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DISTRIBUTED TESTING AND ESTIMATION UNDER SPARSE HIGH DIMENSIONAL MODELS.

Heather Battey1,2, Jianqing Fan1,3, Han Liu1

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|July 24, 2018
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Summary
This summary is machine-generated.

This study introduces a divide-and-conquer algorithm for hypothesis testing and parameter estimation. It shows how to aggregate subsample statistics to achieve near-full sample efficiency, even with many subsamples.

Keywords:
62F10Divide and conquerPrimary 62F05debiasingmassive datasecondary 62F12thresholding

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Area of Science:

  • Statistics
  • Computational Statistics
  • Machine Learning

Background:

  • The divide-and-conquer algorithm is widely used for large-scale data analysis.
  • Assessing the statistical efficiency of such algorithms is crucial for reliable inference.

Purpose of the Study:

  • To develop novel test statistics and point estimators using a unified likelihood-based framework.
  • To determine the maximum number of subsamples (k) for negligible efficiency loss in divide-and-conquer methods.
  • To ensure estimators match oracle rates in low and high-dimensional settings.

Main Methods:

  • Aggregation of statistics from k subsamples of size n/k.
  • Development of a unified likelihood-based framework for hypothesis testing and parameter estimation.
  • Analysis in both low-dimensional and sparse high-dimensional settings.

Main Results:

  • Proposed new test statistics and point estimators.
  • Established conditions on k for negligible efficiency loss as sample size (n) grows.
  • Demonstrated that estimators achieve oracle inferential efficiencies and estimation rates.

Conclusions:

  • The proposed divide-and-conquer approach maintains statistical efficiency comparable to using the full sample.
  • The findings provide theoretical guarantees for the performance of aggregated estimators in large-scale statistical inference.