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Related Concept Videos

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The population standard deviation is rarely known in many day-to-day examples of statistics. When the sample sizes are large, it is easy to estimate the population standard deviation using a confidence interval, which provides results close enough to the original value. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
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Related Experiment Video

Updated: May 31, 2025

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Self-Normalized Moderate Deviations for Degenerate U-Statistics.

Lin Ge1, Hailin Sang2, Qi-Man Shao3

  • 1Division of Arts and Sciences, Mississippi State University at Meridian, Meridian, MS 39307, USA.

Entropy (Basel, Switzerland)
|January 24, 2025
PubMed
Summary
This summary is machine-generated.

This study analyzes self-normalized moderate deviations for degenerate U-statistics. We establish a key result on probability bounds for these statistics, with applications to the law of the iterated logarithm.

Keywords:
degenerate U-statisticslaw of the iterated logarithmmoderate deviationself-normalization

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Area of Science:

  • Probability Theory
  • Statistics
  • Stochastic Processes

Background:

  • Focuses on degenerate U-statistics of order 2, a complex area within statistical theory.
  • Examines properties of symmetric kernel functions defined as infinite sums of products of functions.

Purpose of the Study:

  • To investigate self-normalized moderate deviation principles for degenerate U-statistics.
  • To derive precise probability bounds for these statistics under specific conditions.

Main Methods:

  • Utilizes the properties of independent and identically distributed (i.i.d.) random variables.
  • Applies techniques related to the domain of attraction of normal laws for kernel functions.
  • Employs conditions on the sum of lambda coefficients and truncated function properties.

Main Results:

  • Establishes a moderate deviation theorem for degenerate U-statistics of order 2.
  • Derives the asymptotic behavior of the logarithm of probabilities involving these statistics.
  • Obtains a law of the iterated logarithm as a direct application.

Conclusions:

  • Provides a significant theoretical advancement in the study of U-statistics.
  • The findings offer deeper insights into the probabilistic behavior of complex statistical estimators.
  • The derived law of the iterated logarithm has implications for the asymptotic normality of related processes.