Lan Xu and Guy Nusholtz—FCA US LLC, USA

## Abstract

A parametric model obtained by fitting a set of data to a function generally uses a procedure such as maximum likelihood or least squares. In general this will generate the best estimate for the distribution of the data overall but will not necessarily generate a reasonable estimation for the tail of the distribution unless the function fitted resembles the underlying distribution function. A distribution function can represent an estimate that is significantly different from the actual tail data, while the bulk of the data is reasonably represented by the central part of the fitted distribution. Extreme value theory can be used to improve the predictive capabilities of the fitted function in the tail region.

In this study the peak-over-threshold approach from the extreme value theory was utilized to show that it is possible to obtain a better fit of the tail of a distribution than the procedures that use the entire distribution only. Additional constraints, on the current use of the extreme value approach with respect to the selection of the threshold (an estimate of the beginning of the tail region) that minimize the sensitivity to individual data samples associated with the tail section as well as contamination from the central distribution are used. Once the threshold is determined, the maximum likelihood method was used to fit the exceedances with the Generalized Pareto Distribution to obtain the tail distribution. The approach was then used in the analysis of airbag inflator pressure data from tank tests, crash velocity distribution and mass distribution from the field crash data (NASS).

From the examples, the extreme (tail) distributions were better estimated with the Generalized Pareto Distribution, than a single overall distribution, along with the probability of the occurrence for a given extreme value, or a rare observation such as a high speed crash. It was concluded that the peak-over-threshold approach from extreme value theory can be a useful tool in the vehicle crash, biomechanics and injury tolerance data analysis and in estimation of the occurrence probability of an extreme phenomenon given a set of accurate observations.

Xu L, Nusholtz G. Application of Extreme Value Theory to Crash Data Analysis. *Stapp Car Crash J. *2018 Nov;61:287-298. doi: 10.4271/2017-22-0011.

**Pages:** 12

**Event:** 61^{st} Stapp Car Crash Conference