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REVIEW 2 major objections 3 minor 22 references

Collaborative Inference for Sparse High-Dimensional Models with Non-Shared Data

T0 review · 2 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper demonstrates that a gradient-shifted collaborative score test reaches the oracle pooled-data chi-square limit in sparse high-dimensional models with non-shared data.

desk verdict Real new theory for a distributed score test, but the §2.4 communication-efficient protocol computes a different statistic than the one Theorem 2 covers. read the letter →

arxiv 2504.19924 v2 pith:37662GNU submitted 2025-04-28 stat.ME

Yifan Gu , Hanfang Yang , Songshan Yang , Hui Zou This is my paper · ORCID
classification stat.ME MSC 62F0362F1262J07
keywords collaborativescoretestnon-shareddatadistributedinferencehigh-dimensionalsparsemodelslinearhypothesistestingoraclepropertycommunicationefficiencytwo-stagepenalization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a collaborative score-type test (CST) for testing linear hypotheses about a high-dimensional sparse model when the data are split across sites that refuse to share raw information. The central claim is that the CST is oracle: its statistic converges in distribution to the same chi-square limit as a score test that knew the support of the nuisance parameters, so it inherits the pooled-data power function. If that claim holds, many sites with unbalanced sample sizes can cooperate at the full efficiency of a traditional shared-data score test while exchanging only gradients and estimated parameters. The paper also shows the method tolerates a diverging number of constraints and target parameters, and it relaxes the usual restriction that the number of sites stay negligible.

What carries the argument

The carrying identity is the Kiefer-Bahadur representation of the traditional score statistic, $\nabla\hat{L}(\hat{\beta}_{\mathrm{global}}) \approx C_a \Psi^{-1}(C_a^\top J^{-1}\nabla\hat{L}(\beta^*) - h)$, which shows which gradient components a score test actually needs and motivates the gradient-shifted surrogate loss that approximates them without pooling data. The proof machinery then combines a two-stage partial penalization algorithm—an $\ell_1$-penalized first stage followed by a reweighted folded-concave second stage under the null constraint—with restricted strong convexity and a weak support-stability bound ($\max_{j\notin S^{\mathrm{a}}}\|J_{j,S^{\mathrm{a}}}(J_0)^{-1}\|_1 \le A_0$) to recover the nuisance support and control the three approximation errors $R_1, R_2, R_3$ in the Taylor expansion. The oracle property is obtained by coupling this estimator error control with a Lyapunov-type normality condition and consistent variance estimation.

What would settle it

Simulate a null logistic or linear regression with large $p$, sparse support of size $s$, and a master sample $n_1$ such that $(s+d)^3\log p$ is comparable to $n_1$ rather than $o(n_1)$, then compare the empirical distribution of $T_S$ with $\chi^2_r$ across many replications; if the distribution visibly drifts away from the chi-square reference as $p$ grows while other conditions hold, the master-sample scaling is doing the load-bearing work. The paper's own Corollary 2 makes the same prediction, since it states convergence requires $(s+d)^3(\log p+t_N)=o(n_1)$.

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Extended reading notes

Core claim

Under Conditions 1 and 2, the paper proves both the oracle test (OCST) and the feasible test (CST) satisfy $\sup_x |P(T_S \le x) - P(\chi^2(r, N h^\top V^{-1} h) \le x)| \to 0$, where $T_S = N\|\hat{\Omega}\nabla_{\hat{b}}\hat{L}(\hat{\beta})\|_2^2$. This says that the feasible statistic, built without knowing the nuisance support, shares the limiting noncentral chi-square of the oracle shared-data score test and therefore has the same asymptotic power against local alternatives. The construction uses a surrogate loss $\tilde{L}(\beta;\hat{\beta}^{(0)}) = \hat{L}_1(\beta) + \langle \nabla\hat{L}(\hat{\beta}^{(0)}) - \nabla\hat{L}_1(\hat{\beta}^{(0)}), \beta\rangle$, which approximates the global loss to first order using only local gradients, followed by a two-stage partial penalization that first drives an $\ell_1$-penalized estimator into a neighborhood of the truth and then applies a reweighted folded-concave penalty under the null constraint to remove nuisance bias.

Load-bearing premise

The entire chi-square calibration rests on the condition that the master site's local Hessian is uniformly close to the population Hessian (Assumption 4); for generalized linear models this becomes the requirement that the master sample size satisfy $(s+d)^3(\log p + t_N)=o(n_1)$, so if no single site is large enough, the Hessian approximation error $R_3$ does not vanish and the claimed limiting distribution can fail.

Editorial extensions

If this is right

  • Sites that can never share raw data can run hypothesis tests whose asymptotic power equals the pooled-data score test, exchanging only gradients and iterates.
  • The number of data sources $m$ no longer needs to be negligible relative to the sample size, and local sample sizes can be unbalanced, removing a binding restriction of one-shot divide-and-conquer tests.
  • General linear hypotheses with multivariate constraints or contrasts can be tested while both the number of constraints $r$ and the number of target parameters $d$ diverge.
  • With local variance estimators, the CST can be assembled at a communication cost of $\mathcal{O}(mp)$ bits per round, with iteration counts growing logarithmically in $m$ and doubly logarithmically in $p$ for GLMs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The master role need not be fixed; rotating the master or aggregating Hessian approximations from several larger sites could relax the master-sample condition of Assumption 4 in unbalanced federations, though the paper does not pursue this.
  • The same gradient-shifted surrogate loss could be inverted to build confidence regions for $\theta$ by testing a grid of hypothesized values, extending the method from hypothesis testing to interval inference.
  • Because the approximation error $R_3$ is a product of the Hessian mismatch and the estimator error, one could build a finite-sample diagnostic that flags calibration as suspect whenever $\|\nabla^2\hat{L}_1(\beta^*) - J\|_{\max}\cdot\|\hat{\beta}-\beta^*\|_2$ is not $o(N^{-1/2})$.
  • Applying the same CST construction to smoothed quantile losses, which the paper notes are not covered by its current assumptions, is a natural next step and is already hinted at in the discussion of distributed quantile regression.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. This paper proposes a collaborative score-type test (CST) for testing linear hypotheses H0: Cθ* = t in high-dimensional sparse models when data are stored in m non-shared sites. The construction is based on a Kiefer-Bahadur decomposition of the pooled-data score statistic and a two-stage partial penalization scheme on a surrogate loss that combines the master site's loss with aggregated local gradients. The main theoretical results (Theorems 1 and 2) show that the CST statistic TS in (9) converges in distribution to a (possibly noncentral) chi-square with r degrees of freedom under the null and local alternatives, matching the oracle score test with known nuisance support. Corollaries 2 and 3 specialize the conditions to GLMs. Simulations for linear and logistic regression examine Type I error and power, and a comparison with divide-and-conquer testing is included, along with a real taxi-data application.

Significance. If the results are correct, the paper makes a useful contribution to distributed high-dimensional inference by providing a test for general linear hypotheses with a chi-square limiting distribution that matches the pooled-data oracle test, while using only gradient communication. The theoretical framework appears internally consistent, and the numerical results support the oracle property in the tested settings. The paper also correctly relaxes the balanced-sample-size requirement of earlier divide-and-conquer tests, at least when the master site is sufficiently large. However, the communication-efficient implementation described in Section 2.4 relies on an algebraic identity that is false for matrix-valued local variance estimators; the actual scalar-aggregated statistic is not covered by the theorems. Because communication efficiency is a central advertised feature, this issue is load-bearing.

major comments (2)
  1. [Section 2.4, Eq. (9)] The claim that T_S = Σ_{k=1}^m n_k ‖Ω̂_k ∇_{bb} L̂(β̂)‖_2^2 is algebraically false for matrix-valued Ω̂_k. With bΩ = Σ n_k Ω̂_k / N, the left side equals N ‖ Σ (n_k/N) Ω̂_k g ‖^2, whereas the right side is Σ n_k ‖ Ω̂_k g ‖^2 for g = ∇_{bb} L̂(β̂); these differ by N times the weighted variance of Ω̂_k g. If any site with small n_k is included in the local variance estimators, this discrepancy is O_p((d+s)/n_min) and need not vanish, so the scalar-sum protocol implements a statistic T_comm whose limiting distribution is not covered by Theorem 2. The paper should either prove an asymptotic equivalence theorem for the actual protocol under explicit conditions on the minimum site size used in variance estimation, or modify the protocol (e.g., compute bΩ on the master using transmitted local Hessian and covariance matrices, or use a master-only estimator) and state the resulting communication cost.
  2. [Assumption 4 / Corollary 2] The key condition (s+d)^3(log p + t_N) = o(n1) in Corollary 2 requires the master site's sample size to dominate the cube of the effective dimension. This is restrictive when data are partitioned across many sites with comparable sizes: if n1 is not sufficiently large, Assumption 4 fails and the remainder term R3 in (3) cannot be controlled, so the chi-square limit is not guaranteed. Since the paper motivates its method by allowing flexible site sizes (including 'arbitrarily small' non-master sites, Section 2.2), the master-site scaling should be presented as an explicit limitation, with a discussion of alternatives (e.g., aggregating local Hessians when communication allows) or an analysis of the unbalanced case without this scaling.
minor comments (3)
  1. [Section 2.2] The phrase 'the line segment jointing' should read 'joining'; similar typos appear elsewhere (e.g., 'corvariates' in Table 4 and 'shrinkage the approximation error' in Remark 6).
  2. [Section 3] The main text contains no proof sketches for Theorems 1 and 2; given the complexity of Conditions 1 and 2, a short proof outline or a table mapping conditions to lemmas in the Supplementary Material would aid verification.
  3. [Table 4] The entries 'DC % %' for the multivariate hypotheses should be explained, since the DC test is only defined for univariate hypotheses.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the CST chi-square limit follows from the asymptotic normality of the aggregated score and a plug-in variance estimator, not from a fitted or self-referential input.

full rationale

The central claim, Theorem 2, is that the CST statistic TS is asymptotically chi-square with r degrees of freedom. This is derived, not imposed: the paper starts from the Kiefer-Bahadur representation for the traditional score statistic, approximates the global score by a surrogate collaborative loss, and then invokes asymptotic normality of the aggregated score together with consistency of the plug-in variance estimator in Condition 2(v). The null distribution is therefore obtained from a CLT on the score vector and Slutsky-type arguments for the estimated normalization. The oracle property is supported by standard folded-concave penalization theory, with citations such as Zou and Li (2008) and Fan, Xue and Zou (2014) supplying technical lemmas rather than the target result. No parameter is fitted to a subset of the data and then renamed as a prediction: the variance estimators are plug-in quantities whose consistency is explicitly assumed. The gradient-enhanced surrogate loss is presented as a construction motivated by the Kiefer-Bahadur decomposition, not as an unverified ansatz whose conclusion is built into its definition. The apparent algebraic identity in Section 2.4 concerning the averaged variance estimators could be a correctness concern, but it is not a circularity concern because it does not make the theorem equivalent to its inputs. Overall, no circular step is present in the claimed derivation chain.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

No new model entities are introduced. The central claim rests on a standard high-dimensional M-estimator framework plus the master-site homogeneity and beta-min conditions. These are data assumptions rather than free parameters fitted to the data. The main unshipped ingredient is a rigorously justified communication-limited estimator of the variance matrices J0 and K0; the paper's practical recipe is informal.

free parameters (2)
  • Regularization strengths λ_{I,k}, λ_{II} = HBIC-tuned in simulations; theoretical rates (12)-(13)
    Sparsity tuning controls support recovery and bias; practical choice may depart from the theoretical 'sufficiently large constants'.
  • Iteration counts K_I, K_II = Max 10 with stopping rule (difference < 0.001)
    Theory requires logarithmic lower bounds; the fixed cap may not satisfy Condition 1(vii) in all settings.
assumptions (7)
  • domain assumption The loss ℓ is convex and twice continuously differentiable with a unique minimizer β* (Section 2.1).
    All Taylor expansions and the score-statistic derivation assume smooth convex loss.
  • domain assumption Assumption 4 (homogeneity): the local Hessian at the master is uniformly close to the population Hessian, ∥∇²bL1(β*) − J∥_max ≤ δ∞ with δ∞ < 1.
    The surrogate loss uses the master's Hessian without a Hessian shift; this is the main distributional condition on the data partition.
  • domain assumption Assumption 2 (identifiability): eigenvalues of J and K0 are bounded away from 0 and above.
    Ensures the variance matrix V is invertible.
  • domain assumption Condition 1(iii) (beta-min): |γ*_S| ≥ (α0+α1)λII for the nuisance support.
    Required for support recovery and the oracle property; small nuisance coefficients break the equivalence of CST and OCST.
  • domain assumption Assumption 7: local alternatives satisfy ∥h∥_2 ≤ sqrt(r/N) with h = Cθ*−t.
    The chi-square limit is derived for local alternatives; the test may be conservative or non-standard far from H0.
  • domain assumption Assumption 8/Proposition 1 (GLM): sub-Gaussian covariates and noise, bounded derivatives, n1 ≳ (d+s) log p.
    These imply the concentration, restricted strong convexity, and restricted Lipschitz conditions used in the theory.
  • domain assumption Condition 1(iv) (Lyapunov): E∥Ω∇ℓ(β*;z)∥_2^3 = o(N^{1/2}).
    Needed for asymptotic normality of the score statistic.

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Cite this review

Pith. "Pith review of Collaborative Inference for Sparse High-Dimensional Models with Non-Shared Data." pith.science (2026). https://pith.science/paper/37662GNU

@misc{pith2026250419924,
  author       = {Pith},
  title        = {Pith review of: Collaborative Inference for Sparse High-Dimensional Models with Non-Shared Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/37662GNU}},
  note         = {Machine review of arXiv:2504.19924}
}
read the original abstract

In modern data analysis, statistical efficiency improvement is expected via effective collaboration among multiple data holders with non-shared data. In this article, we propose a collaborative score-type test (CST) for testing linear hypotheses, which accommodates potentially high-dimensional nuisance parameters and a diverging number of constraints and target parameters. Through a careful decomposition of the Kiefer-Bahadur representation for the traditional score statistic, we identify and approximate the key components using aggregated local gradient information from each data source. In addition, we employ a two-stage partial penalization strategy to shrink the approximation error and mitigate the bias from the high-dimensional nuisance parameters. Unlike existing methods, the CST procedure involves constrained optimization under non-shared and high-dimensional data settings, which requires novel theoretical developments. We derive the limiting distributions for the CST statistic under the null hypothesis and the local alternatives. Besides, the CST exhibits an oracle property and achieves the global statistical efficiency. Moreover, it relaxes the stringent restrictions on the number of data sources required in the current literature. Extensive numerical studies and a real example demonstrate the effectiveness and validity of our proposed method.

Figures

Figures reproduced from arXiv: 2504.19924 by the authors.

Figure 1
Figure 1. QQ plots of the p-values of the CST against the theoretical quantiles of the uniform [0, 1] distribution under the null hypothesis in the linear regression setting. The black solid line represents the equal line. fails to control the Type-I error, whereas the CST performs consistently well across all settings. This is consistent with our theoretical results and demonstrates that our proposed tests are more cooperati… view at source ↗
Figure 2
Figure 2. QQ plots of the p-values of the CST against the theoretical quantiles of the uniform [0, 1] distribution under the null hypothesis in the logistic regression setting. The black solid line represents the equal line [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. (a) Empirical type I error of the CST and the DC test (Battey et al., 2018) when the number of data sites m varies across {5, 10, 15, 20, 25, 30, 40}. (b) Empirical power of the CST and the DC test under the local alternatives when the number of data sites m = 10. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

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Reference graph

Works this paper leans on

22 extracted references · 21 canonical work pages

  1. [1]

    Collaborative Inference for Sparse High-Dimensional Models with Non-Shared Data Yifan Gu1, Hanfang Yang1, Songshan Yang1, and Hui Zou2 1Renmin University of China 2University of Minnesota Abstract: In modern data analysis, statistical efficiency improvement is expected via effective collaboration among multiple data holders with non-shared data. In this a...

    arXiv 2017

  2. [2]

    Assumption

    Assumptions 5 and 6 concern the restricted Lipschitz Hessian matrix and restricted strongly convexity (RSC) of the empirical loss function, which are standard for high-dimensional problems (Loh and Wainwright, 2017; Fan et al., 2018; Jordan et al., 2019). Assumption

    work page 2017

  3. [3]

    In Stage I, we use the standardℓ1 penalization to obtain a relatively accurate estimator: bβ(I,k)∈ argmin β=(θT,γT)T eL(β;bβ(I,k−1)) +λI,k∥γ∥1, k = 1,...,K I. (6) 9 In Stage II, starting from bβ(II,0) = bβ(I,KI), a reweighted ℓ1 penalization combined with linear constraints is used fork = 1,...,K II, in order to alleviate the non-negligible bias due to th...

    work page 2008

  4. [4]

    For hypotheses where the DC test yields significantp-values (i.e., below 0.05), the CST show even lowerp-values

    The DC test can be applied only to univariate hypotheses. For hypotheses where the DC test yields significantp-values (i.e., below 0.05), the CST show even lowerp-values. Moreover, while the DC test fails to reject the hypothesis ‘DoW.Thurs = 0’, the CST produces ap-value of 0.012. These results all support that the CST is more powerful 1publicly availabl...

    work page 2013

  5. [5]

    Taylor’s expansion

    All the proofs are deferred to the Supplementary Material. Notation. We use bold uppercase letters to represent matrices. For k ≥ 2, Ik and Ok represent ank×k identity matrix and zero matrix respectively. For any positive semi-definite matrix A, we useλmax(A) and λmin(A) to denote its maximum and minimum eigenvalues. For every integer k ≥ 1, write [k] = {...

    work page 2017

  6. [6]

    13 Assumption 4 (Homogeneity)

    We refer to Assumption 3.2 in Ning and Liu (2017) for more discussions on this assumption for high-dimensional inference. 13 Assumption 4 (Homogeneity). There exist δ∞ and δ2 ∈ (0,

    work page 2017

  7. [7]

    Assumption 4 introduces a homogeneity condition to control variations in loss functions across local data sites, akin to the conditions in Fan et al

    such that∥∇2bL(β∗)− J∥max∨ ∥∇2bL1(β∗)− J∥max≤δ∞, and∥∇2 bbL1(β∗)− J0∥2≤δ2. Assumption 4 introduces a homogeneity condition to control variations in loss functions across local data sites, akin to the conditions in Fan et al. (2023) (see Assumption 3.2 therein). Addi- tionally, we require that the Hessian matrix of the loss function on the master machine c...

    work page 2023

  8. [8]

    The positive definiteness of the Hessian essentially ensures thatb′′(xT iβ∗) is bounded away from zero with high probability, which is mild for sub-Gaussian designs

    (i) There exist some positive constantsσl,σu,Jl and Ju such that 0 < σl < λmin(Σ)≤λmax(Σ)<σ u <∞ and Jl <λ min(J)≤λmax(J)<J u; (ii) The covariatexi is aσx-sub-Gaussian vector with bounded coordinates∥xi∥∞≤ b1, and the random error∇ℓ(β∗;zi) = [b′(xT iβ∗)−yi]xi is aσe-sub-Gaussian vector; that is, for all α∈ Rp, E[exp(αTxi)]≤ exp(σ2 x∥α∥2 2/2) and E[exp(αT∇...

    work page 2012

Show all 22 references
  1. [9]

    It limits our consideration to local alternatives with a bounded radius for∥h∥2 and also requires the regularity of the constraint matrixC

    Assumption 7 focuses on the local alternative parameterh = Cθ∗−t, where the sequence of vectorsh→ 0 asN→∞ . It limits our consideration to local alternatives with a bounded radius for∥h∥2 and also requires the regularity of the constraint matrixC. Typical examples include Cid ...

    2012

  2. [10]

    16 For example, we can use the local ℓ1-penalized estimator computed on the master machine

    Condition (ii) requires that the initial estimator lies in anℓ1 cone centered atβ∗, which is a standard result forℓ1-penalized estimators (Bickel et al., 2009; Negahban et al., 2012). 16 For example, we can use the local ℓ1-penalized estimator computed on the master machine. C...

    2009

  3. [11]

    A common form of the irrepresentable condition can be formulated as follows: for a given valuea0∈ (0, 1), maxj∈Sc∥JjAJ−1 AA∥1≤a0, whereA is the true support of the specific problem

    Notably, Condition (iv) is significantly weaker than the irrepresentable condition, 17 which is both sufficient and nearly necessary for the model consistency of the Lasso (Zhao and Yu, 2006). A common form of the irrepresentable condition can be formulated as follows: for a g...

    2006

  4. [12]

    Corollary 1 shows that the asymptotic power function of the CST isP(χ2(r,eN) > χ2 α(r)), whereeN = NhTV−1h. This result aligns with the power functions of traditional score tests in low-dimensional scenarios with shared data (Boos, 1992), demonstrating the efficiency gains fro...

    2018

  5. [14]

    In particular, we examine the empirical Type I error and power analysis of the CST across different kinds of linear hypotheses and collaborative settings

    4 Numerical studies In this section, we conduct simulation studies to investigate the finite sample performance of the CST. In particular, we examine the empirical Type I error and power analysis of the CST across different kinds of linear hypotheses and collaborative settings...

    2018

  6. [15]

    Linear regression

    (18) We set the true valueβ∗ = (h1, 0, 0, 1, 1, 0T p−5)T and β∗ = (0, 0, 0, 1 +h2, 1, 0T p−5)T for the first two hypotheses and the last hypothesis, respectively, whereh1 and h2 represent the degree of violation of the null hypothesis. Linear regression. Simulated data are gen...

    2018

  7. [17]

    In the distributed system, the company with the most trips serves as the master site, withn1 = 15, 282local samples

    Then, the dataset for the first week of December 2023 containsN = 67, 758 trips. In the distributed system, the company with the most trips serves as the master site, withn1 = 15, 282local samples. In addition, quantitative covariates are standardized to have mean zero and var...

    2023

  8. [18]

    Specifically, we apply the sure independent ranking and screening (Zhu et al., 2011, SIRS) across all the170 covariates using theR package VariableScreening available in CRAN

    To formulate the testing hypotheses, we conduct preliminary analysis using taxi data from the last 10 days of November stored at the master site. Specifically, we apply the sure independent ranking and screening (Zhu et al., 2011, SIRS) across all the170 covariates using theR ...

    2011

  9. [20]

    Distributed testing and estimation under sparse high dimensional models

    The p-values based on the CST and the DC test. The top 8 corvariates {PT.Credit.Card, PT.Cash, DoW.Thurs, DoW.Tues, PCA.76, DoW.Wed, DoW.Sat} and the last 3 covariates {PCA.7, DCA.11, DCA.22} are selected by the preliminary ranking and screening procedure. Estimated coefficien...

    2022

  10. [841]

    Strong oracle optimality of folded concave penalized estimation

    Fan, J., Xue, L., and Zou, H. (2014). “Strong oracle optimality of folded concave penalized estimation.” The Annals of Statistics, 42(3), 819–849. He, X., Pan, X., Tan, K.M., and Zhou, W.X. (2023). “Smoothed quantile regression with large- scale inference.” Journal of Economet...

    2014

  11. [2001]

    Theℓ1 penalty (Tibshirani,

    and the minimax concave penalty (MCP) (Zhang, 2010). Theℓ1 penalty (Tibshirani,

    2010

  12. [2014]

    Third, partial penalization guarantees that no penalties are imposed onθ, the parameters of interest in our hypothesis

    and requires novel technical developments. Third, partial penalization guarantees that no penalties are imposed onθ, the parameters of interest in our hypothesis. This can avoid undesirable bias induced by the sparsity penalty on the final estimator ofθ∗ (Shi et al., 2019). Mo...

    2019

  13. [2023]

    This filtering process leads tom = 10 companies, representing the top ten service providers in

    In the data pre-processing, we remove trips with missing information, such as those with zero payment or unknown location, and retain trips from taxis and companies that record more than 1, 000 and 100, 000 trips in 2023, respectively, based on the unique ID numbers. This filt...

    2023

  14. [3645]

    A survey of tuning parameter selection for high-dimensional regression

    32 Wu, Y. and Wang, L. (2020). “A survey of tuning parameter selection for high-dimensional regression.” Annual Review of Statistics and its Application, 7(1), 209–226. Zhang, C.H. (2010). “Nearly unbiased variable selection under minimax concave penalty.”The Annals of Statist...

    2020

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