# Conservative Estimation of Tail Probabilities from Limited Sample Data.

## Abstract

Several sparse-sample uncertainty quantification (UQ) methods are compared for conser- vative but not overly conservative estimation of small tail probabilities involving responses that lay beyond specified thresholds in the tails of probability distributions. Sixteen very differently shaped distributions (or probability density functions, PDFs) and tail probability magnitudes ranging from 10 -5 to 10 -1 are considered in order for the study to be relevant to a wide range of risk analysis and quantification of margins and uncertainty (QMU) problems. The emphasis of the study is on limited data regimes ranging from N = 2 to 20 samples, reflective of most ex- perimental and some expensive computational situations. Relatively simple sparse-sample UQ methods tested for this regime involve statistical tolerance interval "Equivalent Normar and related "Ensemble of Normals" and "Superdistributiorr (SD) approaches. (The independently derived SD is effectively equivalent to the Bayesian posterior predictive distribution given the assumptions of the derivation.) The performance of the methods was generally improved for N > 5 samples with a generalized Jackknife resampling technique, which determines a tail probability estimate by averaging estimates from smaller sub-samples. Several quantitative metrics for method conservatism and accuracy of tail probability estimation are used to assess and rank themore »

- Authors:

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1605343

- Report Number(s):
- SAND2020-2828

684818

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Jekel JR, Charles Fredrick Romero, Vicente J.
```*Conservative Estimation of Tail Probabilities from Limited Sample Data.*. United States: N. p., 2020.
Web. doi:10.2172/1605343.

```
Jekel JR, Charles Fredrick Romero, Vicente J.
```*Conservative Estimation of Tail Probabilities from Limited Sample Data.*. United States. https://doi.org/10.2172/1605343

```
Jekel JR, Charles Fredrick Romero, Vicente J. Sun .
"Conservative Estimation of Tail Probabilities from Limited Sample Data.". United States. https://doi.org/10.2172/1605343. https://www.osti.gov/servlets/purl/1605343.
```

```
@article{osti_1605343,
```

title = {Conservative Estimation of Tail Probabilities from Limited Sample Data.},

author = {Jekel JR, Charles Fredrick Romero, Vicente J.},

abstractNote = {Several sparse-sample uncertainty quantification (UQ) methods are compared for conser- vative but not overly conservative estimation of small tail probabilities involving responses that lay beyond specified thresholds in the tails of probability distributions. Sixteen very differently shaped distributions (or probability density functions, PDFs) and tail probability magnitudes ranging from 10 -5 to 10 -1 are considered in order for the study to be relevant to a wide range of risk analysis and quantification of margins and uncertainty (QMU) problems. The emphasis of the study is on limited data regimes ranging from N = 2 to 20 samples, reflective of most ex- perimental and some expensive computational situations. Relatively simple sparse-sample UQ methods tested for this regime involve statistical tolerance interval "Equivalent Normar and related "Ensemble of Normals" and "Superdistributiorr (SD) approaches. (The independently derived SD is effectively equivalent to the Bayesian posterior predictive distribution given the assumptions of the derivation.) The performance of the methods was generally improved for N > 5 samples with a generalized Jackknife resampling technique, which determines a tail probability estimate by averaging estimates from smaller sub-samples. Several quantitative metrics for method conservatism and accuracy of tail probability estimation are used to assess and rank the methods' performance over many random trials for each test PDF and probability magnitude. A variant of Bootstrap resampling was also tried, but did not significantly improve tail probability estimates in most cases. Detailed results are presented from over 100-million tests over the above factors that provide useful granular information on which methods or combination of methods perform best in various areas of the factor space.},

doi = {10.2172/1605343},

url = {https://www.osti.gov/biblio/1605343},
journal = {},

number = ,

volume = ,

place = {United States},

year = {2020},

month = {3}

}