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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 term "bootstrap" originated in the 19th century as a metaphor for self-improvement or achieving something independently, without external assistance. This concept extends to statistical bootstrapping, a self-contained method for estimating population parameters through resampling, even though it can be computationally intensive. Developed by the American statistician Dr. Bradley Efron in 1979, bootstrapping provides a robust way to perform inference when the original sample size is...
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An R-Based Landscape Validation of a Competing Risk Model
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Robust estimation of risks from small samples.

Simon H Tindemans1, Goran Strbac2

  • 1Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK s.tindemans@imperial.ac.uk.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|October 21, 2017
PubMed
Summary

This study introduces a robust Bayesian method for data-driven risk analysis in systems with limited data, like the electricity grid. It provides reliable error bounds for probability distribution inference, even with small, ill-behaved samples.

Keywords:
Bayesian inferenceDirichlet processimprecise probabilitiesnon-parametric methodsrare event analysisresampling methods

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Area of Science:

  • Statistics
  • Risk Analysis
  • Bayesian Inference

Background:

  • Data-driven risk analysis requires inferring probability distributions from data.
  • Highly reliable systems, such as the electricity grid, often have limited relevant data, increasing the impact of estimation errors.
  • Accurate risk assessment is crucial for system reliability and optimization.

Purpose of the Study:

  • To present a robust non-parametric Bayesian method for inferring probability distributions from limited data.
  • To establish rigorous error bounds for risk analysis, particularly for small and ill-behaved datasets.
  • To offer a reliable method for systems where data scarcity is a significant challenge.

Main Methods:

  • Developed a non-parametric Bayesian approach for probability distribution inference.
  • Utilized the concept of intervals between ordered observations for probability allocation.
  • Implemented a computational resampling method termed Bayesian interval sampling.

Main Results:

  • The proposed method achieves rigorous error bounds, even with small sample sizes.
  • It demonstrates effectiveness for ill-behaved distributions, outperforming common alternative approaches.
  • Bayesian interval sampling provides a straightforward computational solution.

Conclusions:

  • The Bayesian interval sampling method offers a robust solution for data-driven risk analysis in data-scarce environments.
  • It ensures strict error bounds, enhancing the reliability of probability distribution inference.
  • This approach is particularly valuable for critical infrastructure like the electricity grid.