1. Wang Yao Department of Statistics Rutgers University wyao@eden.rutgers.edu Mentor: Professor Regina Y. Liu DIMACS -- July 17, 2008 Extreme Value Theory (EVT): Application to Runway Safety Extreme Value Theory (EVT): Application to Runway Safety

2. Motivation X s Q: How to determine s such that: P(X> s ).0000001= allow multiple runway usage to ease air traffic congestion! Cut-off point:Require all landings to be completed before the cut-off point with certain “guarantee” Task: (Extremely small!) *

3. Difficulty (why Extreme Value Theory ) Extremely small tail probability e.g. p= 0.0000001 Few or no occurrences (observations) in reality e.g. Even with sample size=2000 Possible Solution: Extreme Value Theory (EVT) Difficulty: No observations!

4. Overview of EVT Random sample from unknown distr. fun. F Order statistics Fréchet distribution heavy tail Weibull distribution finite end point, e.g. uniform dist. Tail index ↔ Characterizes tail thickness of F Gumbel distribution in between, e.g. normal dist.

5. Extreme Quantile Take with, then Let For want to find s.t. Estimated by

6. Learning from Some Known Distributions Generate random samples For p= 0.001, estimate the p-th upper quantile Analysis: Bootstrap Method: a resampling technique for obtaining limiting distribution of any estimator e.g. Normal, Exponential, Chi-square,… P=0.001DistributionEstimated True Error Case A N(0,1) 4.37951 3.09023 1.28928 Case B Exp(1) 5.99578 6.907755 0.911975 Case CChi-square(3) 14.1312 16.2662 2.135 Small sample sizeMethod of moments

7. Real Data underling distribution/model unknown! Task: Applying Bootstrap Method to find a proper k (Bootstrap method: completely nonparametric approach and does not need to know the underlying distribution) e.g. Landing distance:

8. Analyze landing data collected from airport runways Apply bootstrap method with proper choice of k Determine the suitable cut-off point -- estimate the tail index, and extreme quantile Remarks : Yet to be completed Important project with real application. Well motivated and requires new interesting statistical methodology I learned some interesting new subjects, e.g. EVT, bootstrap method. Statistics is a practical field and theoretically challenging.

9. Questions? Questions? Acknowledgment: Thanks to DIMACS REU!