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establishes a connection between the conditional moment of any measurable real function of the ob served data for the compliers and the unconditional moment. The monotonicity assumption rules o ut the existence of defiers, a latent subgroup for whom the treatment choice is always different fro m the assigned treatment. N... | https://arxiv.org/abs/2504.02096v1 |
the two-step estimation procedure. Section 5establishes the asymptotic properties of the proposed esti mator. Simulation results and an em- pirical application regarding the effect of job training prog rams on unemployment duration are described in Sections 6and7, respectively. The proofs, technical details, extra simul... | https://arxiv.org/abs/2504.02096v1 |
standard monotonic- ity assumption that rules out the existence of defiers. This a ssumption is automatically satisfied when individuals in the control group ( W= 0) cannot access the treatment. Finally, the first part of Assumption (A3)requires non-trivial assignment of the instrument, while t he second part is a standar... | https://arxiv.org/abs/2504.02096v1 |
dependence between TandC, given (Z,X⊤) for the compliers, we propose to use a bivariate copula C. The joint distribution function of TandC, conditional on ( Z,X⊤) for the compliers, can therefore be modeled as P(T≤t,C≤c|Z=z,X=x,G=co) =C/parenleftbig FT|Z,X,G(t|z,x,co),FC|Z,X,G(c|z,x,co)/parenrightbig , 7 witht,c >0 and... | https://arxiv.org/abs/2504.02096v1 |
0 for all ( z,x⊤) withk= 1,2. (C3) For all ( µ,η,ξk,Λk)∈ M×H× Ξ×L,k= 1,2, such that lim t→0λ1(t) λ2(t)= 1, we have that: lim y→0ζ2,ξ1/parenleftbig FT|Z,X,G,µ, Λ1(y|z,x,co),FC|Z,X,G,η(y|z,x,co)/parenrightbig ζ2,ξ2/parenleftbig FT|Z,X,G,µ, Λ2(y|z,x,co),FC|Z,X,G,η(y|z,x,co)/parenrightbig= 1 for all ( z,x⊤)⇐⇒ξ1=ξ2. 9 (C4) ... | https://arxiv.org/abs/2504.02096v1 |
the identifiability requirements of Deresa and Van Keilegom (2024a) also satisfy our pro- posed conditions, thereby relaxing the identifiability res ult by removing the untestable covariate restriction. Moreover, Lemma 1verifies the identifiability conditions for additional copu la functions to allow for more modeling flexi... | https://arxiv.org/abs/2504.02096v1 |
respectively. 4.2 Estimating the baseline cumulative hazard function The non-parametric estimator for Λ( · |G=co) will be constructed using martingale ideas from Theorem 1.3.1 by Fleming and Harrington (1991). Firstly, let Ii(y) = /BD(Yi≤y,∆1,i= 1) and ˜Ii(y) = /BD(Yi≥y). Following a similar martingale construction as ... | https://arxiv.org/abs/2504.02096v1 |
Λ to zero to arrive at the following equations: E[U(Si,θ∗,Λ∗)|G=co] = 0, where U/parenleftbig Si,θ∗,Λ∗/parenrightbig =∂ ∂θlogfY,∆1,∆2|Z,X,G,θ∗,Λ∗(Yi,∆1,i,∆2,i|Zi,Xi,co), Again, we can use Theorem 3.1 from Abadie(2003) to rewrite equation this in the following way: E/bracketleftBig κ∗(Si)×U/parenleftbig Si,θ∗,Λ∗/parenri... | https://arxiv.org/abs/2504.02096v1 |
constant ¯λfor allt∈[0,¯τ]. (C12) The functions S(t|z,x,co),ζ1/parenleftbig FT|Z,X,G(t|z,x,co),FC|Z,X,G(t|z,x,co)/parenrightbig and ζ2/parenleftbig FT|Z,X,G(t|z,x,co),FC|Z,X,G(t|z,x,co)/parenrightbig exist and are twice continuously differentiable with respect to θover Θ. Also, all derivatives of order two (with respect... | https://arxiv.org/abs/2504.02096v1 |
would be misspecified. It follows that we can determine Zby usingGandW. The next step depends on the choice of the copula and censoring distribution. As an examp le, we consider the following design (Frank - Weibull): P(T≤t,C≤c|Z=z,X=x,G=g) =Cξg/parenleftbig FT|Z,X,G(t|z,x,g),FC|Z,X,G(c|z,x,g)/parenrightbig , withCξga F... | https://arxiv.org/abs/2504.02096v1 |
as explained in Section 4.3withJ= 100 and a maximum amount of iterations equal to 120. The results for the scenario with low dependence can be found in Table 3. Each of these three designs has around 30 to 40 percent dependent and 5 to 10 perce nt administrative censoring. Note that the design in the middle column is b... | https://arxiv.org/abs/2504.02096v1 |
η20.083 0.091 0.123 0.850 0.103 0.111 0.151 0.786 0.129 0.174 0.217 0.922 η3-0.047 0.187 0.193 0.966 -0.096 0.225 0.244 0.898 -0.118 0.339 0.359 0.966 ν-0.073 0.046 0.086 1.000 -0.019 0.079 0.082 0.948 -0.067 0.076 0.101 0.926 τ0.022 0.115 0.116 0.998 -0.068 0.119 0.137 0.826 0.033 0.081 0.088 0.988 proposed estimator ... | https://arxiv.org/abs/2504.02096v1 |
estimator maintains good performance with respect to bias, even when the sample size i s as small as 100. In contrast, the bias of thenaive estimator remains constant as the samplesize in creases. Furthermore, theRMSE of the naiveestimatorappearstoplateau, showinglittleimprove mentwithincreasingsamplesize. Overall, the... | https://arxiv.org/abs/2504.02096v1 |
applied for JTPA were randomly assigned to either a treatment group, which was eligible for JTPA services, or a control gro up, which was not eligible for 18 months. However, due to some local program staff not strictly adhering to the randomization guidelines, about 3% of the control group members received J TPA servic... | https://arxiv.org/abs/2504.02096v1 |
for participation). This treatment variable can be confounded due to individuals moving themselves between the treatment and control groups in a non-random way. The covariates considered include the p articipant’s age (standardized), educational attainment (high school diploma or GED) and rac e (categorized as white or... | https://arxiv.org/abs/2504.02096v1 |
mator indicate that there is a strong positive dependence between TandC, conditional on the treatment and the measured covariates. 22 Table 4: Estimation results using the proposed estimator fo r the two models with the highest log- likelihood. The bootstrap standard error (BSE) is based on 5 00 bootstrap resamples. Fr... | https://arxiv.org/abs/2504.02096v1 |
Boundsonaverage andquantile treatment effects on duration outcomes under censoring, sele ction, and noncompliance. Journal of Business & Economic Statistics , 38(4):901–920. Bloom, H. S., Orr, L. L., Bell, S. H., Cave, G., Doolittle, F., Lin, W., and Bos, J. M. (1997). The Benefits and Costs of JTPA Title II-A Programs: ... | https://arxiv.org/abs/2504.02096v1 |
warp-s peed method for conducting Monte Carlo experiments involving bootstrap estimators. Econometric Theory , 29(3):567–589. Hiabu, M., Lo, S. M., and Wilke, R. A. (2025). Identifiability and estimation of the competing risks model under exclusion restrictions. Statistica Neerlandica , 79(1):e70003. 25 Huang, X.andZhan... | https://arxiv.org/abs/2504.02096v1 |
Fine, J. P. (2021). Estim ation of causal quantile effects with a binary instrumental variable and censored data. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 83(3):559–578. W¨ uthrich, K. (2020). A comparison of two quantile models wi th endogeneity. Journal of Business & Economic Stat... | https://arxiv.org/abs/2504.02096v1 |
On the Geometry of Receiver Operating Characteristic and Precision-Recall Curves Reza Sameni Department of Biomedical Informatics, Emory University Department of Biomedical Engineering, Georgia Institute of Technology rsameni@dbmi.emory.edu https://sameni.info/ April 4, 2025 Abstract We study the geometry of Receiver O... | https://arxiv.org/abs/2504.02169v1 |
or may not have been normalized to represent actual probabilities of association with the two classes (based on a given training set) in general, or calibrated for a given dataset. The objective is to determine whether a given sample point ‘s’ belongs to the negative or positive class based on its score. Here, we focus... | https://arxiv.org/abs/2504.02169v1 |
τ) + FN( τ) =T.(8) Using (8), the performance metrics and class prior probabilities can be approximated as: tpr(τ)≈TP(τ) P,fpr(τ) = 1−tnr(τ)≈FP(τ) N, tnr(τ)≈TN(τ) N,fnr(τ) = 1−tpr(τ)≈FN(τ) P, πp≈P T, π n≈N T.(9) These empirical approximations converge to their distribution-based counterparts as T→ ∞ , under the assumpt... | https://arxiv.org/abs/2504.02169v1 |
values: τ=F−1 p 1−tpr(τ) =F−1 n 1−fpr(τ) , (12) which results in tpr( τ) = 1−(Fp◦F−1 n) 1−fpr(τ) providing a compact formulation of the ROC curve [2, Sec 2.2.4]: tpr(τ) = 1−G 1−fpr(τ) (13) parameterized by the decision threshold τ. From (13), the monotonically increasing shape of the ROC curve is apparent. It a... | https://arxiv.org/abs/2504.02169v1 |
score distributions but also on the class prior probabilities in the dataset. As compared with the ROC curve, it is also a more complex function of G(·). 6 4.4 Accuracy The accuracy of a binary classifier at decision threshold τis the probability of correct classification, com- puted as the weighted sum of the true pos... | https://arxiv.org/abs/2504.02169v1 |
classifier globally or locally dominates another in terms of both ROC and PR curves, there is no guarantee that it also dominates in measures such as accuracy and Fβ, which depend on both Fp(·) andFn(·), not merely on G1(·) and G2(·). In fact, two classifiers can have identical ROC/PR curves but different accuracies or... | https://arxiv.org/abs/2504.02169v1 |
making the problem more complicated. Similar constraints arise in other application domains as well. Note that in the scenario described above, Mmay or may not scale with the total number of subjects (T). For example, in an overwhelmed healthcare setting operating at maximum capacity, Mmay not scale easily with an incr... | https://arxiv.org/abs/2504.02169v1 |
need for explicitly modeling the trade-offs between model performance and annotation cost. One way to address this is to quantify and integrate ML-based design metrics and socioeconomic factors into bi-objective or multiobjective optimization frameworks. For example, in classifier design one could maximize a cost funct... | https://arxiv.org/abs/2504.02169v1 |
PR characteristics may still require calibration to ensure its score outputs reflect true class probabilities in practical settings. 10 8 Case Studies 8.1 A Random Classifier A trivial corner case for (13) is when fn(·) =fp(·), i.e., the distribution of the score is not affected by the sample point’s class label. We ca... | https://arxiv.org/abs/2504.02169v1 |
1−Fn(τ) = 1−Φτ−µn σn , tpr(τ) = 1−Fp(τ) = 1−Φτ−µp σp .(38) Eliminating τyields a convenient parametric form of the ROC curve. Letting u= (τ−µn)/σnwe obtain fpr(u) = 1−Φ(u),tpr(u) = 1−Φ αu−b , (39) where α:=σn/σp, b := (µp−µn)/σp. (40) Hence the ROC curve can be parameterized as ROC( u): fpr(u),tpr(u) = 1−Φ(u),... | https://arxiv.org/abs/2504.02169v1 |
to the identity line (random classifier). As αincreases, the curves become steeper, representing classifiers with more confident separation between positive-negative classes. Increasing bhelps improve separability of the two classes. Negative bimplies that the class labels or decision rule should be switched (positive ... | https://arxiv.org/abs/2504.02169v1 |
varies α∈ {0.5,1.0,1.5,2.0}, demonstrating how dispersion affects PR behavior. The third experiment (Fig. 7c) tests four ( α, b) combinations under different class ratios. As expected, increasing class imbalance (i.e., decreasing πp/πn) shifts precision downward, even when recall remains high. Fig. 8 shows other exampl... | https://arxiv.org/abs/2504.02169v1 |
University, for their valuable feedback, fruitful discussions and insightful comments on this work. References [1] H. L. Van Trees, Detection, estimation, and modulation theory, part I: detection, estimation, and linear modulation theory . John Wiley & Sons, 2004. [2] W. J. Krzanowski and D. J. Hand, ROC curves for con... | https://arxiv.org/abs/2504.02169v1 |
Testing Independence and Conditional Independence in High Dimensions via Coordinatewise Gaussianization Jinyuan Changa,b, Yue Dua, Jing Hec, and Qiwei Yaod aJoint Laboratory of Data Science and Business Intelligence, Southwestern University of Finance and Economics, Chengdu, China bAcademy of Mathematics and Systems Sc... | https://arxiv.org/abs/2504.02233v1 |
marginal distributions of X,YandZare continuous, we transform each component of X,YandZto a standard normal random variable by its distribution function. Let U,Vand Wbe the transformed vectors of, respectively, X,YandZ. We adopt the maximum absolute pairwise sample covariance between the components of Uand those of Vas... | https://arxiv.org/abs/2504.02233v1 |
al. (2020). All the tests aforementioned require certain moment conditions on Xand Y. To alleviate the moment restrictions, a projection correlation based test is considered by Zhu et al. (2017), and some rank-based tests are presented by Heller et al. (2013), Shi et al. (2022) and Deb and Sen (2023). 3 The independenc... | https://arxiv.org/abs/2504.02233v1 |
5 provides a computationally efficient multiplier bootstrap scheme for computing the critical values of the proposed tests. Section 6 investigates the associated theoretical properties of the proposed tests. Sections 7 and 8 evaluate the finite-sample per- formance of the proposed tests via, respectively, extensive sim... | https://arxiv.org/abs/2504.02233v1 |
·),ˆFY,k(·) =n−1Pn s=1I(Ys,k≤ ·) and ˆFZ,l(·) =n−1Pn s=1I(Zs,l≤ ·). Multiplying them by n(n+ 1)−1in (5) is to guarantee |ˆUi,j|<+∞,|ˆVi,k|<+∞and |ˆWi,l|<+∞. In Sections 3 and 4, we will propose testing procedures for (1) and (2) based on coordinatewise Gaussianization. 3 Testing for Independence Note that γi≡Ui⊗Viis ad... | https://arxiv.org/abs/2504.02233v1 |
Step 1 estimates FX,j(·),FY,k(·) and FZ,l(·) based on WD1, Step 2 estimates fjandgkbased on WD2, and Step 3 calculates the test statistic and critical value based on WD3. See Section 4.1.1 for details. Section 4.1.2 will propose a data-driven procedure to select ( n1, n2, n3) in practice. 4.1.1 Testing Procedure Given ... | https://arxiv.org/abs/2504.02233v1 |
the proposed test. On the other hand, due ton1, n2≫n3, the approximation errors caused by ( ˆU(w) i,ˆV(w) i,ˆW(w) i) to (Ui,Vi,Wi) in Step 1 and ( ˆfj,ˆgk) to ( fj, gk) in Step 2 will be negligible in constructing the theoretical properties of ˜Gn. Hence, we mainly focus on the selection of n3. In practice, we always s... | https://arxiv.org/abs/2504.02233v1 |
the critical value ˆ cv(b) cind,αin the same manner as ˆ cv cind,αdefined in (12) but with replacing {˜ηi}i∈D3by{˜η(b) i}n1+n2+˜ℓ i=n1+n2+1. 10: Calculate ab(˜ℓ) =I{˜G(b) ˜ℓ>ˆ cv(b) cind,α}. 11: end for 12:end for 13: For each ˜ℓ∈[n−n1−n2], calculate ¯ a(˜ℓ) =B−1PB b=1ab(˜ℓ). Output: nopt 3= arg min ˜ℓ∈[n−n1−n2]|¯a(˜ℓ)... | https://arxiv.org/abs/2504.02233v1 |
mentary material indicates that Theorem 1 actually holds provided that log d≪n˜c1for some constant ˜ c1∈(0,1). Assuming p≲nκ1andq≲nκ2is just to simplify the presentation. Write Σ= Cov( γi) := (Σ i,j)d×dandλ(d, α) = (2 log d)1/2+{2 log(1 /α)}1/2. Theorem 2 shows that the proposed independence test is consistent under ce... | https://arxiv.org/abs/2504.02233v1 |
model defined above is ( ϑ, C)-smooth, if all functions occurring in its definition are ( ϑ, C)-smooth according to Definition 1. 14 Definition 3. LetF(m, m ∗, l, K, ϑ, L, C, ˜C) be the set of functions f:Rm→R, which satisfy the following conditions: fsatisfies a ( ϑ, C)-smooth generalized hierarchical interaction mode... | https://arxiv.org/abs/2504.02233v1 |
j∈[d]Θ1/2 j,j with ˜ϵ2 nlogd→ ∞ asn→ ∞ , then it holds that P(˜Gn>ˆ cvcind,α)→1asn→ ∞ . As long as |E(εiδ⊤ i)|∞≥Cn−κ/2(logd)1/2under the alternative hypothesis H1in (4) for some universal constant C > 1, Theorem 4 implies that the proposed conditional independence test based on nonparametric regressions is a consistent... | https://arxiv.org/abs/2504.02233v1 |
some constant ˜c6∈(0,1). Together with Theorem 5, we know that, even if the dimensions of X,YandZ diverge exponentially with the sample size n, the proposed conditional independence test based on linear regressions can still correctly control the Type I error at the significance level α∈(0,1) and also have power approa... | https://arxiv.org/abs/2504.02233v1 |
a magnitude randomly drawn from U(0,1). To ensure positivity, let R∗= (1 + υ)Ip+q+∆with υ={−λmin(Ip+q+∆) + 0.05}I{λmin(Ip+q+∆)≤0}. Then, for j∈[p] and l∈[q], let Xj=φjandYl=φp+l. Example 5. Write ϑ= (ϑ1, . . . , ϑ p+q)⊤. For j∈[p] and l∈[q], let Xj=ϑ1/3 jandYl=ϑ1/3 p+l. Under the null hypothesis H0in (1), generate ϑ∼ N... | https://arxiv.org/abs/2504.02233v1 |
tests return invalid results for more than 20% in the 2000 repetitions due to the heavy tails of the data, the associated results are reported by NA. Such a phenomenon indicates that these five tests may not work for the heavy-tailed data. The results of the JdCov R test for n= 100 and p=q= 1600 are omitted, since the ... | https://arxiv.org/abs/2504.02233v1 |
+˜Yk−mI(k∈[q]\[m]) + 3τkI(k∈[K]) with φj={0.7(Z3 j/5 +Zj/2) + tanh( νj)}I(j∈[m]) and ˜ φk={(Z3 k/4 + Zk)/3 +uk}I(k∈[m]). We set K∈ {0, p/10, p/5}. When K= 0,X⫫Y|Z. Otherwise, X̸⫫Y|Z. Example 9. Generate Z1, . . . , Z m,˜X1, . . . , ˜Xp−m,˜Y1, . . . , ˜Yq−m, ν1, . . . , ν m, u1, . . . , u mi.i.d.∼ N (0,1). Draw τ1, . . ... | https://arxiv.org/abs/2504.02233v1 |
CI-FNN test, ˆfjand ˆgkare estimated by (11) with the parameters ( ℓ, K, m ∗, M∗) = (0 ,1,1,32). We set n1=⌊n/3⌋,n2=⌊n/2⌋andn3=nopt 3, where nopt 3is selected by Algorithm 1 with B= 500. In the CI-Lasso test, the Lasso estimators ˆαjandˆβkare obtained by calling the R-functions glmnet and cv.glmnet in the glmnet with t... | https://arxiv.org/abs/2504.02233v1 |
nominal level in all the settings, since good approximation for the null distributions of the RCIT, RCoT and cdCov tests requires considerable sample size (Runge, 2018; Strobl et al., 2019; Wang et al., 2015). The GCM test has good size control in the simulation settings except Example 6. For Examples 6 and 8, the GCM ... | https://arxiv.org/abs/2504.02233v1 |
total number of rejections of the null. Controlling the FDR at the level 0.01 here ensures the expected number of false discoveries does not exceed 55 ×0.01 = 0 .55<1 asymptotically. This motivates us to further investigate the conditional independence structure among the 11 sectors. More specifically, we can use a net... | https://arxiv.org/abs/2504.02233v1 |
findings to those of the CI-FNN test with Rademacher multiplier. See Figure S1 in the supplementary material for the conditional 27 dependence network of the 11 sectors based on the associated 55 p-values summarized in Table S3 in the supplementary material. We have also repeated the above-mentioned analysis for invest... | https://arxiv.org/abs/2504.02233v1 |
7.4 NA NA NA 100.0 100.0 100.0 7.4 8.2 NA NA 7.6 NA NA NA K=p/10 100.0 100.0 100.0 6.2 9.0 NA NA 11.4 NA NA NA 100.0 100.0 100.0 10.7 17.7 NA NA 13.1 NA NA NA 1600K= 0 0.0 0.0 6.2 5.3 5.1 4.8 5.5 6.1 5.5 4.4 5.1 0.0 1.0 6.3 5.0 5.1 4.2 4.8 − 5.2 5.0 5.8 K=p/20 100.0 100.0 100.0 4.5 4.8 NA NA 7.5 NA NA NA 100.0 100.0 10... | https://arxiv.org/abs/2504.02233v1 |
4.5 5.1 5.2 9.6 5.0 5.1 4.6 100.0 100.0 100.0 5.8 5.1 5.7 5.9 − 5.2 5.1 5.2 Example 4100null 0.1 1.3 7.3 4.9 5.0 4.9 5.0 5.4 6.0 5.7 4.9 0.4 1.6 5.6 5.2 4.9 4.9 4.9 4.4 5.5 5.9 5.2 alternative 76.6 84.2 89.4 12.8 5.8 12.5 12.9 5.1 7.2 4.7 5.8 95.2 96.4 97.1 26.5 6.3 26.8 25.6 5.6 8.9 5.8 6.0 400null 0.0 0.4 7.5 3.9 4.9... | https://arxiv.org/abs/2504.02233v1 |
99.8 17.1 100.0 100.0 100.0 99.7 100.0 100.0 0.1 NA 24.3 22.1 40.6 K=p/5 69.6 100.0 95.6 100.0 99.7 100.0 0.3 NA 99.5 99.5 18.7 100.0 100.0 100.0 100.0 100.0 100.0 0.0 NA 24.9 21.9 35.5 1600K= 0 0.0 0.0 1.2 0.1 9.0 8.9 0.0 NA 99.4 99.2 45.4 0.5 0.3 1.5 1.9 8.5 6.7 0.0 3.2 25.0 25.0 73.8 K=p/10 42.5 100.0 82.0 100.0 97.... | https://arxiv.org/abs/2504.02233v1 |
76.4 100.0 98.2 100.0 100.0 100.0 7.6 NA 100.0 100.0 73.0 100.0 100.0 100.0 100.0 100.0 100.0 4.6 NA 35.6 42.2 86.9 Example 9100K= 0 0.1 0.6 2.9 2.3 7.1 5.3 3.8 4.0 99.9 100.0 98.7 0.7 1.7 1.7 2.9 5.1 6.2 4.0 3.6 40.4 37.2 100.0 K=p/10 12.3 100.0 12.8 100.0 44.7 100.0 9.0 0.0 99.9 100.0 37.6 85.9 100.0 85.6 100.0 89.6 ... | https://arxiv.org/abs/2504.02233v1 |
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30, 146–149. Leung, D. and Drton, M. (2018). Testing independence in high dimensions with sums of rank correlations. Ann. Statist. , 46, 280–307. Liu, H., Lafferty, J. and Wasserman, L. (2009). The nonparanormal: Semiparametric estimation of high dimensional undirected graphs. J. Mach. Learn. Res. , 10, 2295–2328. Lyon... | https://arxiv.org/abs/2504.02233v1 |
of distances. Ann. Statist. , 35, 2769–2794. Sz´ ekely, G. J. and Rizzo, M. L. (2013). The distance correlation t-test of independence in high dimension. J. Multivariate Anal. , 117, 193–213. Sz´ ekely, G. J. and Rizzo, M. L. (2014). Partial distance correlation with methods for dissimilar- ities. Ann. Statist. , 42, 2... | https://arxiv.org/abs/2504.02233v1 |
, provided that logd≪n1/8(logn)−1/4. A.1 Proof of Proposition 1 The following Lemmas 1–4 are needed in the proof of Proposition 1, with their proofs given in Appendices F–I, respectively. Select M1=√κ1lognfor some constant κ1∈(1,2), and define U∗ i,j=Ui,jI(|Ui,j| ≤M1) +M1·sign(Ui,j)I(|Ui,j|> M 1), V∗ i,k=Vi,kI(|Vi,k| ≤... | https://arxiv.org/abs/2504.02233v1 |
under the null hypothesis H0in (3), we have sup x>0 P(Hn> x)−P(|ˆξ|∞> x|Xn,Yn) ≤sup x>0 P(Hn> x)−P(|ξ|∞> x) + sup x>0 P(|ξ|∞> x)−P(|ˆξ|∞> x|Xn,Yn) S4 ≤sup x>0 P(|ξ|∞> x)−P(|ˆξ|∞> x|Xn,Yn) +o(1) (A.4) provided that log d≪n1/7(logn)−1/3. Let ξext= (ξ⊤,−ξ⊤)⊤= (ξext 1, . . . , ξext 2d)⊤andˆξext= (ˆξ⊤,−ˆξ⊤)⊤= (ˆξext 1, . . ... | https://arxiv.org/abs/2504.02233v1 |
(1975), for any u >0, it holds that P |ˆξ|∞>E(|ˆξ|∞|Xn,Yn) +u|Xn,Yn ≤exp −u2 2 max j∈[d]ˆΣj,j . (A.6) For any v1>0, consider the event E1(v1) =( max j∈[d]|ˆΣ1/2 j,j−Σ1/2 j,j| Σ1/2 j,j≤v1) . Since min j∈[d]Σj,j≥c1, by Lemma 4, we have max j∈[d]|ˆΣ1/2 j,j−Σ1/2 j,j| Σ1/2 j,j=Op{n−1/2(logn)(log d)1/2log3/2(dn)} provided... | https://arxiv.org/abs/2504.02233v1 |
−√nµl∗ 1+{1 + (log d)−1}λ(d, α) max j∈[d]Σ1/2 j,j+v2 ≤P1√nnX i=1{Ui,j∗Vi,k∗−E(Ui,j∗Vi,k∗)} ≤ − νn(logd)1/2logn =P1√nnX i=1Ui,j∗Vi,k∗−E(Ui,j∗Vi,k∗)p Var(Ui,j∗Vi,k∗)≤ −νn(logd)1/2lognp Var(Ui,j∗Vi,k∗) →0. Together with (A.9) and (A.11), under the alternative hypothesis H1in (3), it holds that P(Hn>ˆ cvind,α)≥1−2d−νn... | https://arxiv.org/abs/2504.02233v1 |
the null hypothesis H0in (4), we have sup x>0 P(˜Gn> x)−P(|˜ζ|∞> x|Xn,Yn,Zn) =op(1) provided that log˜d≪min nϑ/(4ϑ+m∗)−κ/4(logn)−1−ϱ/(8ϑ), n2κ/15(logn)−14/15, n1/16(logn)−7/8, n4ϑ/(68ϑ+17m∗)(logn)−16/17−ϱ/(34ϑ) , m≪min nϑ{4ϑ/(4ϑ+m∗)−κ}/ϱ(logn)−4ϑ/ϱ−1/2{log(˜dn)}−4ϑ/ϱ, n(1−κ)/2(logn)−1{log(˜dn)}−3/2, nκ/4(logn)−1{log(... | https://arxiv.org/abs/2504.02233v1 |
3, we need Lemmas 9–11 with their proofs given in Appendices N–P, respectively. Lemma 9. Assume (8)and Condition 2hold. Then 1 nnX i=1(ˆεi,jˆδi,k−εi,jδi,k) =√ 2π(n−1) n(n+ 1)nX s=1˜δ4,k(Us,j) +˜δ5,j(Vs,k) + Rem 1(j, k) with max j∈[p], k∈[q]|Rem 1(j, k)|=Op{sn−7/10log3/2(˜dn)}+Op{s1/2n−13/20(logn)−3/4log(˜dn)} S15 prov... | https://arxiv.org/abs/2504.02233v1 |
n1/8(s2logn)−1/4}ands≲n1/4(logn)−5/2. Together with (C.4), under the null hypothesis H0in (4), we have sup x>0 P(ˆGn> x)−P(|ˆζ|∞> x|Xn,Yn,Zn) =op(1) provided that log ˜d≪min{n1/10(slogn)−1/2, n1/8(s2logn)−1/4}ands≲n1/5(logn)−2. Hence, we complete the proof of Proposition 3. 2 C.2 Proof of Theorem 5 The proof of Theorem... | https://arxiv.org/abs/2504.02233v1 |
Together with (C.5) and (C.7), under the alternative hypothesis H1in (4), it holds that P(ˆGn>ˆ cv∗ cind,α)≥1−2˜d−√ 2un/2−u2 n/16−o(1) provided that log˜d≪min{n1/6(s2logn)−1/3, n1/10(logn)−1/2, s−1n1/5logn}, s≪min{n1/4(logn)−2, n1/5(log˜d)1/2(logn)−1/2}. (C.9) S20 Recall ˜d=p∨q∨mwith p≲nκ1,q≲nκ2andm≲nκ3. Ifs≪n1/5(logn)... | https://arxiv.org/abs/2504.02233v1 |
the restrictions (D.5) hold automatically. Hence, we complete the proof of Theorem 7(ii). 2 D.3 Proof of Theorem 7(iii) Parallel to (D.1), by Proposition 3, under the null hypothesis H0in (4), we have sup x>0 P(ˆGn> x)−P(|ˆζ†|∞> x|Xn,Yn,Zn) ≤sup x>0 P(|ˆζ|∞> x|Xn,Yn,Zn)−P(|ˆζ†|∞> x|Xn,Yn,Zn) +op(1) (D.6) provided that ... | https://arxiv.org/abs/2504.02233v1 |
L1∈(0,∞)such that, if hi1,i2are bounded canonical kernels, then P nX i1,i2=1hi1,i2 ψ(1) i1, ψ(2) i2 ≥t ≤L1exp −1 L1mint2 E2,t D,t2/3 B2/3,t1/2 A1/2 for any t >0, where A= max i1,i2∈[n]sup x,y∈S hi1,i2(x, y) , E2=nX i1,i2=1E h2 i1,i2 ψ(1) i1, ψ(2) i2 , B2= max i2∈[n]sup y∈SnX i1=1E{1} h2 i1,i2 ψ(1) i1, y ... | https://arxiv.org/abs/2504.02233v1 |
for any n≥2 and 0 < x≤2(n−1)−1, P |ˆF(i) X,j(Xi,j)−FX,j(Xi,j)|> x ≤1≤˜Cexp{−4Cn/(n−1)2} ≤˜Cexp(−Cnx2). Hence, for any n≥2 and x >0, it holds that P |ˆF(i) X,j(Xi,j)−FX,j(Xi,j)|> x ≤(¯C∨˜C) exp(−Cnx2). Analogously, we can also establish the same upper bound for P{|ˆF(i) Y,k(Yi,k)−FY,k(Yi,k)|> x}. We complete the proof... | https://arxiv.org/abs/2504.02233v1 |
1)X 1≤i1̸=i2≤nϖ1(Ti1,Ti2), by Inequalities 2 and 3, we have P{|I111(j, k)| ≥x} ≤C1P C1 X 1≤i1̸=i2≤nϖ1 T(1) i1,T(2) i2 ≥n(n+ 1)x√ 2π ≤C2exp −1 C2minn2M2x2 eM2 2/2,nx M1eM2 2/2,nx2/3 M2/3 1eM2 2/3,nx1/2 M1/2 1eM2 2/4 for any x >0 under the null hypothesis H0in (3). Recall d=pq. Notice that above inequality holds ... | https://arxiv.org/abs/2504.02233v1 |
the proof of (F.9). 2 F.3 Convergence rate of max j∈[p], k∈[q]|I2(j, k)| Notice that Φ−1(x) is infinitely differentiable at any x∈(0,1). We have I2(j, k) =∞X l=11 n·l!nX i=1 (Φ−1)(l){FX,j(Xi,j)}n n+ 1ˆFX,j(Xi,j)−FX,j(Xi,j)l ×V∗ i,kI(M2<|Ui,j| ≤M1) . LetK(Ui,j, p, n) = 4 n−1/2[Φ(Ui,j){1−Φ(Ui,j)}]1/2log1/2(pn) + 7n−1... | https://arxiv.org/abs/2504.02233v1 |
1/2, max i∈[n], j∈[p]Var{I(|Ui,j|> M 1)}≲M−1 1e−M2 1/2. (F.24) Identical to the derivation of (F.17), we have max j∈[p] 1 nnX i=1I(|Ui,j|> M 1) =Op(M−1 1e−M2 1/2) (F.25) provided that log p≲ne−M2 1/2M−1 1. Hence, it holds that max j∈[p], k∈[q]|I3(j, k)|=Op(M−1 1e−M2 1/2logn) provided that log p≲ne−M2 1/2M−1 1. Recall M... | https://arxiv.org/abs/2504.02233v1 |
Cauchy-Schwarz inequality, we then have max i∈[n], j∈[p], k∈[q]E eU2 i,j/2eV2 i,k/2I(|Ui,j| ≤M2)I(|Vi,k| ≤M2) ≤max i∈[n], j∈[p] E eU2 i,jI(|Ui,j| ≤M2) 1/2max i∈[n], k∈[q] E eV2 i,kI(|Vi,k| ≤M2) 1/2≲M−1 2eM2 2/2, max i∈[n], j∈[p], k∈[q]Var eU2 i,j/2eV2 i,k/2I(|Ui,j| ≤M2)I(|Vi,k| ≤M2) ≤max i∈[n], j∈[p] E e2U2 ... | https://arxiv.org/abs/2504.02233v1 |
have P(Hc 4)≤4(qn)−2. Hence, apply- ing the same arguments in Section F.2.2 for deriving the convergence rate of max j∈[p], k∈[q]|I12(j, k)|, we can show max j∈[p], k∈[q]|J12(j, k)|=Op{n−1M1/2 1M−1 2eM2 2/4log(dn)} provided that log( dn)≪min{ne−M2 1/2M−1 1, ne−M2 2M−2 2}. Using the similar arguments, we can also show s... | https://arxiv.org/abs/2504.02233v1 |
H2TH4, by (G.12), max j∈[p], k∈[q]|J14(j, k)| ≤ max j∈[p], k∈[q]|J141(j, k)|+ max j∈[p], k∈[q]|J142(j, k)| + max j∈[p], k∈[q]|J143(j, k)|+ max j∈[p], k∈[q]|J144(j, k)|. SinceP(Hc 2)≤4(pn)−2andP(Hc 4)≤4(qn)−2, applying the similar arguments in Section F.2.2 for deriving the convergence rate of max j∈[p], k∈[q]|I12(j, k)... | https://arxiv.org/abs/2504.02233v1 |
+1 nnX i=1´Ui,j´Vi,kI(|Ui,j|> Q)I(|Vi,k|> M 1) | {z } K33(j,k)+E{´Ui,j´Vi,kI(M1<|Ui,j|,|Vi,k| ≤Q)}| {z } K34(j,k). Recall Ui,j, Vi,k∼ N (0,1),´Ui,j=Ui,j−M1·sign(Ui,j) and ´Vi,k=Vi,k−M1·sign(Vi,k). By Cauchy-Schwarz inequality, we have max i∈[n], j∈[p], k∈[q]Var{´Ui,j´Vi,kI(M1<|Ui,j|,|Vi,k| ≤Q)} ≤max i∈[n], j∈[p] E[{M1... | https://arxiv.org/abs/2504.02233v1 |
M 1). Let ˆU∗ i,j=ˆUi,j−U∗ i,j,ˆV∗ i,k=ˆVi,k−V∗ i,k,˜Ui,j=Ui,j−U∗ i,j,˜Vi,k=Vi,k−V∗ i,k. Then, we have ˆUi,j−Ui,j=ˆU∗ i,j−˜Ui,jandˆVi,k−Vi,k=ˆV∗ i,k−˜Vi,k. Hence, it holds that R11= max j,l∈[p], k,t∈[q] 1 nnX i=1(ˆU∗ i,j−˜Ui,j)Ui,lVi,kVi,t ≤ max j,l∈[p], k,t∈[q] 1 nnX i=1˜Ui,jUi,lVi,kVi,t | {z } R111+ max j,l∈[p], k,t∈... | https://arxiv.org/abs/2504.02233v1 |
i,j={Ui,j−M1·sign(Ui,j)}I(|Ui,j|> M 1). Using the similar arguments for deriving the convergence rate of R 111in Section I.1.1, we can also show R121=Op{e−M2 1/2log3/2(dn)}+Op{n−1(logd) log2(dn)} provided that log d≲nM−1 1e−M2 1/2. Analogous to the derivation of R 112in Section I.1.1, we have R122=Op{n−1/2(logn)(log d)... | https://arxiv.org/abs/2504.02233v1 |
convergence rate of R 15, by the symmetry, we only need to consider the convergence rates of the following terms: R151= max j,l∈[p], k,t∈[q] 1 nnX i=1˜Ui,j˜Ui,l˜Vi,k˜Vi,t ,R152= max j,l∈[p], k,t∈[q] 1 nnX i=1ˆU∗ i,j˜Ui,l˜Vi,k˜Vi,t R153= max j,l∈[p], k,t∈[q] 1 nnX i=1ˆU∗ i,jˆU∗ i,l˜Vi,t˜Vi,k ,R154= max j,l∈[p], k,t∈[q] ... | https://arxiv.org/abs/2504.02233v1 |
= max j∈[p], k∈[q]|R′ 33(j, k)|provided that log( dn)≲Q2. Furthermore, max j∈[p], k∈[q]|R′ 34(j, k)|≲max i∈[n], j∈[p] E{I(|Ui,j|> Q)}1/2+ max i∈[n], k∈[q] E{I(|Vi,k|> Q)}1/2 ≲Q−1/2e−Q2/4. By selecting Q=˜Clog1/2(dn) for some sufficiently large constant ˜C >0, it holds that max j∈[p], k∈[q]|R′ 3(j, k)|=Op{n−1/2(logd... | https://arxiv.org/abs/2504.02233v1 |
i,jVi,kI(|Vi,k| ≤Q) | {z } R′ 431+ max j∈[p], k∈[q] 1 nnX i=1ˆU∗ i,jVi,kI(|Vi,k|> Q) | {z } R′ 432. S61 Recall ˆU∗ i,j=ˆUi,j−U∗ i,j. By (I.13), it holds that R′ 431≤Qmax j∈[p]1 nnX i=1|ˆUi,j−U∗ i,j|=Op{n−1/2Q(logn)(log p)1/2}. Analogous to the derivation of (H.4), we also have R′ 432=op(n−1) provided that log( pn)≲Q2. ... | https://arxiv.org/abs/2504.02233v1 |
j∈[p], k∈[q]|K′ 11(j, k)|=Op{n−3/4(logn)1/2log(dn)} (J.3) provided that log d≪n1/2(logn)−1. Hence, we have max j∈[p], k∈[q]|K′ 1(j, k)|=Op{n−5/8(logn)−1/4log1/2(dn)}+Op{n−3/5(logn)1/2} provided that log d≲n1/4(logn)−3/2. Analogously, we can also show such convergence rate holds for max j∈[p], k∈[q]|K′ 2(j, k)|. We comp... | https://arxiv.org/abs/2504.02233v1 |
prove Lemma 6, we need Lemmas K1–K3, with their proofs given in Sections K.1–K.3, respectively. Recall ˜d=p∨q∨m. S66 Lemma K1. Iflog(˜dn)≪n1−κ(logn)−1/2, then max j∈[p], k∈[q] 1 n3X t∈D3(ˆU(w) t,j−Ut,j)δt,k =Op{n−κlog2(˜dn)}+Op{n−1/2log(˜dn)} = max j∈[p], k∈[q] 1 n3X t∈D3(ˆV(w) t,k−Vt,k)εt,j . Lemma K2. Under Condition... | https://arxiv.org/abs/2504.02233v1 |
P(Hc 5) =P[ i∈D3, j∈[p] |ˆFX,j(Xi,j)−FX,j(Xi,j)|> K(Ui,j, p, n 1) ≤X i∈D3pX j=1E P 1 n1X s∈D1 I(Us,j≤Ui,j)−Φ(Ui,j) > K(Ui,j, p, n 1) Ui,j ≤2n3pmax i∈D3, j∈[p]E exp −n1K2(Ui,j, p, n 1) 4Φ(Ui,j){1−Φ(Ui,j)} + exp −n1K(Ui,j, p, n 1) 2 ≤4(n1p)−2. (K.4) Restricted on H5, for any integer l≥0, it holds that |ˆF(... | https://arxiv.org/abs/2504.02233v1 |
=Op{n−κm2log(˜dn)}+Op{n−1/2mlog(˜dn)} provided that log( ˜dn)≪n1−κ(logn)−1/2. Identically, we can also show such convergence rate holds for max j∈[p], k∈[q]|n−1 3P t∈D3{gk(ˆW(w) t)−gk(Wt)}εt,j|. Hence, we complete the proof of Lemma K2. 2 K.2.1 Convergence rate of max j∈[p], k∈[q]|H1(j, k)| We first show that for any f... | https://arxiv.org/abs/2504.02233v1 |
n−1/2 1[Φ(Wi,j){1−Φ(Wi,j)}]1/2log1/2(mn1) + 7n−1 1log(mn1). Define the event H6=\ i∈D3, j∈[m] |ˆFZ,j(Zi,j)−FZ,j(Zi,j)| ≤K(Wi,j, m, n 1) . (K.13) S74 Restricted on H6, given some constant M2∈(0, M1), it holds that |H11|=C3Q n3X t∈D3mX j=1|Φ−1{ˆF(w) Z,j(Zt,j)} −Φ−1{FZ,j(Zt,j)}|I(|Wt|∞≤M1) ≤∞X l=1C3Q n3·l!X t∈D3mX j=1|(Φ... | https://arxiv.org/abs/2504.02233v1 |
t∈D3mX j=1|W∗ t,j−Wt,j|I(|Wt|∞≤Q1) | {z } H131 +C3Q n3X t∈D3mX j=1|W∗ t,j−Wt,j|I(|Wt|∞> Q 1) | {z } H132. Since Wi,j∼ N (0,1) and |W∗ i,j−Wi,j| ≤ |Wi,j|I(|Wi,j|> M 1), using the similar arguments for the derivation of (K.17), it holds that EmX j=1|W∗ t,j−Wt,j|I(|Wt|∞≤Q1) ≤EmX j=1|W∗ t,j−Wt,j| ≲me−M2 1/2, S77 VarmX... | https://arxiv.org/abs/2504.02233v1 |
= 0, then E(δt,k|Zt) = 0. Notice that ˆfjis specified in (11) based on the data in WD1∪ W D2. Since WD1,WD2andWD3are independent, for any t∈ D 3, we have E {ˆfj(ˆW(w) t)−fj(ˆW(w) t)}δt,k|WD1,WD2 =E {ˆfj(ˆW(w) t)−fj(ˆW(w) t)} ×E(δt,k|WD1,WD2,Zt)|WD1,WD2 =E {ˆfj(ˆW(w) t)−fj(ˆW(w) t)} ×E(δt,k|Zt)|WD1,WD2 = 0. Recall... | https://arxiv.org/abs/2504.02233v1 |
t)}δt,kI(|δt,k| ≤Q)−˜µ1,j QK(n, m, ˜d) >x QK(n, m, ˜d) ≤pmax j∈[p]P 1 n3X t∈D3{ˆfj(ˆW(w) t)−fj(ˆW(w) t)}δt,kI(|δt,k| ≤Q) QK(n, m, ˜d) −˜µ1,j QK(n, m, ˜d) >x QK(n, m, ˜d),G1 +P(Gc 1) S82 ≤pmax j∈[p]E P 1 n3X t∈D3{ˆfj(ˆW(w) t)−fj(ˆW(w) t)}δt,kI(|δt,k| ≤Q)−˜µ1,j QK(n, m, ˜d) >x QK(n, m, ˜d),G1 WD1,WD2 +P(Gc 1) ≤˜... | https://arxiv.org/abs/2504.02233v1 |
P{˘G12(j)> x|WD1} =P sup h∈T˜βnH(ℓ) 1 n2X i∈D2 {h(ˆW(w) i)−ˆU(w) i,j}2−E {h(ˆW(w) i)−ˆU(w) i,j}2|WD1 > x WD1 ≤P sup a∈G2 1 n2X i∈D2 a(ˆW(w) i,ˆU(w) i,j)−E a(ˆW(w) i,ˆU(w) i,j)|WD1 > x WD1 ≤8E N1x 8,G2,{(ˆW(w) i,ˆU(w) i,j)}i∈D2 WD1 ×exp −n2x2 128(4 ˜β2 n)2 (K.30) for any x >0. Recall ˆW(w) i∈[−√2 logn1... | https://arxiv.org/abs/2504.02233v1 |
nκ. Let K(Wt,j,˜d, n 1) = 4 n−1/2 1[Φ(Wt,j){1−Φ(Wt,j)}]1/2log1/2(˜dn1) + 7n−1 1log(˜dn1). S87 Using the similar arguments for the derivation of the convergence rate of |H11|in Section K.2.2 for the proof of Lemma K2, it holds that EmX j=1|ˆW(w) t,j−Wt,j|I(|Wt|∞≤M1)mY k=1I{|ˆFZ,k(Zt,k)−FZ,k(Zt,k)| ≤K(Wt,k,˜d, n 1)} WD1... | https://arxiv.org/abs/2504.02233v1 |
K5, and ˜C7>0is a universal constant only depending on ( ˜m,˜N). Lemma K7. Let˜m≥1be a general integer, and f:R˜m→Rbe a(ϑ, C)-smooth function with (ϑ, C)given in Lemma K4, where ϑ=˜ϑ+sfor˜ϑ∈N0ands∈(0,1]. Let p˜ϑbe the Taylor polynomial of the total degree ˜ϑaround x0withx0= (x0,1, . . . , x 0,˜m)⊤∈R˜m, i.e., p˜ϑ(x) =X ... | https://arxiv.org/abs/2504.02233v1 |
m∗+N X j=1˜µjσ4m∗X l=1λj,lσm∗X k=1θj,l,kϕ⊤ kx+θj,l,0 +λj,0 =(N+1)(Mn+1)m∗·Cm∗ m∗+N X j=1˜µjσ4m∗X l=1λj,lσmX v=1m∗X k=1ϕk,vθj,l,kxv+θj,l,0 +λj,0 =(N+1)(Mn+1)m∗·Cm∗ m∗+N X j=1˜µjσ4m∗X l=1λj,lσmX v=1˜θj,l,vxv+˜θj,l,0 +λj,0 , S92 where ˜θj,l,v=Pm∗ k=1ϕk,vθj,l,kand˜θj,l,0=θj,l,0. Recall ˜ αn=¯C(cηn)−1m˜ϑMm∗+2+ϑ(... | https://arxiv.org/abs/2504.02233v1 |
N). Using the similar arguments for deriving (K.46), we have |t(x)−f(x)| ≤2˜C20M−ϑ nm2N+3a2N+3 n,x∈¯Dc. Moreover, it holds that P X∈[ j∈[m∗],k∈[K]Dj,k ∪[ k∈[K]Dk ≤X j∈[m∗],k∈[K]P(X∈Dj,k) +X k∈[K]P(X∈Dk) ≤X j∈[m∗],k∈[K]cηn 2Km∗+X k∈[K]cηn 2K=cηn. Hence, we have Lemma K4 holds for ℓ=¯l. Based on the mathematical in... | https://arxiv.org/abs/2504.02233v1 |
˜aj,0 ≤(˜N+1)C˜m ˜m+˜NX j=1|˜bj|σ 2˜C3C1˜m·˜an R( ˜m+ 1)·˜R−B(1−ς) +˜C3˜an R( ˜m+ 1) ≤(˜N+1)C˜m ˜m+˜NX j=1|˜bj|σ¯C5˜m˜an˜R R( ˜m+ 1)−B(1−ς) ≤(˜N+1)C˜m ˜m+˜NX j=1|˜bj|σ¯C5˜an˜R R−B(1−ς) , where ¯C5>0 is a universal constant only depending on ( ˜ m,˜N). Recall R= (˜Mn+ 1)ϑ,˜R= (˜Mn+ 1)ϑ(˜N+1),B= (˜Mn+ 1)˜m+1+ϑ(˜N+... | https://arxiv.org/abs/2504.02233v1 |
∂j1x1···∂j˜mx˜m(xi,0) ×(x1−xi,0,1)j1···(xm−xi,0,˜m)j˜m . (K.52) Notice that pi,˜ϑ(x) =X j1,...,j ˜m∈{0}∪[˜ϑ], j1+···+j˜m≤˜ϑ1 j1!···j˜m!×∂j1+···+j˜mf ∂j1x1···∂j˜mx˜m(xi,0) ×j1X k1=0Ck1 j1xk1 1(−xi,0,1)j1−k1 ···j˜mX k˜m=0Ck˜m j˜mxk˜m ˜m(−xi,0,˜m)j˜m−k˜m =X j1,...,j ˜m∈{0}∪[˜ϑ], j1+···+j˜m≤˜ϑj1X k1=0···j˜mX k˜m=0... | https://arxiv.org/abs/2504.02233v1 |
˜G4(j,k). As we will show in Sections L.1–L.3, max j∈[p], k∈[q]|˜G1(j, k)|=Op{n−κlog2(˜dn)}+Op{n−1/2(logn)1/2log1/2(˜dn)}, (L.1) max j∈[p], k∈[q]|˜G2(j, k)|=Op{n−κ(logn) log2(˜dn)}+Op{n−1/2(logn) log( ˜dn)} = max j∈[p], k∈[q]|˜G3(j, k)| (L.2) S102 provided that log( ˜dn)≪n1−κ(logn)−1/2, and max j∈[p], k∈[q]|˜G4(j, k)|=... | https://arxiv.org/abs/2504.02233v1 |
j∈[p]|ˆfj(ˆW(w) t)| ≤ ˜βn, using the similar arguments, we can show such result also holds for max j∈[p], k∈[q]|˜G3(j, k)|. Due to ˜βn= (log n) log1/2(˜dn), then (L.2) holds. 2 L.3 Proof of (L.3) Notice that ˜G4(j, k) =1 n3X t∈D3{ˆfj(ˆW(w) t)−fj(ˆW(w) t)}{ˆgk(ˆW(w) t)−gk(ˆW(w) t)} | {z } ˜G41(j,k) +1 n3X t∈D3{fj(ˆW(w) ... | https://arxiv.org/abs/2504.02233v1 |
Recall ˜d=p∨q∨m,n3≍nκfor some constant 0 < κ < 1,P(|εi,j|> x)≤C1e−x2/4and P(|δi,k|> x)≤C1e−x2/4for any x >0. Identical to the arguments for deriving the convergence rate of R 2in Section I.2 for R 2defined in (I.1), we have ˜S1=Op{n−κ/2(log˜d)1/2}+Op{n−κ(log˜d) log2(˜dn)}. (M.2) Notice that max k∈[p], t∈[q]|E(εi,kδi,t)... | https://arxiv.org/abs/2504.02233v1 |
three disjoint subsets of [ n] with |D1|=n1≍n,|D2|=n2≍nand |D3|=n3≍nκfor some constant 0 < κ < 1 and n1+n2+n3=n. Using the similar arguments for the derivation of (K.27), by (K.26), we have max j∈[p] 1 n3X i∈D3 |ˆfj(ˆW(w) i)−fj(ˆW(w) i)| −E |ˆfj(ˆW(w) i)−fj(ˆW(w) i)| WD1,WD2 =Op{n−κ/2−ϑ/(4ϑ+m∗)(m2logn)(ϑ+2m∗˜ϑ+3m∗)... | https://arxiv.org/abs/2504.02233v1 |
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