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Updated: Sep 1, 2025

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THE PROHOROV METRIC FRAMEWORK AND AGGREGATE DATA INVERSE PROBLEMS FOR RANDOM PDEs.

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  • 1Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh, NC, USA.

Communications in Applied Analysis
|August 12, 2022
PubMed
Summary

This study presents a computational method for estimating probability measures using only population-level data. It establishes the existence and consistency of least squares estimators for measure estimation, applicable to random partial differential equations (PDEs).

Keywords:
34A5546S5062G0793E24aggregate dataexistence and approximation of estimatorsindividual datainverse problems

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Area of Science:

  • Statistics
  • Computational Mathematics
  • Probability Theory

Background:

  • Estimation of probability measures often requires individual-level data.
  • Aggregate data presents unique challenges for parameter estimation.
  • Existing computational methods have been developed over several decades.

Purpose of the Study:

  • To address nonparametric estimation of probability measures using aggregate data.
  • To summarize and theoretically validate existing computational estimation methods.
  • To apply these methods to problems involving random partial differential equations (PDEs).

Main Methods:

  • Summarizing a long-standing computational method for measure estimation.
  • Presenting theoretical results on least squares estimation.
  • Developing general estimators for ordinary and generalized least squares.

Main Results:

  • Established the existence and consistency of least squares estimates.
  • Demonstrated the applicability of the method to measure estimation problems.
  • Showcased specific applications to random PDEs.

Conclusions:

  • The proposed method provides a consistent framework for nonparametric estimation with aggregate data.
  • Theoretical guarantees support the use of least squares estimators.
  • The approach is effective for complex problems like random PDEs.