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Related Concept Videos

Absolute and Local Extreme Values01:22

Absolute and Local Extreme Values

The highest and lowest values of a function, relative to a reference axis, are known as extreme values. These include absolute maximum and absolute minimum values, which represent the highest and lowest points the function reaches across its entire domain. Within a restricted portion of the function, the highest and lowest values are referred to as local maximum and local minimum values, respectively.Periodic functions, such as sine and cosine, show extreme values at infinitely many points due...
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The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
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Entropy and the Second Law of Thermodynamics

Consider an isolated system in which a hot object is placed in contact with a cold one. This is an irreversible process that eventually leads both objects to reach the same equilibrium temperature. It is crucial to note that the constituents of any substance exhibit increased disorder at higher temperatures. As a cold substance absorbs heat, its constituents become more disordered. The energy transfer from a hotter object to a cooler one increases the system's disorder or randomness. This...
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Related Experiment Video

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Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180&#176; Curved Artery Test Section
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A local maximal inequality under uniform entropy.

Aad van der Vaart1, Jon A Wellner

  • 1Department of Mathematics, Faculty of Sciences, Vrije Universiteit De Boelelaan 1081a, 1081 HV Amsterdam, aad@cs.vu.nl.

Electronic Journal of Statistics
|March 17, 2012
PubMed
Summary

This study provides an upper bound for empirical process means, crucial for understanding function class behavior. The findings improve convergence rates for minimum contrast estimators in statistical analysis.

Area of Science:

  • Statistical Learning Theory
  • Empirical Processes
  • Functional Data Analysis

Background:

  • Empirical processes are fundamental in non-parametric statistics.
  • Understanding the supremum of empirical processes is key for generalization bounds.
  • Variance bounds are essential for controlling the behavior of statistical estimators.

Purpose of the Study:

  • To derive a novel upper bound for the mean of the supremum of an empirical process.
  • To establish a bound dependent on the uniform entropy integral and variance.
  • To demonstrate the application of this bound to minimum contrast estimators.

Main Methods:

  • Derivation of an upper bound using techniques from empirical process theory.
  • Characterization of the bound using the uniform entropy integral at a given variance level δ.

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  • Application to analyze the convergence rates of minimum contrast estimators.

    • Main Results:

    • An explicit upper bound for the mean of the supremum of the empirical process is derived.
    • The bound is shown to be dependent on the uniform entropy integral of the function class.
    • The derived bound provides a theoretical guarantee for the performance of minimum contrast estimators.

    Conclusions:

    • The derived upper bound offers a refined tool for analyzing empirical processes.
    • This work contributes to the theoretical understanding of statistical estimation.
    • The results have implications for the efficiency and convergence properties of estimators in statistical learning.