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On many occasions, physicists, other scientists, and engineers need to make estimates of a particular quantity. These are sometimes referred to as guesstimates, order-of-magnitude approximations, back-of-the-envelope calculations, or Fermi calculations. The physicist Enrico Fermi was famous for his ability to estimate various kinds of data with surprising precision. Estimating does not mean guessing a number or a formula at random. Instead, estimation means using prior experience and sound...
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When the population standard deviation is unknown and the sample size is large, the sample standard deviation s is commonly used as a point estimate of σ. However, it can sometimes under or overestimate the population standard deviation. To overcome this drawback, confidence intervals are determined to estimate population parameters and eliminate any calculation bias accurately. However, this only applies to random samples from normally distributed populations. Knowing the sample mean and...
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In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. 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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The accurate values of population parameters such as population proportion, population mean, and population standard deviation (or variance) are usually unknown. These are fixed values that can only be estimated from the data collected from the samples. The estimates of each of these parameters are sample proportion, the sample mean, and sample standard deviation (or variance). To obtain the values of these sample statistics, data are required that have particular distribution and central...
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It isn't easy to measure a parameter such as the mean height or the mean weight of a population. So, we draw samples from the population and calculate the mean height or mean weight of the individuals in the sample. This sample data acts as a representative measure of the population parameter. These sample statistics are known as estimates. 
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Ensemble Estimation of Information Divergence †.

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  • 1Genetics Department and Applied Math Program, Yale University, New Haven, CT 06520, USA.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces a novel nonparametric estimator for information divergence between random variables. It overcomes limitations of existing methods, offering improved mean squared error convergence, especially in high dimensions.

Keywords:
bayes error ratecentral limit theoremconvergence ratesdifferential entropydivergencenonparametric estimation

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Area of Science:

  • Information Theory
  • Statistical Inference
  • Machine Learning

Background:

  • Nonparametric estimation of information divergence functionals is crucial for analyzing relationships between random variables.
  • Existing methods often require restrictive assumptions about density support sets or complex boundary calculations.
  • A need exists for robust estimators that perform well across diverse support set conditions.

Purpose of the Study:

  • To develop a nonparametric divergence estimator that does not require prior knowledge of the support set boundary.
  • To generalize ensemble estimation theory for improved divergence estimation rates.
  • To propose an empirical estimator for Rényi-α divergence with enhanced performance.

Main Methods:

  • Derivation of mean squared error (MSE) convergence rates for a leave-one-out kernel density plug-in estimator.
  • Generalization of optimally weighted ensemble estimation theory.
  • Development of an empirical estimator for Rényi-α divergence.

Main Results:

  • The proposed estimator achieves improved MSE convergence rates for general bounded density support sets without boundary knowledge.
  • A new divergence estimator achieves the parametric rate for sufficiently smooth densities.
  • The empirical Rényi-α divergence estimator demonstrates superior performance and robustness to tuning parameters compared to standard methods.

Conclusions:

  • The developed methods provide a more flexible and accurate approach to estimating information divergence functionals.
  • The proposed Rényi-α divergence estimator offers significant advantages in mean squared error, particularly in high-dimensional settings.
  • The estimator's robustness and performance are validated through simulations and application to Bayes error rate estimation.