pith. sign in

arxiv: 2606.08981 · v1 · pith:43VCNL7Cnew · submitted 2026-06-08 · 📊 stat.ME

Divide-and-shrink: An efficient and heterogeneity-agnostic approach for transfer estimation using summary statistics

Ruoyu Wang , Xihong Lin This is my paper

Pith reviewed 2026-06-27 15:57 UTC · model grok-4.3

classification 📊 stat.ME
keywords transfer estimationsummary statisticspopulation heterogeneityshrinkage estimationmodel-free methodsclosed-form estimatorquadratic error
0
0 comments X

The pith

The dShrink estimator combines target and external summary statistics into a closed-form solution guaranteed to reduce expected quadratic error versus target-only estimation under arbitrary heterogeneity.

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

The paper presents the divide-and-shrink (dShrink) method for estimating parameters in a target population by fusing summary statistics from the target and one or more external sources. The approach derives an explicit formula that requires no model assumptions, no user-chosen tuning parameters, and no access to individual-level data. A central result establishes that the dShrink estimator has strictly lower or equal expected quadratic error than the estimator based solely on target data, with larger gains when populations are similar or true parameter values are near zero. The method stays effective when external covariance matrices are unavailable and extends to multiple sources plus side information.

Core claim

dShrink is a model-free, tuning-free procedure that produces a closed-form estimator by dividing the problem into target and source components and shrinking their combination; the resulting estimator is guaranteed to outperform the target-only estimator in expected quadratic error for arbitrary population heterogeneity and applies across a wide class of parameter estimation tasks.

What carries the argument

The divide-and-shrink combination rule, which forms a weighted average of target and source summary statistics with weights derived directly from the observed summaries themselves.

If this is right

  • The estimator applies to many parameter estimation problems without requiring a specific statistical model.
  • Performance remains strong even when covariance information is missing from the external summaries.
  • Gains increase when target and source populations are similar or when true parameter values are near zero.
  • The method accommodates side information and summary statistics from multiple external populations simultaneously.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same closed-form shrinkage step could be examined for high-dimensional parameters where the number of variables exceeds sample size.
  • Publicly released summary statistics from large cohorts could be aggregated with dShrink without needing data-use agreements for raw records.
  • Direct comparisons with standard meta-analysis estimators under mismatched population sizes would quantify the robustness difference.

Load-bearing premise

The provided summary statistics from each external source accurately represent that source population and can be combined directly with target summaries without introducing extra bias or incompatibility.

What would settle it

A simulation in which external summary statistics are deliberately drawn from populations whose parameters differ from the reported values in a way that violates direct combinability would show whether the quadratic-error guarantee still holds.

Figures

Figures reproduced from arXiv: 2606.08981 by Ruoyu Wang, Xihong Lin.

Figure 1
Figure 1. Figure 1: MSEs of βbT , βbC , βbprt, and βbds in the simulation under the setting of (5) with H = Ip, p = 5, nT = nS = 300, βT = αT = µ + b, βS = αS = µ − b, µ = (0.05, 0.02, 0.1, 0.1, 0.1)T/ √ 5, b = tη, η = (0.2, 0.3, 0.3, 0.3, 0.3)T/ √ 5 and different t’s. estimator underperforms βbT , especially at small values of t. We delve into the reasons behind the observed phenomena concerning the performance of the SURE e… view at source ↗
Figure 2
Figure 2. Figure 2: MSEs of λbs,1, λbs,2 λbSURE,1, and λbSURE,2 defined in (7) and (10) with respect to their pop￾ulation counterparts λs,1, λs,2, λopt,1, and λopt,2 in the simulation under the setting of (5) with H = Ip, p = 5, nT = nS = 300, βT = αT = µ+b, αS = µ−b, µ = (0.05, 0.02, 0.1, 0.1, 0.1)T/ √ 5, b = tη, η = (0.2, 0.3, 0.3, 0.3, 0.3)T/ √ 5 and different t’s. hand, although ∥λbSURE,2δbC − λopt,2δbC ∥ ≤ ∥δbC ∥|λbSURE,… view at source ↗
Figure 3
Figure 3. Figure 3: MSEs of the estimator using only the data from the target population βbT , the JS esti￾mator βbJS, the data enriched linear regression, and the dShrink estimator βbds in the simulation under the setting of (5) with H = Ip, p = 5, nT = nS = 300, µ = (0.05, 0.02, 0.1, 0.1, 0.1)T/ √ 5, b = tη, η = (0.2, 0.3, 0.3, 0.3, 0.3)T/ √ 5 and different t’s. is replaced by βbS reduces to that of data enriched linear reg… view at source ↗
Figure 4
Figure 4. Figure 4: MSEs of the regression functions derived from βbT , βbC , βbprt, βbSURE, and βbds in simu￾lation studies. (a) pX = 10, qX = 6, nT = 100, and nS = 500; (b) pX = 30, qX = 20, nT = 300, and nS = 1000. The calibration estimator βbC performs well when t is small, i.e., when target and source populations are approximately homogeneous. Its error grows rapidly as t increases, i.e., as the 32 [PITH_FULL_IMAGE:figu… view at source ↗
Figure 5
Figure 5. Figure 5: compares the dShrink estimator with the constrained maximum likelihood (CML) method [Chatterjee et al., 2016], the generalized integration method (GIM) [Zhang et al., 2020], the penalized constrained maximum likelihood (PCML) method [Zhai and Han, 2022], and the JS type estimator (JS Comb) proposed in [Han et al., 2024] based on 5000 simulation runs. 0.050 0.075 0.100 0.125 0.150 0.00 0.25 0.50 0.75 1.00 t… view at source ↗
Figure 6
Figure 6. Figure 6: Average coverage rates and average widths over different components of the standard CIs based on βbT and the CIs (15) in simulation studies. The average widths are multiplied by ten. (a) pX = 10, qX = 6, nT = 100, and nS = 500; (b) pX = 30, qX = 20, nT = 300, and nS = 1000. competitors in terms of MSE and coverage rate, and reduces the width of the CIs compared to the target population-based estimator in m… view at source ↗
Figure 7
Figure 7. Figure 7: Confidence intervals of the individualized treatment effects for the covariate values of 10 randomly selected patients from the randomized trial [Elmunzer et al., 2012] constructed using the target population-based estimator and the dShrink estimator calculate the point estimates and their CIs for the individualized treatment effects of individual patients based on their covariate values in the data set of… view at source ↗
read the original abstract

Knowledge transfer across data sources holds great promise for improving the estimation of target population parameters by leveraging the growing availability of data from different sources. However, the effectiveness of knowledge transfer is often challenged by the complex and pervasive heterogeneity between data sources and the lack of access to individual-level data. This paper proposes the divide-and-shrink (dShrink) method, a transfer estimation method that estimates target population parameters in a closed form using summary statistics from a target population and some external source populations while accounting for population heterogeneity. The dShrink estimator is guaranteed to outperform the estimator based solely on the target population in terms of expected quadratic error under arbitrary population heterogeneity. The gain can be substantial when the target and source populations are similar, or the underlying true parameter values are near zero. Notably, dShrink is model-free, requires no user-specified tuning parameters, robust to various types of heterogeneity between data sources, and applies to a broad range of parameter estimation problems. dShrink remains effective even when the covariance matrix is not accessible for the external summary statistics and offers flexibility in incorporating side information and summary statistics from multiple source populations. Simulations and real data analyses demonstrate the superior performance of the dShrink estimator and its potential as a robust tool for transfer estimation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes the divide-and-shrink (dShrink) estimator, a closed-form method that combines summary statistics from a target population with those from external source populations to estimate target parameters. It claims that dShrink is guaranteed to achieve expected quadratic risk no larger than the target-only estimator under arbitrary heterogeneity, is model-free, requires no tuning parameters, remains effective without external covariance matrices, and accommodates multiple sources or side information.

Significance. If the risk inequality holds as stated, the result would provide a practical, tuning-free tool for transfer estimation in settings where only summary statistics are available, with potential applicability across parameter estimation problems in statistics and related fields. The model-free character and explicit robustness claim to heterogeneity are notable strengths if the supporting derivation is complete.

major comments (1)
  1. [Abstract] Abstract: the central guarantee that dShrink has expected quadratic error ≤ target-only estimator for arbitrary heterogeneity rests on the external summary statistics being unbiased (or bias-correctable) representations of their source parameters that can be linearly combined without incompatibility. The derivation of the risk bound must explicitly state and justify this compatibility assumption; if it is implicit, the guarantee does not automatically extend to fully arbitrary heterogeneity that includes unmodeled bias, differing sampling designs, or non-commensurable parameter spaces.
minor comments (1)
  1. The abstract refers to simulations and real-data analyses demonstrating superior performance, but the main text should include explicit statements of the simulation designs (e.g., heterogeneity levels, sample sizes) and data sources to allow independent verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for this constructive comment on the assumptions underlying our risk bound. We agree that the abstract and derivation would benefit from an explicit statement of the compatibility conditions, and we will revise the manuscript to address this.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central guarantee that dShrink has expected quadratic error ≤ target-only estimator for arbitrary heterogeneity rests on the external summary statistics being unbiased (or bias-correctable) representations of their source parameters that can be linearly combined without incompatibility. The derivation of the risk bound must explicitly state and justify this compatibility assumption; if it is implicit, the guarantee does not automatically extend to fully arbitrary heterogeneity that includes unmodeled bias, differing sampling designs, or non-commensurable parameter spaces.

    Authors: We agree with the referee that the risk inequality is derived under the assumption that the external summary statistics are unbiased (or bias-corrected) estimators of their respective source parameters and that these estimators are commensurable with the target parameter space so that linear combination is valid. The phrase 'arbitrary population heterogeneity' in the manuscript is intended to mean arbitrary differences in the underlying true parameter values across populations, not arbitrary biases or incompatibilities in the supplied summary statistics themselves. The derivation in Section 3 proceeds from the unbiasedness of all input estimators and the quadratic risk decomposition; no further modeling assumptions on the data-generating processes are used. We will revise the abstract and add an explicit statement of this assumption (with justification) to the beginning of the theoretical development section. This change will make the scope of the guarantee precise without altering the stated results or proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; guarantee derived independently from quadratic risk comparison

full rationale

The paper derives a closed-form dShrink estimator and proves its expected quadratic risk is always at most that of the target-only estimator under arbitrary heterogeneity. This guarantee follows directly from the algebraic construction of the shrinkage weights and does not reduce to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation. The method is explicitly model-free with no user tuning parameters, and the abstract states the performance bound holds without additional compatibility assumptions beyond the provided summaries. No enumerated circularity patterns are present in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract does not introduce or rely on any free parameters, axioms, or invented entities; the approach is explicitly model-free with no user-specified tuning parameters.

pith-pipeline@v0.9.1-grok · 5750 in / 1100 out tokens · 23337 ms · 2026-06-27T15:57:30.518836+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Tuning-Free Efficient Estimation for Multi-Source Data via Covariance-Aware Shrinkage

    stat.ME 2026-06 unverdicted novelty 6.0

    Proposes a covariance-aware tuning-free shrinkage framework and sequential algorithm for multi-source estimation that attains oracle risk asymptotically and improves on single-step methods.

Reference graph

Works this paper leans on

86 extracted references · cited by 1 Pith paper

  1. [1]

    American Journal of Epidemiology , volume=

    Systematic review and meta-analyses of perinatal death and maternal exposure to tobacco smoke during pregnancy , author=. American Journal of Epidemiology , volume=. 2016 , publisher=

    2016

  2. [2]

    , author=

    On Data-Enriched Logistic Regression. , author=. Mathematics , volume=

  3. [3]

    2019 , publisher=

    High-dimensional Statistics: A Non-asymptotic Viewpoint , author=. 2019 , publisher=

    2019

  4. [4]

    Electronic Journal of Statistics , volume=

    Turning the information-sharing dial: efficient inference from different data sources , author=. Electronic Journal of Statistics , volume=. 2024 , publisher=

    2024

  5. [5]

    Journal of Machine Learning Research , volume=

    Transfer learning with uncertainty quantification: Random effect calibration of source to target (recast) , author=. Journal of Machine Learning Research , volume=

  6. [6]

    Electronic Journal of Statistics , volume=

    Data enriched linear regression , author=. Electronic Journal of Statistics , volume=

  7. [7]

    Statistical Science , volume=

    Bayesian transfer learning , author=. Statistical Science , volume=. 2025 , publisher=

    2025

  8. [8]

    The Annals of Applied Statistics , volume=

    Targeting underrepresented populations in precision medicine: a federated transfer learning approach , author=. The Annals of Applied Statistics , volume=

  9. [9]

    Journal of the American Statistical Association , pages=

    Federated adaptive causal estimation (face) of target treatment effects , author=. Journal of the American Statistical Association , pages=. 2025 , publisher=

    2025

  10. [10]

    Cambridge Books , year=

    Asymptotic Statistics , author=. Cambridge Books , year=

  11. [11]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Causal inference with invalid instruments: post-selection problems and a solution using searching and sampling , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2023 , publisher=

    2023

  12. [12]

    arXiv preprint arXiv:2501.18577 , year=

    Prediction-Powered Inference with Imputed Covariates and Nonuniform Sampling , author=. arXiv preprint arXiv:2501.18577 , year=

    arXiv

  13. [13]

    Proceedings of the National Academy of Sciences , volume=

    The rise and fall of excess male infant mortality , author=. Proceedings of the National Academy of Sciences , volume=. 2008 , publisher=

    2008

  14. [14]

    Nature Genetics , volume=

    Valid inference for machine learning-assisted GWAS , author=. Nature Genetics , volume=

  15. [15]

    Advances in Neural Information Processing Systems , volume=

    Task-Agnostic Machine Learning-Assisted Inference , author=. Advances in Neural Information Processing Systems , volume=

  16. [16]

    Journal of Machine Learning Research , volume=

    The correlation-assisted missing data estimator , author=. Journal of Machine Learning Research , volume=

  17. [17]

    Biometrika , volume=

    Data integration: exploiting ratios of parameter estimates from a reduced external model , author=. Biometrika , volume=. 2023 , publisher=

    2023

  18. [18]

    1994 , publisher=

    A survey of componentwise perturbation theory , author=. 1994 , publisher=

    1994

  19. [19]

    Statistics in Medicine , volume=

    Multivariate meta-analysis: potential and promise , author=. Statistics in Medicine , volume=. 2011 , publisher=

    2011

  20. [20]

    arXiv preprint arXiv:2308.05883 , year=

    Empirical bayes estimation with side information: A nonparametric integrative tweedie approach , author=. arXiv preprint arXiv:2308.05883 , year=

    arXiv

  21. [21]

    Journal of the American Statistical Association , volume=

    Adaptive sparse estimation with side information , author=. Journal of the American Statistical Association , volume=. 2020 , publisher=

    2020

  22. [22]

    Nature Genetics , volume=

    Dynamic incorporation of multiple in silico functional annotations empowers rare variant association analysis of large whole-genome sequencing studies at scale , author=. Nature Genetics , volume=. 2020 , publisher=

    2020

  23. [23]

    Biometrics , volume=

    Integrating external summary information in the presence of prior probability shift: an application to assessing essential hypertension , author=. Biometrics , volume=. 2024 , publisher=

    2024

  24. [24]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Robust angle-based transfer learning in high dimensions , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2025 , publisher=

    2025

  25. [25]

    The Annals of Statistics , volume=

    Transfer learning for nonparametric classification: Minimax rate and adaptive classifier , author=. The Annals of Statistics , volume=

  26. [26]

    Journal of the American Statistical Association , number=

    Doubly robust augmented model accuracy transfer inference with high dimensional features , author=. Journal of the American Statistical Association , number=. 2024 , publisher=

    2024

  27. [27]

    Journal of the American Statistical Association , pages=

    Semi-supervised triply robust inductive transfer learning , author=. Journal of the American Statistical Association , pages=. 2024 , publisher=

    2024

  28. [28]

    Biometrics , volume=

    Efficient data integration under prior probability shift , author=. Biometrics , volume=. 2024 , publisher=

    2024

  29. [29]

    The Annals of Mathematical Statistics , pages=

    Non-optimality of preliminary-test estimators for the mean of a multivariate normal distribution , author=. The Annals of Mathematical Statistics , pages=. 1972 , publisher=

    1972

  30. [30]

    Improving prediction of linear regression models by integrating external information from heterogeneous populations:

    Han, Peisong and Li, Haoyue and Park, Sung Kyun and Mukherjee, Bhramar and Taylor, Jeremy MG , journal=. Improving prediction of linear regression models by integrating external information from heterogeneous populations:. 2024 , publisher=

    2024

  31. [31]

    Green, Edwin J and Strawderman, William E , journal=. A. 1991 , publisher=

    1991

  32. [32]

    The Annals of Statistics , pages=

    Estimation of the mean of a multivariate normal distribution , author=. The Annals of Statistics , pages=. 1981 , publisher=

    1981

  33. [33]

    Biometrics , volume=

    Combining primary cohort data with external aggregate information without assuming comparability , author=. Biometrics , volume=. 2021 , publisher=

    2021

  34. [34]

    Journal of the American Statistical Association , pages=

    Robust inference for federated meta-learning , author=. Journal of the American Statistical Association , pages=. 2025 , publisher=

    2025

  35. [35]

    Quantitative Economics , volume=

    Model averaging, asymptotic risk, and regressor groups , author=. Quantitative Economics , volume=. 2014 , publisher=

    2014

  36. [36]

    arXiv preprint arXiv:2303.17765 , year=

    Learning from similar linear representations: Adaptivity, minimaxity, and robustness , author=. arXiv preprint arXiv:2303.17765 , year=

    arXiv

  37. [37]

    Journal of Machine Learning Research , volume=

    Fused Lasso Approach in Regression Coefficients Clustering--Learning Parameter Heterogeneity in Data Integration , author=. Journal of Machine Learning Research , volume=

  38. [38]

    Journal of Machine Learning Research , volume=

    Augmented transfer regression learning with semi-non-parametric nuisance models , author=. Journal of Machine Learning Research , volume=

  39. [39]

    Stochastic Processes and their Applications , volume=

    Multivariate regression estimation local polynomial fitting for time series , author=. Stochastic Processes and their Applications , volume=. 1996 , publisher=

    1996

  40. [40]

    Journal of Econometrics , volume=

    Convergence rates and asymptotic normality for series estimators , author=. Journal of Econometrics , volume=. 1997 , publisher=

    1997

  41. [41]

    Finite sample theory , author=

    Parametric estimation. Finite sample theory , author=. The Annals of Statistics , volume=

  42. [42]

    The Annals of Probability , pages=

    Rates of convergence in the martingale central limit theorem , author=. The Annals of Probability , pages=. 1981 , publisher=

    1981

  43. [43]

    Journal of Statistical Planning and Inference , volume=

    On the dependence of the Berry--Esseen bound on dimension , author=. Journal of Statistical Planning and Inference , volume=. 2003 , publisher=

    2003

  44. [44]

    Statistics in Medicine , volume=

    Simultaneous selection and incorporation of consistent external aggregate information , author=. Statistics in Medicine , volume=. 2023 , publisher=

    2023

  45. [45]

    Journal of the Royal Statistical Society Series A: Statistics in Society , volume=

    A re-evaluation of random-effects meta-analysis , author=. Journal of the Royal Statistical Society Series A: Statistics in Society , volume=. 2009 , publisher=

    2009

  46. [46]

    arXiv preprint arXiv:2210.00200 , year=

    Semiparametric Efficient Fusion of Individual Data and Summary Statistics , author=. arXiv preprint arXiv:2210.00200 , year=

    arXiv

  47. [47]

    Journal of the American Statistical Association , volume=

    Multivariate meta-analysis of heterogeneous studies using only summary statistics: efficiency and robustness , author=. Journal of the American Statistical Association , volume=

  48. [48]

    Journal of the American Statistical Association , volume=

    Combining multiple observational data sources to estimate causal effects , author=. Journal of the American Statistical Association , volume=. 2020 , publisher=

    2020

  49. [49]

    Statistics in Medicine , volume=

    A unified approach for synthesizing population-level covariate effect information in semiparametric estimation with survival data , author=. Statistics in Medicine , volume=. 2020 , publisher=

    2020

  50. [50]

    Biometrika , volume=

    Using covariate-specific disease prevalence information to increase the power of case-control studies , author=. Biometrika , volume=. 2015 , publisher=

    2015

  51. [51]

    Journal of the American Statistical Association , volume=

    Efficient estimation of the Cox model with auxiliary subgroup survival information , author=. Journal of the American Statistical Association , volume=. 2016 , publisher=

    2016

  52. [52]

    JAMA Internal Medicine , volume=

    The skyrocketing cost of rectal indomethacin , author=. JAMA Internal Medicine , volume=. 2020 , publisher=

    2020

  53. [53]

    2013 , publisher=

    The bootstrap and Edgeworth expansion , author=. 2013 , publisher=

    2013

  54. [54]

    1992 , publisher =

    Quadratic forms in random variables: theory and applications , author=. 1992 , publisher =

    1992

  55. [55]

    Annals of the Institute of Statistical Mathematics , volume=

    Asymptotic expansions for the distribution of quadratic forms in normal variables , author=. Annals of the Institute of Statistical Mathematics , volume=. 1988 , publisher=

    1988

  56. [56]

    Bernoulli , volume=

    Improved minimax estimation of a multivariate normal mean under heteroscedasticity , author=. Bernoulli , volume=

  57. [57]

    Journal of the American Statistical Association , volume=

    The statistical consequences of preliminary test estimators in regression , author=. Journal of the American Statistical Association , volume=. 1973 , publisher=

    1973

  58. [58]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Elastic integrative analysis of randomised trial and real-world data for treatment heterogeneity estimation , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2023 , publisher=

    2023

  59. [59]

    Journal of the American Statistical Association , volume=

    On pooling data , author=. Journal of the American Statistical Association , volume=. 1948 , publisher=

    1948

  60. [60]

    Electronic Journal of Statistics , volume=

    Pretest estimation in combining probability and non-probability samples , author=. Electronic Journal of Statistics , volume=. 2023 , publisher=

    2023

  61. [61]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    A unified approach to regression analysis under double-sampling designs , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2000 , publisher=

    2000

  62. [62]

    Journal of the American Statistical Association , volume=

    Calibration estimators in survey sampling , author=. Journal of the American Statistical Association , volume=. 1992 , publisher=

    1992

  63. [63]

    arXiv preprint arXiv:2209.04977 , year=

    Semi-supervised Triply Robust Inductive Transfer Learning , author=. arXiv preprint arXiv:2209.04977 , year=

    arXiv

  64. [64]

    Biometrics , volume=

    Synthesizing external aggregated information in the presence of population heterogeneity: A penalized empirical likelihood approach , author=. Biometrics , volume=. 2022 , publisher=

    2022

  65. [65]

    Journal of the American Statistical Association , pages=

    Transfer learning under high-dimensional generalized linear models , author=. Journal of the American Statistical Association , pages=. 2022 , publisher=

    2022

  66. [66]

    Journal of the American Statistical Association , pages=

    Estimation and inference for high-dimensional generalized linear models with knowledge transfer , author=. Journal of the American Statistical Association , pages=. 2023 , publisher=

    2023

  67. [67]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Transfer learning for high-dimensional linear regression: Prediction, estimation and minimax optimality , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2022 , publisher=

    2022

  68. [68]

    Management Science , volume=

    Predicting with proxies: Transfer learning in high dimension , author=. Management Science , volume=. 2021 , publisher=

    2021

  69. [69]

    arXiv preprint arXiv:2007.12922 , year=

    Improved inference for heterogeneous treatment effects using real-world data subject to hidden confounding , author=. arXiv preprint arXiv:2007.12922 , year=

    arXiv 2007

  70. [70]

    arXiv preprint arXiv:2202.12891 , year=

    Combining observational and randomized data for estimating heterogeneous treatment effects , author=. arXiv preprint arXiv:2202.12891 , year=

    arXiv

  71. [71]

    arXiv preprint arXiv:2308.14836 , year=

    Data fusion using weakly aligned sources , author=. arXiv preprint arXiv:2308.14836 , year=

    arXiv

  72. [72]

    arXiv preprint arXiv:2302.13428 , year=

    Methods for Integrating Trials and Non-Experimental Data to Examine Treatment Effect Heterogeneity , author=. arXiv preprint arXiv:2302.13428 , year=

    arXiv

  73. [73]

    World Journal of Gastroenterology: WJG , volume=

    Is rectal indomethacin effective in preventing of post-endoscopic retrograde cholangiopancreatography pancreatitis? , author=. World Journal of Gastroenterology: WJG , volume=. 2014 , publisher=

    2014

  74. [74]

    New England Journal of Medicine , volume=

    A randomized trial of rectal indomethacin to prevent post-ERCP pancreatitis , author=. New England Journal of Medicine , volume=. 2012 , publisher=

    2012

  75. [75]

    Nature genetics , volume=

    Multi-ancestry genetic study of type 2 diabetes highlights the power of diverse populations for discovery and translation , author=. Nature genetics , volume=. 2022 , publisher=

    2022

  76. [76]

    Nature genetics , volume=

    Fine-mapping type 2 diabetes loci to single-variant resolution using high-density imputation and islet-specific epigenome maps , author=. Nature genetics , volume=. 2018 , publisher=

    2018

  77. [77]

    Diabetes , volume=

    Transferability and fine mapping of type 2 diabetes loci in African Americans: the Candidate Gene Association Resource Plus Study , author=. Diabetes , volume=. 2013 , publisher=

    2013

  78. [78]

    Statistics in Medicine , volume=

    A random-effects regression model for meta-analysis , author=. Statistics in Medicine , volume=. 1995 , publisher=

    1995

  79. [79]

    Nature communications , volume=

    Genome-wide association analyses identify 143 risk variants and putative regulatory mechanisms for type 2 diabetes , author=. Nature communications , volume=. 2018 , publisher=

    2018

  80. [80]

    Nature , volume=

    Identification of type 2 diabetes loci in 433,540 East Asian individuals , author=. Nature , volume=. 2020 , publisher=

    2020

Showing first 80 references.