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Generalized Regression Estimators with High-Dimensional Covariates.

Tram Ta1, Jun Shao1,2, Quefeng Li3

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Summary
This summary is machine-generated.

The generalized regression estimator improves survey data precision using auxiliary variables. This study shows it remains efficient and robust even with a large, increasing number of covariates.

Keywords:
Asymptotic efficiencyLASSOauxiliary informationhigh dimensionmodel-assistedsurvey sampling

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Area of Science:

  • Statistics
  • Survey Methodology
  • Econometrics

Background:

  • Survey data often includes auxiliary variables with known population totals.
  • These covariates can enhance the precision of population total estimations.
  • The generalized regression estimator (GREG) leverages covariates within a model-assisted framework.

Purpose of the Study:

  • To investigate the performance of the GREG when the number of covariates (p) diverges with sample size (n).
  • To compare the GREG's efficiency against the Horvitz-Thompson estimator under diverging covariate scenarios.
  • To assess the robustness and variance estimation consistency of the GREG.

Main Methods:

  • Utilizing weighted least squares for parameter estimation when p < n.
  • Employing the LASSO method for parameter estimation when the model parameter is sparse.
  • Analyzing asymptotic properties under an assisted model and specific covariate distribution conditions.

Main Results:

  • The GREG demonstrates asymptotic efficiency gains over the Horvitz-Thompson estimator when p diverges with n.
  • The GREG is robust against model misspecification.
  • Consistency of variance estimation for the GREG is established.

Conclusions:

  • The GREG is a powerful tool for improving survey estimation precision, especially with high-dimensional covariates.
  • The findings support the use of GREG in complex survey designs where covariate numbers grow.
  • The study validates theoretical results through simulations and a practical example.