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Lossless Transformations and Excess Risk Bounds in Statistical Inference.

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Summary
This summary is machine-generated.

This study introduces excess minimum risk in statistical inference, defining lossless transformations and developing a consistent test for them. It also provides information-theoretic bounds for δ-lossless transformations across various applications.

Keywords:
classificationdeep learninginformation bottleneckinformation-theoretic boundsportfolio selectionregressionstatistical inference with lossstrongly consistent test

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Area of Science:

  • Statistical Inference
  • Information Theory
  • Machine Learning

Background:

  • Excess minimum risk quantifies information loss in statistical estimation.
  • Understanding transformations that preserve essential information is crucial for efficient inference.

Purpose of the Study:

  • To define and characterize lossless transformations in statistical inference.
  • To develop a statistical test for identifying lossless transformations.
  • To establish information-theoretic bounds on excess risk for general loss functions.

Main Methods:

  • Definition of excess minimum risk and lossless transformations.
  • Construction of a partitioning test statistic for lossless hypothesis testing.
  • Derivation of information-theoretic upper bounds on excess risk.
  • Introduction of the concept of δ-lossless transformations.

Main Results:

  • Characterization of lossless transformations.
  • Strong consistency of the partitioning test for i.i.d. data.
  • Uniform information-theoretic upper bounds on excess risk.
  • Sufficient conditions for universally δ-lossless transformations.

Conclusions:

  • Lossless transformations are identified and tested, with implications for data analysis.
  • Information-theoretic bounds provide a general framework for assessing transformation efficiency.
  • The concepts are broadly applicable to diverse fields including deep learning and econometrics.