA, thus proving QnA→QAand concluding the proof. For a set AletAdenote the closure of A. For sets A, B it can be shown that ∂(A∩B)⊆ B∩∂A ∪ A∩∂B . 38 Apply... | https://arxiv.org/abs/2503.22369v1 |
Optimal treatment regimes for the net benefit of a treatment François Petit1, Gérard Biau3, and Raphaël Porcher1,2 1Université Paris Cité and Université Sorbonne Paris Nord, Inserm, INRAE, Center for Research in Epidemiology and StatisticS (CRESS), F-75004 Paris, France 2Centre d’Épidémiologie Clinique, Assistance Publ... | https://arxiv.org/abs/2503.22580v1 |
between the probability that a randomly selected participant in the experimental group achieves a better outcome than a randomly selected participant in the control group and the probability of the reverse. This approach involves hierarchi- cally comparing multiple outcomes between paired patients, with each pair consi... | https://arxiv.org/abs/2503.22580v1 |
benefit IPB (r0,r1)allows to define a notion of optimality for individualized treatment rules aiming at optimizing treatment decisions with respect to a hierarchy of outcomes. In this setting, the estimation of optimal treatment strategies reduces to accurately estimating IPB (r0,r1). While the IPB provides a useful me... | https://arxiv.org/abs/2503.22580v1 |
σ(Yi, Vj)to generalize the definition of σij. In the simple case of a single binary or continuous outcome, YandVare scalars, and the ordering simply relates to the natural ordering of R, adding the consid- 3 eration of whether larger values of Y(resp. V) are beneficial or detrimental to the individuals. The average tre... | https://arxiv.org/abs/2503.22580v1 |
i Unfavorable Any — Not considered −1 (1,1)or(0,0)V1 j▷ ◁ Y1 i Tie/neutral Y2 i−V2 j<−δ V2 j≻Y2 i Favorable +1 (1,1)or(0,0)V1 j▷ ◁ Y1 i Tie/neutral Y2 i−V2 j> δ V2 j≺Y2 i Unfavorable −1 (1,1)or(0,0)V1 j▷ ◁ Y1 i Tie/neutral |Y2 i−V2 j| ≤δ V2 j▷ ◁ Y2 i Tie/neutral 0 4 3 Individualized pairwise comparisons 3.1 General ide... | https://arxiv.org/abs/2503.22580v1 |
patient with characteristics X(resp. U) with observed outcome Y(resp. V) has two potential outcomes Y(0)(resp. V(0)) and Y(1)(resp. V(1)) representing the outcome s/he would achieve if, possibly contrary to fact, s/he had received treatment option A= 0orA= 1 respectively. An ITR is a map r:X → { 0; 1}which assigns to e... | https://arxiv.org/abs/2503.22580v1 |
predicts for each pair (X, U)the pseudo- outcomes corresponding to the status of the pair (favorable, neutral, unfavorable) and deduce from it an estimate of ∆(r0,r1)which in turn provides an estimates of the optimal ITR. We recall the data generating mechanism that we are considering. 3.3.1 Model of the data generatin... | https://arxiv.org/abs/2503.22580v1 |
variables and defining a meaningful distance metric for mixed data types presents a significant challenge. Therefore, in the next section, we propose a classification-based approach to estimate the optimal ITR. 3.4 Classification-based approach We consider data from patients in the control group (Xi, Yi)1≤i≤mand from p... | https://arxiv.org/abs/2503.22580v1 |
A simpler alternative is to construct k= min( m, n)iid pairs of the form (Xi, Yi, Ui, Vi)1≤i≤k, yielding an iid sample of kvariables (Xi, Ui, σ(Yi, Vi)). This dataset, denoted D′ k, allows learning ∆(x, u) but is inefficient, as it utilizes only min(m, n)of the mnavailable pairs. To address these limitations, we propos... | https://arxiv.org/abs/2503.22580v1 |
defined in Table 2 using δ= 3. In both cases, we computed the oracle ∆(r0,r1)and the corresponding optimal individualized treatment rule (ITR). Details of the data generation mechanism are provided in Appendix E. 10 Data Generation: For each scenario, we generated two datasets of 2 million individuals each: one for tra... | https://arxiv.org/abs/2503.22580v1 |
Matthew correlation coefficient (S.E) 0.62 (0.05) 0.70 (0.02) 0.64 (0.04) 0.74 (0.02) Specificity (S.E) 0.81 (0.03) 0.86 (0.02) 0.84 (0.03) 0.88 (0.02) Sensitivity (S.E) 0.73 (0.07) 0.80 (0.03) 0.77 (0.06) 0.83 (0.03) 11 Table 4: Confusion Matrix for the different scenarios and trial sizes Treatment Trial of size 400 T... | https://arxiv.org/abs/2503.22580v1 |
and experimental ( E) groups, respectively, with outcomes (Y1 i, V1 j)and(Y2 i, V2 j)with threshold δ= 5. Outcome with higher priority: Outcome with lower priority: OHS at 6 months EuroQol score at 6 months (Y1 i, V1 j) Ordering Label (Y2 i, V2 j) Ordering Label σij V1 j−Y1 i>0V1 j≻Y1 i Favorable Any — Not considered +... | https://arxiv.org/abs/2503.22580v1 |
main- taining a computational complexity comparable to the S-learner. While it accurately estimates the Conditional Average Treatment Effect (CATE) in binary outcomes, it offers neither a distinct advantage nor a theoreti- cal disadvantage. However, its memory footprint is substantial, scaling quadratically with N=n+m,... | https://arxiv.org/abs/2503.22580v1 |
variables collected from prerandomisation scan Expert reader’s assessment of acute ischaemic change Scan completely normal 9.6% 8.0% Scan not normal but no sign of acute ischaemic change 48.9% 53.2% Signs of acute ischaemic change % 41.4% 38.7% 15 likely closely related to the win ratio. However, the latter is a ratio ... | https://arxiv.org/abs/2503.22580v1 |
benefit-risk of new treatments using generalised pairwise comparisons: the case of erlotinib in pancreatic cancer. Br J Cancer 2015; 112:971–6. [15] Buyse M, Saad ED, Peron J, Chiem JC, De Backer M, Cantagallo E, Ciani O. The net benefit of a treatment should take the correlation between benefits and harms into account... | https://arxiv.org/abs/2503.22580v1 |
Let f:X →R be a locally integrable function. If µ(x)>0, then xis a Lebesgue point of f. This setting applies in particular to the situation where Xis at most countable and µis a discrete probability measure. In this setting, we can endow Xwith the discrete distance i.e., d(x, y) = 0 ifx=yandd(x, y) = 1 otherwise . (iii... | https://arxiv.org/abs/2503.22580v1 |
implies in particular that the value ∆(r0,r1)(x, x)is well-defined or in other word that IPB (x)is well- defined. The Hypothesis H.1 can be understood as claiming that the pairwise benefit of a pair is the average of the pairwise benefit of the neighboring pairs. We need to verify that the functions ∆(ri,rj)represent t... | https://arxiv.org/abs/2503.22580v1 |
(x, u)∈supp( µ⊗µ) ∆(ri,ri)(x, u) =−∆(ri,ri)(u, x). Setting x=u, this implies ∆(ri,ri)(x, x) = 0 . Lemma B.5. Letrandsbe two ITRs. Then E(σ(Y(r), V(s))|X, U) =r(X)s(U)∆(r1,r1)(X, U) +s(U)(1−r(X))∆(r0,r1)(X, U) +s(U)(1−r(X))∆(r0,r1)(X, U) + (1 −s(U))r(X)∆(r1,r0)(X, U) + (1−r(X))(1−s(U))∆(r0,r0)(X, U)a.s. 20 Proof. Writin... | https://arxiv.org/abs/2503.22580v1 |
possible to recover the notion of proportion in favor of treatment from the notion of proportion in favor of the rule by setting r=r0ands=r1and by assuming that Xis a discrete random variable with value in X={1, . . . , K }endowed with a distance dX(for instance the discrete distance). 22 We now provide closed formulas... | https://arxiv.org/abs/2503.22580v1 |
2||x−u||. Hence, for every η >0, we have for εsufficiently small, |∆(r0,r1)(x, u)−∆(r0,r1)(u, x)| ≤η. This implies thatE(|∆(r0,r1)(X, U)−∆(r0,r1)(U, X)||(X, U)∈Dε)≤ηwhich proves that limε→0(Λε−Γε) = 0 . This implies that limε→0Λεexists if and only if limε→0Γεexists and, if so, they are equal. This conclude the proof. (... | https://arxiv.org/abs/2503.22580v1 |
limit along εand the integration against α. By the Lebesgue differentiability of gα(β)at zero, for almost every α∈Rn lim ε→0h(α, ε) =gα(0) = [s(α)−r(α)]∆(r0,r1)(α, α)f2(α). We can further assume that ε≤2, thus||β|| ≤1. As the support of fis compact, we consider the set K={x∈Rn;d(x,supp( f))≤1} where d(x,supp( f)) = inf... | https://arxiv.org/abs/2503.22580v1 |
n, (iv)supm,n(cmenmax i,j(vmi,nj))<∞. Then the corresponding two-samples conditionnal U-statistics TNsatisfies E|TN(x, u)−T(x, u)|2→0at all (x, u)insupp( µ⊗µ). In particular, for all (x, u)∈supp( µ⊗µ), TN(x, u)→T(x, u)in probability . Remark C.2. We wish to emphasize that the above theorem does not only hold for almost... | https://arxiv.org/abs/2503.22580v1 |
borel mesurable function. Let T(X, U) =E(σ(Y, V)|X, U). For 1≤i≤m,1≤j≤n, set Z(i)(j)(x, u) =σ(Y(i)(x), V(j)(u))−T(X(i)(x), U(j)(u)). For1≤i, p≤mand1≤j, q≤nwithi̸=pandj̸=q, the random variables Z(i)(j)(x, u)andZ(p)(q)(x, u) are independent conditional on X1, . . . X m, U1, . . . , U n. Moreover, E(Z(i)(j)(x, u))|X1, . .... | https://arxiv.org/abs/2503.22580v1 |
β∈Ii(ℓ,n)P(Z= (α, β)|L=ℓ)P(L=ℓ) =ℓ0X ℓ=1k ℓ qℓ k(1−qk)k−lX α∈I(ℓ,m) β∈Ii(ℓ,n)P(Z= (α, β)|L=ℓ) =ℓ0X ℓ=1k ℓ qℓ k(1−qk)k−l−→ k→∞0. (D.3) Combining the expressions (D.1), (D.2) and (D.3), we get that for ksufficiently large P(|gL(x, u,D′ m,n(Z))−∆(x, u)|> δ)≤2ε, which proves the claim. E Simulation In this section, we ... | https://arxiv.org/abs/2503.22580v1 |
scenario, it yields ∆(r0,r1)(x, u) =X y,v∈S25 y225 v2 σ(y, v)p1(x)p2(x, y1)q1(u)q2(u, v1), where y= (y1, y2),v= (v1, v2)andS={0,1} × { 0, . . . , 25}. F List of personalization variables The model used to construct the ITR from the IST-3 uses as outomes the variables ohs6 andeuroqol6 data. It takes into account the... | https://arxiv.org/abs/2503.22580v1 |
Score at randomisation, 11.treatdelay : Time (hour) from stroke to treatment, 12.konprob : Probability of good outcome based on Konig model, 13.R_mca_aspects : Aspects score (max 10) for Middle cerebral artery (R scan), 14.R_tot_aspects : Total aspects score (max 12) including ACA/PCA (R scan). 35 G Single outcome Here... | https://arxiv.org/abs/2503.22580v1 |
Tracy-Widom, Gaussian, and Bootstrap: Approximations for Leading Eigenvalues in High-Dimensional PCA Nina D¨ ornemann∗ Department of Mathematics Aarhus UniversityMiles E. Lopes Department of Statistics University of California, Davis April 1, 2025 Abstract Under certain conditions, the largest eigenvalue of a sample co... | https://arxiv.org/abs/2503.23097v1 |
the best of our knowledge, the problem (1.1) has not been systematically addressed in the literature. But, at first sight, this problem might be conflated with one that has been extensively studied—the problem of detecting “spike” population eigenvalues [18, 44, 29] —and so it is important to clarify why the two proble... | https://arxiv.org/abs/2503.23097v1 |
is possible to explain our problem formulation here with only a bit of notation. For any k= 0,1,...,p−1 such that λk+1(Σ)>0, define the parameter ξn,kto be the unique value in the interval (0 ,1/λk+1(Σ)) that solves the equation 1 pp/summationdisplay j=k+1/parenleftbiggλj(Σ)ξn,k 1−λj(Σ)ξn,k/parenrightbigg2 =n p. (1.3) ... | https://arxiv.org/abs/2503.23097v1 |
to formulating the problem (1.5) in terms of ε. It is due to a very fine-grained effect that can occur when|λ1(Σ)−1/ξn,1|=O(n−1/3), i.e. when λ1(Σ) is in a vanishing interval of radiusO(n−1/3) centered at the threshold 1 /ξn,1. In this exceptional boundary case, the fluctuations of λ1(/hatwideΣ) may not be well approxi... | https://arxiv.org/abs/2503.23097v1 |
that no parameters in the limiting distribution need to be estimated. Likewise, it would be reasonable to consider Rn(κ) as a candidate test statistic for the proposed testing problem (1.5), but as we explain below, this statistic has serious limitations that we seek to overcome. Limitations of gap ratios. The asymptot... | https://arxiv.org/abs/2503.23097v1 |
scale estimation problem, and in Proposi- tion 1 we establish its consistency within the subcritical regime. Consequently, it is not necessary for us to rely on gap ratios, and the proposed test statistic Tnis able to avoid the power losses associated with the statistic Rn(κ) discussed earlier. Furthermore, in Section ... | https://arxiv.org/abs/2503.23097v1 |
sample eigenvalues. Numerical results involving both simulated data and financial data are presented in Section 7. Lastly, all proofs are deferred to Section 8. 2 Preliminaries and setup LetY∈Rn×pbe a random data matrix whose rows consist of ncentered i.i.d. observations inRp. The associated sample covariance matrix is... | https://arxiv.org/abs/2503.23097v1 |
Hn, andσnin the following three paragraphs. Estimation of ξn,0andξn,1.Recall the formula ξn,0=−s0 n(rn) from equation (2.6). Since rnapproximately corresponds to the right edge of the limiting bulk sample spectrum, it is naturally estimated by λ1(/hatwideΣ). Meanwhile, the Stieltjes transform s0 nis the limit of sn=sF/... | https://arxiv.org/abs/2503.23097v1 |
1(c) and the limit (8.5) given later. Thus, the primary technical challenge in establishing Theorem 1 lies in proving Proposition 1. As an added benefit of our work so far, it is possible to develop confidence intervals for the parameters λ1(Σ) andrn. For this purpose, define /hatwideεas the smallest ε >0 for which H0,... | https://arxiv.org/abs/2503.23097v1 |
Specifically, if H1,n(K) holds for some unknown value of K≥1, then a consistent estimate of Kcan be constructed as follows. For eachj= 1,...,p−1, define the jth gap statistic n2/3 /hatwideσn/parenleftig λj(/hatwideΣ)−λj+1(/hatwideΣ)/parenrightig , which specializes to Tnwhenj= 1. The estimate of Kis then defined by /... | https://arxiv.org/abs/2503.23097v1 |
is to normalize λ1(/hatwideΣ⋆),...,λk(/hatwideΣ⋆) so that their fluctuations can be theoretically compared with those of λ1(/hatwideΣ),...,λk(/hatwideΣ). In detail, for any fixed integers k,m≥1 and continuous function g:Rk→Rm, the comparison will be made between the random vectors Vn=g/parenleftbigg n2/3 σn(λ1(/hatwide... | https://arxiv.org/abs/2503.23097v1 |
test statisticsTnandRn(κ) withκ∈{1,10}, and recorded whether or not H0,nwas rejected at a nominal level of α= 0.05. We refer to the proportion of rejections occurring among these 800 trials as the rejection probability. Note also that the critical value for Rn(κ) is given by the (1−α)-quantile of the random variable ma... | https://arxiv.org/abs/2503.23097v1 |
anticipated by the discussion in the introduction (page 6). In particular, this illustrates the sensitivity of the statistic Rn(κ) to the choice of κ. rejection probabilities hypothesis K(λ1(Σ),λ2(Σ),λ3(Σ))TnRn(1)Rn(10) H0,n 0 (1,1,1) 0.05 0.05 0.04 H0,n 0 (1.25,1.25,1.25) 0.04 0.04 0.07 H0,n 0 (1.25,1.25,1) 0.06 0.07 ... | https://arxiv.org/abs/2503.23097v1 |
playing the ground truth values, and the lower row summarizing the bootstrap estimates. Specifically, each lower row contains the means of the bootstrap estimates (with standard deviation in parentheses) for E(Ln),E(Gn),/radicalig var(Ln), and/radicalig var(Gn), as well as the cov- erage probabilities P(Ln≤/hatwideq0... | https://arxiv.org/abs/2503.23097v1 |
that arise from the two datasets when using the proposed statisticTnand Onatski’s statistic Rn(κ) withκ∈{1,10}. Up to two decimal places, the proposed statistic gives p-values that are 0, whereas the p-values for Rn(1) andRn(10) are non-zero. Furthermore, the p-values for Rn(10) are larger than those for Rn(1), which i... | https://arxiv.org/abs/2503.23097v1 |
to the rightmost edge of its support, as recorded below. Lemma 2. [9, Lemma 2.1] Suppose that Assumptions (A1) and (A3) hold, and that H0,n holds for all large n. Then,Fyn,Hnhas a continuous derivative, denoted ρ0 n, onR\{0}, and there exists a constant c>0not depending on nsuch that ρ0 n(λ)≍/radicalig rn−λfor allλ∈[r... | https://arxiv.org/abs/2503.23097v1 |
definition of ηn, then we haven−1+τ=ηn≤1 for alln. Thus, the limit (8.12) holds, and the proof of (8.11) concludes. Before turning to the proof of Lemma 1(b), we need two auxiliary results, which will be proven in Section 8.3. The following lemma shows that the critical thresholds ξn,0and ξn,1are asymptotically equival... | https://arxiv.org/abs/2503.23097v1 |
f∈BL/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 pp/summationdisplay j=1f(˜λj,Q∧/hatwidebn)−f(˜λj,Q∧b0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+ sup f∈BL/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 pp/summationdisplay j=1f(˜λj,Q∧b0)−f(˜... | https://arxiv.org/abs/2503.23097v1 |
0 be defined by 1 /(1 + ˘ε) =λ1(Σ)/hatwideξn.As a preparatory step, we want to show that Tn(˘ε)D→ζ−ζ′, n→∞. (8.23) 27 For eachj∈{1,...,p}, let ˘λj=˜λj,Q∧1 /hatwideξn(1 + ˘ε)=˜λj,Q∧λ1(Σ) and define the associated distribution spectral distribution function ˘Hn(t) =1 pp/summationdisplay j=11{˘λj≤t}. Using an argument ana... | https://arxiv.org/abs/2503.23097v1 |
all convergent subsequences of {ξn,0}converge to the same value. To this end, letJ⊂Nbe a subsequence along which ξn,0converges to some limit ξ′ 0. It suffices to show that ξ′ 0resides in the interval (0 ,1/u) and solves the equation /integraldisplay/parenleftbigg λξ′ 0 1−λξ′ 0/parenrightbigg2 dH(λ) =y, (8.27) because t... | https://arxiv.org/abs/2503.23097v1 |
respect to n, and that H1,n(K)holds for all large n. Then, as n→∞ , the following limit holds for all 1≤j≤K √n ςj,n/parenleftig λj(/hatwideΣ)−ψn(λj(Σ))/parenrightigD→N (0,1). (8.36) Also, if H1,n(K)holds forK≥2, and if 1≤j <K , then asn→∞ , Dn,jD→N (0,1). (8.37) Proof of Proposition 5. This result follows from a comb... | https://arxiv.org/abs/2503.23097v1 |
Next, we turn to the case when K > 1. By Proposition 5, we know that as n→∞ , Dn,1D→N (0,1). (8.45) Moreover, note that Hn,K+1D→Handλ2(Σ)−λK+1(Σ)≳1 under H1,n(K). This gives limn→∞ψn(λj(Σ)) =ψ(λj(Σ)), j∈{1,2}. (8.46) Using (8.39), (8.40), (8.45), (8.46) and the formula Tn=n1/6γ1,n /hatwideσnDn,1+n2/3 /hatwideσn{ψn(λ1(Σ... | https://arxiv.org/abs/2503.23097v1 |
completes the proof. Proof of Theorem 3. Let the function gbe as in the definition of VnandV⋆ n, and let (ζ1,...,ζk) have ak-dimensional Tracy-Widom distribution. It is known from [34, Corol- lary 3.19] that the limit limn→∞d(L(Vn),L(g(ζ1,...,ζk)) = 0 (8.54) is implied by the conditions (8.1), (A1), (A2), and the follo... | https://arxiv.org/abs/2503.23097v1 |
density of the spectrum on the edge for sample covariance matrices with general population. Preprint. https://personal.ntu.edu.sg/gmpan/publications.html . [10] Bao, Z., G. Pan, and W. Zhou (2015). Universality for the largest eigenvalue of sample covariance matrices with general population. The Annals of Statistics 43... | https://arxiv.org/abs/2503.23097v1 |
1231 – 1254. [30] Johnstone, I. M. and D. Paul (2018). PCA in high dimensions: An orientation. Proceedings of the IEEE 106 (8), 1277–1292. [31] Kabundi, A. and F. N. De Simone (2020). Monetary policy and systemic risk-taking in the euro area banking sector. Economic Modelling 91 , 736–758. [32] Karoui, N. E. (2007). Tr... | https://arxiv.org/abs/2503.23097v1 |
spectral distribution of large dimensional random matrices. Journal of Multivariate Analysis 54 (2), 295–309. [51] Tao, T. (2012). Topics in Random Matrix Theory . American Mathematical Society. [52] Wang, S. and M. E. Lopes (2023). A bootstrap method for spectral statistics in high-dimensional elliptical models. Elect... | https://arxiv.org/abs/2503.23097v1 |
Optimal Change Point Detection and Inference in the Spectral Density of General Time Series Models Sepideh Mosaferi Department of Mathematics and Statistics University of Massachusetts Amherst and Abolfazl Safikhani Department of Statistics George Mason University and Peiliang Bai Microsoft Abstract This paper addresse... | https://arxiv.org/abs/2503.23211v2 |
Fryzlewicz (2018) that apply a tail-greedy Haar transformation to consistently estimate the number and locations of multiple change points in the univariate piecewise-constant model, Inspect method (Wang and Samworth, 2018) which proposes a high-dimensional change-point detection method that uses a sparse projection to... | https://arxiv.org/abs/2503.23211v2 |
critical for understanding brain dynamics (Safikhani and Shojaie, 2022). While existing methods in the literature primarily assume specific parametric forms before and after the change point, we take a nonparametric approach by leveraging the well-known Wold decomposition. This allows us to represent the pre- and post-... | https://arxiv.org/abs/2503.23211v2 |
research. Technical details and additional numerical results are available in the Supplementary Material. Notation. Throughout this paper, we use Rto represent the real line. For any vector v∈Rp,∥v∥1,∥v∥2, and∥v∥∞indicate the ℓ1norm, the Euclidean norm, and the maximum norm, respectively. For the sub-Weibull distributi... | https://arxiv.org/abs/2503.23211v2 |
allows the construction of confidence intervals for the unknown change point. 2.3 Definition of sub-Weibull random variable In the following, we provide the definition of an Orlicz norm for random variables and properties of the sub-Weibull distribution. Here, we refer to Section 2.7.1 of Vershynin (2018) and the refer... | https://arxiv.org/abs/2503.23211v2 |
same lag for notational simplicity (see Section 5 for more details on how the lag is selected in practice). Finally, we construct the objective function according to sum of squared error with respect to the time point ⌊Tτ⌋as well as fitted AR( p) model parameters ˆϕ1andˆϕ2: L(τ)def=⌊Tτ⌋X t=1(Xt−ˆϕ′ 1Zt)2+TX t=⌊Tτ⌋+1(Xt... | https://arxiv.org/abs/2503.23211v2 |
series cannot be close to the bound- aries. Condition (b) demonstrates the connection between sample size and lag p, as well as imposing a lower bound on the model parameter differences before/after the change point (Kaul and Michailidis, 2025). As stated, the selected lag pincreases by increasing sample size, and thus... | https://arxiv.org/abs/2503.23211v2 |
setup, we define the refitted least squares estimator as: ˜τ= arg min τ∈{2/T,3/T,..., (T−1)/T}Q(τ;ˆϕ1,ˆϕ2). (8) The following Algorithm 1 summarizes the procedure to obtain the optimal estimator of the change point parameter ⌊Tτ⋆⌋. The main idea is to separate the estimation of the model parameters and the detection of... | https://arxiv.org/abs/2503.23211v2 |
TlT, (12) with probability at least 1 −o(1), where lTis a sequence defined in Condition B(a), and cu>0 is some universal large enough constant. Moreover, the following condition on the tail parameters of the Wold representation is required. Suppose pis the fixed order of autoregressive model, then the tail lags of the ... | https://arxiv.org/abs/2503.23211v2 |
ϕ= 0.5. •Scenario (III): before change point: AR(3) process Xt= 0.9Xt−1−0.5Xt−2+0.3Xt−3+ ϵt; after change point: AR(1) process Xt=ϕXt−1+ϵtwhere ϕ=−0.9. 20 •Scenario (IV): before change point: MA(1) process Xt=ϵt+θϵt−1where θ=−0.9; after change point: AR(3) process Xt= 0.9Xt−1−0.5Xt−2+ 0.3Xt−3+ϵt. •Scenario (V): before ... | https://arxiv.org/abs/2503.23211v2 |
( ⌊T˜τ⌋) 95% CP ( ⌊T˜τ⌋) 99% CP ( ⌊T˜τ⌋) ⌊T/3⌋ 2.680 2.450 4.693 4.403 0.890 0.900 0.940 θ=−0.9⌊T/2⌋ 2.380 2.480 3.945 4.268 0.900 0.920 0.940 ϕ= 0.5⌊2T/3⌋2.470 2.130 3.759 3.318 0.910 0.930 0.980 ⌊4T/5⌋2.030 1.970 3.872 3.678 0.900 0.950 0.970 5.1 Comparisons with some existing methods In this section, we report empir... | https://arxiv.org/abs/2503.23211v2 |
Performance of the model for scenario V with T= 500. Coefficient Truth AB ( ⌊Tˆτ⌋) AB ( ⌊T˜τ⌋) RMSE ( ⌊Tˆτ⌋) RMSE ( ⌊T˜τ⌋) 90% CP ( ⌊T˜τ⌋) 95% CP ( ⌊T˜τ⌋) 99% CP ( ⌊T˜τ⌋) ⌊T/3⌋16.690 17.640 38.977 39.789 0.890 0.920 0.960 ϕ=−0.5⌊T/2⌋10.840 9.490 20.429 18.639 0.930 0.960 0.970 ⌊2T/3⌋8.260 8.390 13.854 14.798 0.910 0.95... | https://arxiv.org/abs/2503.23211v2 |
in the left temporal lobe, where the seizure activity was localized. Based on clinical assessment, the seizure is estimated to have occurred around t ≈85 s. The EEG recordings exhibit time-varying spectral characteristics, with notable changes in both magnitude and volatility occurring 26 Table 6: Comparison of coverag... | https://arxiv.org/abs/2503.23211v2 |
provided by the proposed method. Table 7: Location of estimated change points in the EEG data set with their CIs for the segment (1 ,110) based on our method. True change point is ⌊Tτ⌋= 85. Channel value ( ⌊T˜τ⌋) 70% CI 80% CI 90% CI 95% CI 99% CI P3 90 (64,96) (63,100) (60,107) (58,115) (52,136) T3 86 (85,87) (85,87) ... | https://arxiv.org/abs/2503.23211v2 |
that the true value is 174. pixel value ( ⌊T˜τ⌋) 70% CI 80% CI 90% CI 95% CI 99% CI 48 178 (176,188) (176,191) (176,198) (175,206) (175,226) 78 179 (175,184) (175,186) (175,191) (175,196) (174,209) 110 181 (168,187) (165,189) (160,194) (156,198) (145,211) 174 171 (170,172) (170,172) (170,172) (170,172) (170,333) 209 17... | https://arxiv.org/abs/2503.23211v2 |
models. Statistica Sinica 33 , 1–28. Basseville, M. and I. V. Nikiforov (1993). Detection of abrupt changes: theory and appli- cation , Volume 104. Prentice Hall Englewood Cliffs. Birnbaum, Z. and A. W. Marshall (1961). Some multivariate chebyshev inequalities with ex- tensions to continuous parameter processes. The An... | https://arxiv.org/abs/2503.23211v2 |
R., P. Fearnhead, and I. A. Eckley (2012). Optimal detection of changepoints with a linear computational cost. Journal of the American Statistical Association 107 (500), 1590–1598. Kolar, M. and E. P. Xing (2012). Estimating networks with jumps. Electronic Journal of Statistics 6 , 2069 – 2106. 36 Li, M. and Y. Yu (202... | https://arxiv.org/abs/2503.23211v2 |
H., D. Wang, Z. Zhao, and Y. Yu (2024). Change-point inference in high-dimensional regression models under temporal dependence. The Annals of Statistics 52 (3), 999–1026. Yu, Y. (2020). A review on minimax rates in change point detection and localisation. arXiv preprint arXiv:2011.01857 . 39 Supplementary Material for ... | https://arxiv.org/abs/2503.23211v2 |
similar equation, we can also derive that: ˆϕ1=ˆϕ(2) 1+ Ip−τ⋆ τ−τ⋆ˆΓ(1) 1 Ip+ˆΓ(2)−1 1ˆΓ(1) 1−1ˆΓ(2)−1 1τ⋆ τ−τ⋆ˆΓ(2)−1 1ˆγ(1) 1 −τ⋆ τ−τ⋆ˆΓ(1) 1 Ip+ˆΓ(2)−1 1ˆΓ(1) 1−1ˆΓ(2)−1 1ˆϕ(2) 1. Based on the detection algorithm, we consider the introduced objective function L(τ) and it can be written as follows: L(τ) =⌊Tτ⌋X ... | https://arxiv.org/abs/2503.23211v2 |
=1 T−p+ 1 ⌊Tτ⌋X t=⌊Tτ⋆⌋+1(Xt−ˆϕ′ 1Zt)2−⌊Tτ⌋X t=⌊Tτ⋆⌋+1(Xt−ˆϕ′ 2Zt)2 =1 T−p+ 1⌊Tτ⌋X t=⌊Tτ⋆⌋+1n (ˆη′Zt)2−2ϵtˆη′Zt+ 2(ˆϕ2−ϕ⋆ 2)′ZtZ′ tˆηo .(B.1) This algebraic rearrangement leads to the following result: inf τ∈G(uT,vT) τ≥τ⋆U(τ;ˆϕ1,ˆϕ2)≥ inf τ∈G(uT,vT) τ≥τ⋆1 T−p+ 1⌊Tτ⌋X t=⌊Tτ⋆⌋+1(ˆη′Zt)2− sup τ∈G(uT,vT) τ≥τ⋆2 T... | https://arxiv.org/abs/2503.23211v2 |
a large enough m, we have ⌊T˜τ⌋ − ⌊ Tτ⋆⌋ ≤Tcgm a3/Tumwith probability at least 1 −3a−o(1). Then continuing this recursion procedure for an infinite number of iterations leads to: ⌊T˜τ⌋ − ⌊Tτ⋆⌋ ≤Tc2 a3 T=c2 a3, with probability at least 1 −3a−o(1). It finishes the proof of Theorem 4.1. Proof of Theorem 4.2. Note that un... | https://arxiv.org/abs/2503.23211v2 |
t=⌊Tτ⋆⌋+1∥η⋆′Zt∥2 2, (B.10) s2 n=E TX t=⌊Tτ⋆⌋+1E(Y2 t|Ft−1) = (T− ⌊Tτ⋆⌋)ξ−2 2σ2η⋆Σ2 Zη⋆′, (B.11) then combining (C.3) and (C.4) implies that V2 T s2 T=PT t=⌊Tτ⋆⌋+1∥η⋆′Zt∥2 2 (T− ⌊Tτ⋆⌋)η⋆Σ2 Zη⋆′P→1, (B.12) asT→+∞, which holds based on ergodicity. Next, we focus on validating the Lindeberg condition. s−2 TnX t=⌊Tτ⋆⌋+... | https://arxiv.org/abs/2503.23211v2 |
also a sub- Weibull( γ2) distributed. Then ζt=ϵtη⋆′Ztis real valued sub-Weibull distributed, and by applying general H¨ older inequality, it can be proved by the following: ∥ζt∥k≤ ∥ϵtη⋆′Zt∥k≤ ∥ϵt∥2k∥η⋆′Zt∥2k≤ ∥ϵt∥2k(2k)1 γ2∥η⋆′Zt∥ψγ2 ≤ ∥ϵt∥2k(2k)1 γ2∥η⋆∥2∥Zt∥ψγ2≤(2k)2 γ2KϵKXξ2. Hence, we proved that ζt∼sub-Weibull( γ2/... | https://arxiv.org/abs/2503.23211v2 |
left hand side of both (i) and (ii) as: 1 T⌊Tτ⌋X t=⌊Tτ⋆⌋+1∥η⋆′Zt∥2 2=1 T⌊Tτ⌋X t=⌊Tτ⋆⌋+1pX j=1(η⋆ jZt,j)2, where η⋆ jandZt,jare the j-th coordinate of coefficient vector η⋆and random vector Zt. 57 Note that by using the property of sub-Weibull distribution (see Definition 2.3), we have: pX j=1st,j=pX j=1 (η⋆ jZt,j)2−E(... | https://arxiv.org/abs/2503.23211v2 |
we have: E(Sn+1|Fn) =E(Sn+ϵn+1η⋆′Zn+1|Fn) =Sn+E(ϵn+1|Fn)E(η⋆′Zn+1|Fn) =Sn, hence, we show that Snis a martingale. Moreover, we notice that for any i < j , we have: E(ζiζj) =E(E(ζiζj|Fi)) =E(ϵiη⋆′ZiE(ϵj|Fi)E(η⋆′Zj|Fi)) = 0 . Then we apply Theorem C.4 with ψk≡1 and r= 2 for all ktoSntogether with the result 60 of Lemma C... | https://arxiv.org/abs/2503.23211v2 |
≥κminξ2 2 vT−ca1σ2ruT T′−cuuT ξ2pp log(p∨T)∥ˆη−η⋆∥2 (ii) For the term J2in proof of Lemma 4.1, sup τ∈G(uT,vT) τ≥τ⋆1 T′ ⌊Tτ⌋X t=⌊Tτ⋆⌋+1ϵtˆη′Zt ≤ca122 γ2KϵKXξ2ruT T′+cuKr uTlog(p∨T) T∥ˆη−η⋆∥1, with probability at least 1−a−o(1). (iii) For the term J3in the proof of Lemma 4.1, sup τ∈G(uT,vT) τ≥τ⋆1 T′ ⌊Tτ⌋X t=⌊Tτ⋆⌋+1(ˆϕ2... | https://arxiv.org/abs/2503.23211v2 |
C are satisfied. Furthermore, let R11, R12, R13, R21,andR22be defined based on Lemma C.9. Consider a universal constant 0< cu<+∞, 67 then we have the following bounds (i) sup τ∈G(cuT−1ξ−2 2,0)|R21−R11|=op(1); (ii) sup τ∈G(cuT−1ξ−2 2,0)|R22−R12|=op(1); (iii) sup τ∈G(cuT−1ξ−2 2,0)|R13|=op(1), where each holds with probab... | https://arxiv.org/abs/2503.23211v2 |
we have: ⌊Tτ⌋X t=⌊Tτ⋆⌋+1 ∥η⋆′Zt∥2 2−E∥η⋆′Zt∥2 2 =op(1). Additionally, for any constant r >0and if ξ−2 2E∥η⋆′Zt∥2 2→σ2, then we have: ⌊Tτ⌋X t=⌊Tτ⋆⌋+1∥η⋆′Zt∥2 2P→rσ2. Proof of Lemma C.11. Note that if we assume ξ2→0, then by assuming that τ≥τ⋆, we can show that cu1rξ−2 2≤ ⌊Tτ⋆+rξ−2 2⌋ − ⌊Tτ⋆⌋ ≤cu2rξ−2 2. (C.12) Then us... | https://arxiv.org/abs/2503.23211v2 |
D.1: Performance of the model for Scenario I with T= 500. Coefficient Truth AB ( ⌊Tˆτ⌋) AB ( ⌊T˜τ⌋) RMSE ( ⌊Tˆτ⌋) RMSE ( ⌊T˜τ⌋) 90% CP ( ⌊T˜τ⌋) 95% CP ( ⌊T˜τ⌋) 99% CP ( ⌊T˜τ⌋) ⌊T/3⌋ 6.340 5.070 9.949 7.993 0.950 0.980 0.990 θ=−0.7⌊T/2⌋ 5.880 4.780 9.241 7.529 0.960 0.970 0.990 ϕ=−0.5⌊2T/3⌋7.030 6.110 11.405 10.356 0.93... | https://arxiv.org/abs/2503.23211v2 |
0.920 0.970 ⌊T/3⌋ 3.330 2.920 5.815 4.315 0.960 0.970 0.990 θ= 0.9⌊T/2⌋ 3.040 2.810 4.652 4.208 0.920 0.960 0.990 ϕ=−0.3⌊2T/3⌋3.140 2.560 5.667 4.200 0.940 0.950 0.970 ⌊4T/5⌋3.770 3.390 5.832 5.438 0.860 0.890 0.950 ⌊N/3⌋ 2.790 2.230 4.520 3.375 0.940 0.970 0.970 θ= 0.9⌊N/2⌋ 2.330 2.270 3.754 3.618 0.910 0.920 0.970 ϕ=... | https://arxiv.org/abs/2503.23211v2 |
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