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Using the sample maximum to estimate the parameters of the underlying distribution.

Alex Capaldi1, Tiffany N Kolba1

  • 1Department of Mathematics and Statistics, Valparaiso University, Valparaiso, IN, United States of America.

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Summary
This summary is machine-generated.

This study introduces new methods to estimate parameters for exponential and normal distributions using only sample maxima. These estimators are accurate and precise, with performance varying based on sample size and number of maxima.

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Area of Science:

  • Statistics
  • Probability Theory
  • Extreme Value Theory

Background:

  • Estimating distribution parameters typically requires full datasets.
  • Sample maxima offer a unique, potentially information-rich subset of data.
  • Existing methods may not fully leverage information from sample maxima.

Purpose of the Study:

  • To develop and analyze novel statistical estimators for exponential and normal distributions using only sample maxima.
  • To assess the accuracy and precision of these new estimators.
  • To demonstrate the practical application of these methods in a biological context.

Main Methods:

  • Development of estimators for exponential and normal distribution parameters based on sample maxima.
  • Application of extreme value theory to analyze estimator properties.
  • Monte Carlo simulations to evaluate estimator accuracy and precision.
  • Empirical application to biological data (pollen tube length).

Main Results:

  • For exponential distributions, the mean estimator is unbiased, with variance decreasing as sample size (n) or number of maxima (m) increases.
  • For normal distributions, the mean estimator has negligible bias, and the variance estimator is unbiased.
  • Normal distribution estimator variance decreases with increasing m but increases with increasing n.
  • Successful application to estimate mean pollen tube length in Arabidopsis thaliana.

Conclusions:

  • Novel estimators provide accurate and precise parameter estimation for exponential and normal distributions using sample maxima.
  • The performance characteristics of the estimators are well-defined concerning sample size and number of maxima.
  • The method is applicable to real-world biological problems, demonstrating its utility.