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Asymptotic Normality and Convergence Rates for Tsallis Entropy Estimators via Stabilization Techniques.

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We introduce new goodness-of-fit (GOF) tests using Tsallis entropy for complex models like q-Gaussian distributions. These methods offer reliable statistical measurement across various dimensions and parameters.

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Area of Science:

  • Statistics
  • Information Theory
  • Statistical Modeling

Background:

  • Goodness-of-fit (GOF) testing is crucial for validating statistical models.
  • Tsallis entropy offers a generalized framework for entropy calculations.
  • Existing GOF methods may not adequately address complex, non-Gaussian distributions.

Purpose of the Study:

  • To develop novel GOF procedures based on Tsallis entropy.
  • To specifically target multivariate exponential-power (generalized Gaussian) and q-Gaussian models.
  • To provide a robust statistical framework for model validation in complex scenarios.

Main Methods:

  • The proposed GOF statistic compares closed-form Tsallis entropy under the null hypothesis with a non-parametric k-nearest-neighbor (k-NN) estimator.
  • Theoretical analysis establishes consistency and mean-square convergence of the k-NN estimator.
  • Asymptotic normality is discussed for the regime where q approaches 1.
  • Critical values are calibrated using parametric bootstrap and permutation methods.

Main Results:

  • The study establishes the theoretical underpinnings (consistency, convergence) of the proposed GOF estimator.
  • Extensive Monte Carlo simulations assess empirical size, power, and runtime across different dimensions, k values, and q values.
  • The methods demonstrate practical applicability through an example, highlighting calibration and sensitivity.

Conclusions:

  • The developed Tsallis entropy-based GOF procedures provide a statistically sound and computationally feasible approach for model validation.
  • These methods are particularly effective for complex distributions like q-Gaussian models.
  • The study offers essential tools for accurate statistical measurement and model selection in diverse applications.