# Fitting of Failure Rate Data to a Closed Form Solution using Gamma-Poisson Method of Moments

## Abstract

Due to the responsibility and gravity of estimation of nuclear power plant failure rates, it is required that there be consistent and accurate methodology for describing the likelihood of an event occurring. However, the methodology currently employed can vary dependent on the data, as well as subjective conjecture from the expert performing the analysis. The current implementation of the empirical Bayes method to a Gamma-Poisson (GaP) distribution utilizes algorithms to solve for the parameters that do not provide consistent answers. Additionally, to achieve a distribution of the likelihood, the variance of each parameter of the GaP must be determined. There does not exist an exact solution to one of the parameters variance and is typically estimated using techniques like the Kass-Steffey adjustment. Thus a new approach to the problem is proposed, built upon the method of moments for a negative binomial, based upon the work of Bradlow. Using the method of moments approach, we are able to achieve a closed form estimation of the mean and variance for each parameter in the negative binomial distribution. Due to the relationship between the negative binominal and GaP distribution, results can be made between the distributions. The hyper-priors defined assume beta prime distributionmore »

- Authors:

- Idaho National Laboratory

- Publication Date:

- Research Org.:
- Idaho National Lab. (INL), Idaho Falls, ID (United States)

- Sponsoring Org.:
- USDOE Office of Nuclear Energy (NE)

- OSTI Identifier:
- 1462369

- Report Number(s):
- INL/CON-16-40754

- DOE Contract Number:
- AC07-05ID14517

- Resource Type:
- Conference

- Resource Relation:
- Conference: 2017 International Topical Meeting on Probabilistic Safety Assessment and Analysis (PSA 2017), Pittsburgh, PA, 9/24/2017 - 9/28/2017

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Failure Rate; Gamma-Poisson; Hyper-priors; Negative Binominal; Rare Events

### Citation Formats

```
Kunz, Matthew Ross, and Ewing, Sarah Elizabeth Marie.
```*Fitting of Failure Rate Data to a Closed Form Solution using Gamma-Poisson Method of Moments*. United States: N. p., 2017.
Web.

```
Kunz, Matthew Ross, & Ewing, Sarah Elizabeth Marie.
```*Fitting of Failure Rate Data to a Closed Form Solution using Gamma-Poisson Method of Moments*. United States.

```
Kunz, Matthew Ross, and Ewing, Sarah Elizabeth Marie. Fri .
"Fitting of Failure Rate Data to a Closed Form Solution using Gamma-Poisson Method of Moments". United States. https://www.osti.gov/servlets/purl/1462369.
```

```
@article{osti_1462369,
```

title = {Fitting of Failure Rate Data to a Closed Form Solution using Gamma-Poisson Method of Moments},

author = {Kunz, Matthew Ross and Ewing, Sarah Elizabeth Marie},

abstractNote = {Due to the responsibility and gravity of estimation of nuclear power plant failure rates, it is required that there be consistent and accurate methodology for describing the likelihood of an event occurring. However, the methodology currently employed can vary dependent on the data, as well as subjective conjecture from the expert performing the analysis. The current implementation of the empirical Bayes method to a Gamma-Poisson (GaP) distribution utilizes algorithms to solve for the parameters that do not provide consistent answers. Additionally, to achieve a distribution of the likelihood, the variance of each parameter of the GaP must be determined. There does not exist an exact solution to one of the parameters variance and is typically estimated using techniques like the Kass-Steffey adjustment. Thus a new approach to the problem is proposed, built upon the method of moments for a negative binomial, based upon the work of Bradlow. Using the method of moments approach, we are able to achieve a closed form estimation of the mean and variance for each parameter in the negative binomial distribution. Due to the relationship between the negative binominal and GaP distribution, results can be made between the distributions. The hyper-priors defined assume beta prime distribution which is appropriately informed; the results of this application translate back to the Gamma distribution for easy utilization in SAPHIRE. Additionally several cases are explored using publically available NRC data that considers zero inflated, over dispersed, and Poisson data.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2017},

month = {9}

}